Operator Inequalities of the Jensen, Čebyšev and Grüss Type (SpringerBriefs in Mathematics)

SpringerBriefs in Mathematics

For further volumes: http://www.springer.com/series/10030

Silvestru Sever Dragomir

Operator Inequalities L of the Jensen, CebyL sev and Gr¨uss Type

123

Silvestru Sever Dragomir School of Engineering and Science Victoria University Melbourne, Australia 8001 [email protected] School of Computational and Applied Mathematics University of the Witwatersrand Braamfontein 2000 Johannesburg, South Africa

ISSN 2191-8198 e-ISSN 2191-8201 ISBN 978-1-4614-1520-6 e-ISBN 978-1-4614-1521-3 DOI 10.1007/978-1-4614-1521-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011938141 Mathematics Subject Classification (2010): 47A63, 47A60, 47A30, 26D15 c Silvestru Sever Dragomir 2012  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

This book is dedicated to my beloved children Sergiu & Camelia and granddaughter Sienna Clarisse

Abstract

The main aim of this book is to present recent results concerning inequalities of the ˇ Jensen, Cebyˇ sev and Gr¨uss type for continuous functions of selfadjoint operators on complex Hilbert spaces. It is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas.

vii

Preface

Linear Operator Theory in Hilbert spaces plays a central role in contemporary mathematics with numerous applications for Partial Differential Equations, in Approximation Theory, Optimization Theory, Numerical Analysis, Probability Theory and Statistics and other fields. The main aim of this short book is to present recent results concerning inequalˇ ities of the Jensen, Cebyˇ sev and Gr¨uss type for continuous functions of bounded selfadjoint operators on complex Hilbert spaces. The book is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas. In Chap. 1, we recall some fundamental facts concerning bounded selfadjoint operators on complex Hilbert spaces. The generalized Schwarz’s inequality for positive selfadjoint operators as well as some results for the spectrum of this class of operators are presented. Then we introduce and explore the fundamental results for polynomials in a linear operator, continuous functions of selfadjoint operators as well as the step functions of selfadjoint operators. Using these results, we then introduce the spectral decomposition of selfadjoint operators (the Spectral Representation Theorem) that will play a central role in the rest of the book. This result is used as a key tool in obtaining various new inequalities for continuous functions of selfadjoint operators, functions that are of bounded variation, Lipschitzian, monotonic or absolutely continuous. Another tool that will greatly simplify the error bounds provided in the book is the Total Variation Schwarz’s Inequality for which a simple proof is offered. Jensen’s type inequalities in their various settings ranging from discrete to continuous case play an important role in different branches of Modern Mathematics. A simple search in the MathSciNet database of the American Mathematical Society with the key words “jensen” and “inequality” in the title reveals more than 300 items intimately devoted to this famous result. However, the number of papers where this inequality is applied is a lot larger and far more difficult to find.

ix

x

Preface

In Chap. 2, we present some recent results obtained by the author that deal with different aspects of this well-researched inequality than those recently reported in the book [19]. They include but are not restricted to the operator version of the Dragomir–Ionescu inequality, Slater’s type inequalities for operators and its inverses, Jensen’s inequality for twice differentiable functions whose second derivatives satisfy some upper and lower bounds conditions, Jensen’s type inequalities for log-convex functions and for differentiable log-convex functions. Finally, some Hermite–Hadamard’s type inequalities for convex functions and Hermite– Hadamard’s type inequalities for operator convex functions are presented as well. ˇ Chapter 3 is devoted to Cebyˇ sev and Gr¨uss’ type inequalities. ˇ The Cebyˇsev, or in a different spelling – Chebyshev, inequality which compares the integral/discrete mean of the product with the product of the integral/discrete means is famous in the literature devoted to Mathematical Inequalities. It has been extended, generalized, refined, etc. by many authors during the last century. A simple search utilizing either spellings and the key word “inequality” in the title in the comprehensive MathSciNet database produces more than 200 research articles devoted to this result. The sister inequality due to Gr¨uss which provides error bounds for the magnitude of the difference between the integral mean of the product and the product of the integral means has also attracted much interest since it has been discovered in 1935 with more than 180 papers published, as a simple search in the same database reveals. Far more publications have been devoted to the applications of these inequalities and an accurate picture of the impacted results in various fields of Modern Mathematics is difficult to provide. In this chapter, however, we present only some recent results due to the author for the corresponding operator versions of these two famous inequalities. For the sake of completeness, all the results presented are completely proved and the original references where they have been first obtained are mentioned. The chapters are followed by the list of references used therein and therefore are relatively independent and can be read separately. Melbourne and Johannesburg

Silvestru Sever Dragomir

Contents

1

Functions of Selfadjoint Operators on Hilbert Spaces. . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Bounded Selfadjoint Operators .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Operator Order .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Continuous Functions of Selfadjoint Operators . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 Polynomials in a Bounded Operator . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.2 Continuous Functions of Selfadjoint Operators . . . . . . . . . . . . . . . 1.4 Step Functions of Selfadjoint Operators . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 The Spectral Decomposition of Selfadjoint Operators . . . . . . . . . . . . . . . . 1.5.1 Operator Monotone and Operator Convex Functions .. . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 1 1 4 4 5 7 9 12 14

2 Inequalities of the Jensen Type . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Reverses of the Jensen Inequality . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 An Operator Version of the Dragomir–Ionescu Inequality . . . 2.2.2 Further Reverses . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Some Slater Type Inequalities .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Slater Type Inequalities for Functions of Real Variables . . . . . 2.3.2 Some Slater Type Inequalities for Operators . . . . . . . . . . . . . . . . . . 2.3.3 Further Reverses . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Other Inequalities for Convex Functions.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 Some Inequalities for Two Operators .. . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Some Jensen Type Inequalities for Twice Differentiable Functions .. 2.5.1 Jensen’s Inequality for Twice Differentiable Functions.. . . . . . 2.6 Some Jensen’s Type Inequalities for Log-Convex Functions . . . . . . . . . 2.6.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.2 Jensen’s Inequality for Differentiable Log-Convex Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.3 More Inequalities for Differentiable Log-Convex Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.4 A Reverse Inequality . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

15 15 16 16 17 23 23 24 25 29 29 32 32 35 35 38 42 46 xi

xii

Contents

2.7 Hermite–Hadamard’s Type Inequalities . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.1 Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.2 Some Inequalities for Convex Functions . .. . . . . . . . . . . . . . . . . . . . 2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.2 Some Hermite–Hadamard’s Type Inequalities . . . . . . . . . . . . . . . . 2.8.3 Some Operator Quasi-linearity Properties .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

47 47 48

ˇ 3 Inequalities of the Cebyˇ sev and Gruss ¨ Type . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . ˇ 3.2 Cebyˇ sev’s Inequality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . ˇ 3.2.1 Cebyˇ sev’s Inequality for Real Numbers . . .. . . . . . . . . . . . . . . . . . . . ˇ 3.2.2 A Version of the Cebyˇ sev Inequality for One Operator .. . . . . . 3.2.3 Related Results for One Operator .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Gr¨uss Inequality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.1 Some Elementary Inequalities of Gr¨uss Type . . . . . . . . . . . . . . . . . 3.3.2 An Inequality of Gr¨uss’ Type for One Operator.. . . . . . . . . . . . . . 3.4 More Inequalities of Gr¨uss Type . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.1 Some Vectorial Gr¨uss’ Type Inequalities ... . . . . . . . . . . . . . . . . . . . 3.4.2 Some Inequalities of Gr¨uss’ Type for One Operator .. . . . . . . . . ˇ 3.5 More Inequalities for the Cebyˇ sev Functional .. . . .. . . . . . . . . . . . . . . . . . . . 3.5.1 A Refinement and Some Related Results . .. . . . . . . . . . . . . . . . . . . . ˇ 3.6 Bounds for the Cebyˇ sev Functional of Lipschitzian Functions .. . . . . . 3.6.1 The Case of Lipschitzian Functions . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.2 The Case of .'; ˆ/-Lipschitzian Functions . . . . . . . . . . . . . . . . . . . 3.7 Quasi-Gr¨uss’ Type Inequalities.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7.2 Vector Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7.3 Applications for Gr¨uss’ Type Inequalities .. . . . . . . . . . . . . . . . . . . . 3.8 Two Operators Gr¨uss’ Type Inequalities .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8.1 Some Representation Results . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8.2 Bounds for f of Bounded Variation . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8.3 Bounds for f Lipschitzian . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8.4 Bounds for f Monotonic Non-decreasing . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

71 71 71 71 73 75 77 77 77 80 80 81 84 84 90 90 93 94 94 95 102 104 104 106 114 118 119

58 58 60 63 69

Chapter 1

Functions of Selfadjoint Operators on Hilbert Spaces

1.1 Introduction In this introductory chapter we recall some fundamental facts concerning bounded selfadjoint operators on complex Hilbert spaces. Since all the operators considered in this book are supposed to be bounded, we no longer mention this but understand it implicitly. The generalized Schwarz’s inequality for positive selfadjoint operators as well as some results for the spectrum of this class of operators are presented. Then we introduce and explore the fundamental results for polynomials in a linear operator, continuous functions of selfadjoint operators as well as the step functions of selfadjoint operators. By the use of these results, we then introduce the spectral decomposition of selfadjoint operators (the Spectral Representation Theorem) that will play a central role in the rest of the book. This result is used as a key tool in obtaining various new inequalities for continuous functions of selfadjoint operators which are of bounded variation, Lipschitzian, monotonic or absolutely continuous. Another tool that will greatly simplify the error bounds provided in the book is the Total Variation Schwarz’s Inequality for which a simple proof is offered. The chapter is concluded with some well-known operator inequalities of Jensen’s type for convex and operator convex functions. More results in this spirit can be found in the recent book [1].

1.2 Bounded Selfadjoint Operators 1.2.1 Operator Order Let .H I h:; :i/ be a Hilbert space over the complex numbers field C: A bounded linear operator A defined on H is selfadjoint, i.e. A D A if and only if hAx; xi 2 R for all x 2 H and if A is selfadjoint, then ˇ sev and Gr¨uss Type, S.S. Dragomir, Operator Inequalities of the Jensen, Cebyˇ SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-1521-3 1, © Silvestru Sever Dragomir 2012

1

2

1 Functions of Selfadjoint Operators on Hilbert Spaces

kAk D sup jhAx; xij D kxkD1

sup

jhAx; yij :

(1.1)

kxkDkykD1

We assume in what follows that all operators are bounded on defined on the whole Hilbert space H: We denote by B.H / the Banach algebra of all bounded linear operators defined on H: Definition 1.1. Let A and B be selfadjoint operators on H: Then A  B (A is less or equal to B) or, equivalently, B  A if hAx; xi  hBx; xi for all x 2 H: In particular, A is called positive if A  0: It is well known that for any operator A 2 B.H / the composite operators A A and AA are positive selfadjoint operators on H . However, the operators A A and AA are not comparable with each other in general. The following result concerning the operator order holds (see for instance [2, p. 220]): Theorem 1.2. Let A; B; C 2 B.H / be selfadjoint operators and let ˛; ˇ 2 R. Then 1. 2. 3. 4.

A  AI If A  B and B  C , then A  C I If A  B and B  A, then A D BI If A  B and ˛  0; then A C C  B C C; ˛A  ˛B; A  BI

5. If ˛  ˇ; then ˛A  ˇA: The following generalization of Schwarz’s inequality for positive selfadjoint operators A holds: jhAx; yij2  hAx; xi hAy; yi

(1.2)

for any x; y 2 H: The following inequality is of interest as well, see [2, p. 221]: Theorem 1.3. Let A be a positive selfadjoint operator on H: Then kAxk2  kAk hAx; xi

(1.3)

for any x 2 H: Theorem 1.4. Let An ; B 2 B.H / with n  1 be selfadjoint operators with the property that A1  A2      An      B: Then there exists a bounded selfadjoint operator A defined on H such that An  A  B for all n  1

1.2 Bounded Selfadjoint Operators

3

and lim An x D Ax for all x 2 H:

n!1

An analogous assertion holds if the sequence fAn g1 nD1 is decreasing and bounded below. Definition 1.5. We say that a sequence fAn g1 nD1  B.H / converges strongly to an operator A 2 B.H /; called the strong limit of the sequence fAn g1 nD1 and we denote this by .s/ limn!1 An D A; if limn!1 An x D Ax for all x 2 H: The convergence in norm, i.e. limn!1 kAn  Ak D 0 will be called the “uniform convergence” as opposed to strong convergence. We denote limn!1 An D A for the convergence in norm. From the inequality kAm x  An xk  kAm  An k kxk that holds for all n; m and x 2 H it follows that uniform convergence of the 1 sequence fAn g1 nD1 to A implies strong convergence of fAn gnD1 to A: However, the converse of this assertion is false. It is also possible to introduce yet another concept of “weak convergence” in B.H / by defining .w/ limn!1 An D A if and only if limn!1 hAn x; yi D hAx; yi for all x; y 2 H: The following result holds (see [2, p. 225]): Theorem 1.6. Let A be a bounded selfadjoint operator on H: Then ˛ 1 W D inf hAx; xi D max f˛ 2 R j˛I  A gI kxkD1

˛ 2 W D sup hAx; xi D min f˛ 2 R jA  ˛I gI kxkD1

and kAk D max fj˛ 1 j ; j˛ 2 jg : Moreover, if Sp.A/ denotes the spectrum of A; then ˛ 1 ; ˛ 2 2 Sp.A/ and Sp.A/  Œ˛ 1 ; ˛ 2  : Remark 1.7. We remark that, if A; ˛ 1 ; ˛ 2 are as above, then obviously ˛ 1 D min f j 2 Sp.A/ g DW min Sp.A/I ˛ 2 D max f j 2 Sp.A/ g DW max Sp.A/I kAk D max fjj j 2 Sp.A/ g: We also observe that 1. A is positive iff ˛ 1  0I 2. A is positive and invertible iff ˛ 1 > 0I   1 3. If ˛ 1 > 0; then A is a positive selfadjoint operator and min Sp A1 D   1 ˛ 1 D ˛ 1 2 ; max Sp A 1 :

4

1 Functions of Selfadjoint Operators on Hilbert Spaces

1.3 Continuous Functions of Selfadjoint Operators 1.3.1 Polynomials in a Bounded Operator For two functions ';

W C ! C we adhere to the canonical notation: .' C

/ .s/ WD '.s/ C

.s/;

.'/ .s/ WD '.s/; .' / .s/ WD '.s/ .s/ for sum, scalar multiple and product of these functions. We denote by '.s/ N the complex conjugate of '.s/: As a first class of functions we consider the algebra P of all polynomials in one variable with complex coefficients, namely ( P WD '.s/ WD

n X

) ˛ k s k jn  0; ˛ k 2 C,0  k  n :

kD0

P Theorem 1.8. Let A 2 B.H / and for '.s/ WD nkD0 ˛ k s k 2 P define '.A/ WD   Pn P k 0 N WD nkD0 ˛N k .A /k 2 B.H /: Then the kD0 ˛ k A 2 B.H / A D I and '.A/ mapping '.s/ ! '.A/ has the following properties: (a) (b) (c) (d)

.' C / .A/ D '.A/ C .A/I .'/ .A/ D '.A/I .' / .A/ D '.A/ .A/I Œ'.A/ D '.A/: N

Note that '.A/ .A/ D .A/'.A/ and the constant polynomial '.s/ D ˛ 0 is mapped into the operator. Recall that, a mapping a ! a0 of an algebra U into an algebra U 0 is called a homomorphism if it has the properties: (a) .a C b/0 D a0 C b 0 I (b) .'/0 D a0 I (c) .ab/0 D a0 b 0 : With this terminology, Theorem 1.8 asserts that the mapping which associates with any polynomial '.s/ the operator '.A/ is a homomorphism of P into B.H / satisfying the additional property d. The following result provides a connection between the spectrum of A and the spectrum of the operator '.A/: Theorem 1.9. If A 2 B.H / and ' 2 P, then Sp .'.A// D ' .Sp.A// :

1.3 Continuous Functions of Selfadjoint Operators

5

Corollary 1.10. If A 2 B.H / is selfadjoint and the polynomial '.s/ 2 P has real coefficients, then '.A/ is selfadjoint and k'.A/k D max fj' ./j ;  2 Sp.A/g :

(1.4)

Remark 1.11. If A 2 B.H / and ' 2 P, then 1. '.A/ is invertible iff './ ¤ 0 for all  2 Sp.A/I 2. If '.A/ is invertible, then Sp.'.A/1 / D f'./1 ;  2 Sp.A/g:

1.3.2 Continuous Functions of Selfadjoint Operators Assume that A is a bounded selfadjoint operator on the Hilbert space H: If ' is any function defined on R we define k'kA D sup fj' ./j ;  2 Sp.A/g : If ' is continuous, in particular if ' is a polynomial, then the supremum is actually assumed for some points in Sp.A/ which is compact. Therefore the supremum may then be written as a maximum and (1.4) can be written in the form k'.A/k D k'kA : Consider C .R/ the algebra of all continuous complex-valued functions defined on R. The following fundamental result for continuous functional calculus holds, see for instance [2, p. 232]: Theorem 1.12. If A is a bounded selfadjoint operator on the Hilbert space H and ' 2 C .R/, then there exists a unique operator '.A/ 2 B.H / with the property that whenever f' n g1 nD1  P such that limn!1 k'  ' n kA D 0; then '.A/ D limn!1 ' n .A/: The mapping ' ! '.A/ is a homomorphism of the algebra C .R/ into B.H / with the additional properties Œ'.A/ D '.A/ N and k'.A/k  2 k'kA : Moreover, '.A/ is a normal operator, i.e. Œ'.A/ '.A/ D '.A/ Œ'.A/ : If ' is real valued, then '.A/ is selfadjoint. As examples we notice that, if A 2 B.H / is selfadjoint and '.s/ D eis ; s 2 R then 1 X 1 .iA/k : eiA D kŠ kD0

Moreover, e

iA

is a unitary operator and its inverse is the operator 1 X  iA  1 .iA/k : D eiA D e kŠ kD0

Now, if  2 C n R, A 2 B.H / is selfadjoint and '.s/ D '.A/ D .A  I /1 :

1 s

2 C .R/ ; then

6

1 Functions of Selfadjoint Operators on Hilbert Spaces

If the selfadjoint operator A 2 B.H / and the functions '; 2 C .R/ are given, then we obtain the commutativity property '.A/ .A/ D .A/'.A/: This property can be extended for another operator as follows, see for instance [2, p. 235]: Theorem 1.13. Assume that A 2 B.H / and the function ' 2 C .R/ are given. If B 2 B.H / is such that AB D BA; then '.A/B D B'.A/: The next result extends Theorem 1.9 to the case of continuous functions, see for instance [2, p. 235]: Theorem 1.14. If A is abounded selfadjoint operator on the Hilbert space H and ' is continuous, then Sp .'.A// D ' .Sp.A// : As a consequence of this result we have: Corollary 1.15. With the assumptions in Theorem 1.14 we have: (a) (b) (c) (d)

The operator '.A/ is selfadjoint iff ' ./ 2 R for all  2 Sp.A/I The operator '.A/ is unitary iff j' ./j D 1 for all  2 Sp.A/I The operator '.A/ is invertible iff ' ./ ¤ 0 for all  2 Sp.A/I If '.A/ is selfadjoint, then k'.A/k D k'kA :

In order to develop inequalities for functions of selfadjoint operators we need the following result, see for instance [2, p. 240]: Theorem 1.16. Let A be a bounded selfadjoint operator on the Hilbert space H: The homomorphism ' ! '.A/ of C .R/ into B.H / is order preserving, meaning that, if '; 2 C .R/ are real valued on Sp.A/ and ' ./  ./ for any  2 Sp.A/; then '.A/  .A/ in the operator order of B.H /: (P) The “square root” of a positive bounded selfadjoint operator on H can be defined as follows, see for instance [2, p. 240]: Theorem 1.17. If the operator A 2 B.H / is selfadjoint and positive, then there p exists a unique positive selfadjoint operator B WD A 2 B.H / such that B 2 D A: If A is invertible, then so is B: If A 2 B.H /; then the operatorpA A is selfadjoint and positive. Define the “absolute value” operator by jAj WD A A: Analogously to the familiar factorization of a complex number  D jj ei arg  a bounded normal operator on H may be written as a commutative product of a positive selfadjoint operator, representing its absolute value, and a unitary operator, representing the factor of absolute value one.

1.4 Step Functions of Selfadjoint Operators

7

In fact, the following more general result holds, see for instance [2, p. 241]: Theorem 1.18. For every bounded linear operator A on H; there exists a positive selfadjoint operator B D jAj 2 B.H / and an isometric operator C with the domain DC D B.H / and range RC D C .DC / D A.H / such that A D CB: In particular, we have: Corollary 1.19. If the operator A 2 B.H / is normal, then there exists a positive selfadjoint operator B D jAj 2 B.H / and a unitary operator C such that A D BC D CB: Moreover, if A is invertible, then B and C are uniquely determined by these requirements. Remark 1.20. Now, suppose that A D CB where B 2 B.H / is a positive selfadjoint operator and C is an isometric operator. Then p (a) B D A AI consequently B is uniquely determined by the stated requirements; (b) C is uniquely determined by the stated requirements iff A is one-to-one.

1.4 Step Functions of Selfadjoint Operators Let A be a bounded selfadjoint operator on the Hilbert space H: We intend to extend the order preserving homomorphism ' ! '.A/ of the algebra C .R/ of continuous functions ' defined on R into B.H /; restricted now to real-valued functions, to a larger domain, namely an algebra of functions containing the “step functions” '  ;  2 R, defined by ( 1; for  1 < s  ; '  .s/ WD 0; for  < s < C1: Observe that '  .s/ D '  .s/ and ' 2 .s/ D '  .s/ which will imply that Œ'  .A/ D '  .A/ and Œ'  .A/2 D '  .A/; i.e. '  .A/ will then be a projection. However, since the function '  cannot be approximated uniformly by continuous functions on any interval containing ; then, in general, there is no way to define an operator '  .A/ as a uniform limit of operators ' ;n .A/ with ' ;n 2 C .R/ : The uniform limit of operators can be relaxed to the concept of strong limit of operators (see Definition 1.5) in order to define the operator '  .A/: In order to do that, observe that the function '  may be obtained as a pointwise limit of a decreasing sequence of real-valued continuous functions ' ;n defined by

'  .s/ WD

8 ˆ ˆ < ˆ ˆ :

1; for  1 < s  ; 1  n .s  / ; for   s   C 1=n 0; for  < s < C1:

8

1 Functions of Selfadjoint Operators on Hilbert Spaces

By Theorem 1.4 we observe that the sequence of corresponding selfadjoint operators ' ;n .A/ is non-decreasing and bounded below by zero in the operator order of B.H /: It therefore converges strongly to some bounded selfadjoint operator '  .A/ on H; see [2, p. 244]. To provide a formal presentation of the above, we need the following definition: Definition 1.21. A real-valued function ' on R is called upper semi-continuous if it is a pointwise limit of a non-increasing sequence of continuous real-valued functions on R. We observe that it can be shown that a real-valued functions ' on R is upper semi-continuous iff for every s0 2 R and for every " > 0 there exists a ı > 0 such that '.s/ < ' .s0 / C " for all s 2 .s0  ı; s0 C ı/ : We can introduce now the operator '.A/ as follows, see for instance [2, p. 245]: Theorem 1.22. Let A be a bonded selfadjoint operator on the Hilbert space H and let ' be a non-negative upper semi-continuous function on R. Then there exists a unique positive selfadjoint operator '.A/ such that whenever f' n g1 nD1 is any nonincreasing sequence of non-negative functions in C .R/ ; pointwise converging to ' on Sp.A/; then '.A/ D .s/ lim ' n .A/: If ' is continuous, then the operator '.A/ defined by Theorem 1.12 coincides with the one defined by Theorem 1.22. Theorem 1.23. Let A 2 B.H / be selfadjoint, let ' and be non-negative upper semi-continuous functions on R, and let ˛ > 0 be given. Then the functions ' C ; ˛' and ' are non-negative upper semi-continuous and .' C / .A/ D '.A/ C .A/; .˛'/ .A/ D ˛'.A/ and .' / .A/ D '.A/ .A/: Moreover, if '.s/  .s/ for all s 2 Sp.A/ then '.A/  .A/: We enlarge the class of non-negative upper semi-continuous functions to an algebra by defining R .R/ as the set of all functions ' D ' 1  ' 2 where ' 1 ; ' 2 are non-negative and upper semi-continuous functions defined on R. It is easy to see that R .R/ endowed with pointwise sum, scalar multiple and product is an algebra. The following result concerning functions of operators '.A/ with ' 2 R .R/ can be stated, see for instance [2, pp. 249–250]: Theorem 1.24. Let A 2 B.H / be selfadjoint and let ' 2 R .R/ : Then there exists a unique selfadjoint operator '.A/ 2 B.H / such that if ' D ' 1  ' 2 where ' 1 ; ' 2 are non-negative and upper semi-continuous functions defined on R, then '.A/ D ' 1 .A/  ' 2 .A/: The mapping ' ! '.A/ is a homomorphism of R .R/ into B.H / which is order preserving in the following sense: if '; 2 R .R/ with the property that '.s/  .s/ for any s 2 Sp.A/; then '.A/  .A/: Moreover, if B 2 B.H / satisfies the commutativity condition AB D BA; then '.A/B D B'.A/:

1.5 The Spectral Decomposition of Selfadjoint Operators

9

1.5 The Spectral Decomposition of Selfadjoint Operators Let A 2 B.H / be selfadjoint and let '  defined for all  2 R as follows: ( '  .s/ WD

1 for  1 < s  ; 0 for  < s < C1:

Then for every  2 R the operator E WD '  .A/

(1.5)

is a projection which reduces A: The properties of these projections are summed up in the following fundamental result concerning the spectral decomposition of bounded selfadjoint operators in Hilbert spaces, see for instance [2, p. 256]: Theorem 1.25 (Spectral Representation Theorem). Let A be a bonded selfadjoint operator on the Hilbert space H and let m D min f j 2 Sp.A/ g DW min Sp.A/ and M D max f j 2 Sp.A/ g DW max Sp.A/: Then there exists a family of projections fE g2R , called the spectral family of A; with the following properties: (a) E  E0 for   0 I (b) Em0 D 0; EM D I and EC0 D E for all  2 R; (c) We have the representation Z M AD dE :

(1.6)

m0

More generally, for every continuous complex-valued function ' defined on R and for every " > 0 there exists a ı > 0 such that   n  X   0    ' k Ek  Ek1   " '.A/   

(1.7)

kD1

whenever

8 0 < m D 1 <    < n1 < n D M; ˆ ˆ < k  k1  ı for 1  k  n; ˆ ˆ : 0 k 2 Œk1 ; k  for 1  k  n;

this means that

Z

(1.8)

M

'.A/ D

' ./ dE ; m0

where the integral is of Riemann–Stieltjes type.

(1.9)

10

1 Functions of Selfadjoint Operators on Hilbert Spaces

Corollary 1.26. With the assumptions of Theorem 1.25 for A; E and ' we have the representations Z

M

'.A/x D

' ./ dE x for all x 2 H

(1.10)

m0

and

Z

M

h'.A/x; yi D

' ./ d hE x; y i for all x; y 2 H:

(1.11)

' ./ d hE x; x i for all x 2 H:

(1.12)

j' ./j2 d kE xk2 for all x 2 H:

(1.13)

m0

In particular, Z

M

h'.A/x; xi D m0

Moreover, we have the equality Z k'.A/xk2 D

M

m0

The next result shows that it is legitimate to talk about “the” spectral family of the bounded selfadjoint operator A since it is uniquely determined by the requirements (a), (b) and (c) in Theorem 1.25, see for instance [2, p. 258]. Theorem 1.27. Let A be a bonded selfadjoint operator on the Hilbert space H and let m D min Sp.A/ and M D max Sp.A/: If fF g2R is a family of projections satisfying the requirements (a), (b) and (c) in Theorem 1.25, then F D E for all  2 R where E is defined by (1.5). By the above two theorems, the spectral family fE g2R uniquely determines and in turn is uniquely determined by the bounded selfadjoint operator A: The spectral family also reflects in a direct way the properties of the operator A as follows, see [2, pp. 263–266] Theorem 1.28. Let fE g2R be the spectral family of the bounded selfadjoint operator A: If B is a bounded linear operator on H , then AB D BA iff E B D BE for all  2 R. In particular E A D AE for all  2 R. Theorem 1.29. Let fE g2R be the spectral family of the bounded selfadjoint operator A and  2 R. Then (a)  is a regular value of A; i.e. A  I is invertible iff there exists a  > 0 such that E D EC I (b)  2 Sp.A/ iff E < EC for all  > 0I (c)  is an eigenvalue of A iff E0 < E : The following result will play a key role in many results concerning inequalities for bounded selfadjoint operators in Hilbert spaces. Since we were not able to locate it in the literature, we will provide here a complete proof.

1.5 The Spectral Decomposition of Selfadjoint Operators

11

Theorem 1.30 (Total Variation Schwarz’s Inequality). Let fE g2R be the spectral family of the bounded selfadjoint operator A and let m D min Sp.A/ and M D max Sp.A/: Then for any x; y 2 H the function  ! hE x; yi is of bounded variation and we have the inequality M _ ˝

E./ x; y

˛

 kxk kyk :

(TVSI)

m0

Proof. If P is a non-negative selfadjoint operator on H; i.e. hP x; xi  0 for any x 2 H; then the following inequality is a generalization of the Schwarz inequality in H : jhP x; yij2  hP x; xi hP y; yi

(1.14)

for any x; y 2 H: Now, if d W m  s D t0 < t1 <    < tn1 < tn D M is an arbitrary partition of the interval Œm  s; M , where s > 0, then we have by Schwarz’s inequality for non-negative operators (1.14) that ( n1 ) M _ X ˇ˝ ˝  ˛ˇ ˛ ˇ Et  Et x; y ˇ E./ x; y D sup i C1

d

ms

i

i D0

( n1 ) X h˝  ˛1=2 ˝  ˛1=2 i Eti C1  Eti x; x Eti C1  Eti y; y  sup d

i D0

WD I:

(1.15)

By the Cauchy–Buniakovski–Schwarz inequality for sequences of real numbers we also have that 8" #1=2 " n1 #1=2 9 n1 < X = X ˝ ˝  ˛  ˛ I  sup Eti C1  Eti x; x Eti C1  Eti y; y ; d : i D0

 sup d

" D

8" n1 < X ˝ :

i D0

Eti C1  Eti x; x

˛

#1=2 " n1 X ˝

i D0

M _ ˝ ˛ E./ x; x

ms



Eti C1  Eti y; y

i D0

#1=2 "

M _ ˝

E./ y; y

˛



˛

#1=2 9 = ;

#1=2 (1.16)

ms

for any x; y 2 H: On making use of (1.15) and (1.16) and letting s > 0 we deduce the desired result (TVSI). t u

12

1 Functions of Selfadjoint Operators on Hilbert Spaces

1.5.1 Operator Monotone and Operator Convex Functions We say that a real-valued continuous function f defined on an interval I is said to be operator monotone if it is monotone with respect to the operator order, i.e. if A and B are bounded selfadjoint operators with A  B and Sp.A/; Sp.A/  I; then f .A/  f .B/ : The function is said to be operator convex (operator concave) if for any A, B bounded selfadjoint operators with Sp.A/; Sp.A/  I; we have f Œ.1  / A C B  ./ .1  / f .A/ C f .B/

(1.17)

for any  2 Œ0; 1: Example 1.31. The following examples are well know in the literature and can be found for instance in [1, pp. 7–9] where simple proofs were also provided: 1. The affine function f .t/ D ˛ C ˇt is operator monotone on every interval for all ˛ 2 R and ˇ  0: It is operator convex for all ˛; ˇ 2 R; 2. If f; g are operator monotone, and if ˛; ˇ  0 then the linear combination ˛f C ˇg is also operator monotone. If the functions fn are operator monotone and fn .t/ ! f .t/ as n ! 1; then f is also operator monotone; 3. The function f .t/ D t 2 is operator convex on every interval, however, it is not operator monotone on Œ0; 1/ even though it is monotonic non-decreasing on this interval; 4. The function f .t/ D t 3 is not operator convex on Œ0; 1/ even though it is a convex function on this interval; 5. The function f .t/ D 1t is operator convex on .0; 1/ and f .t/ D  1t is operator monotone on .0; 1/I 6. The function f .t/ D ln t is operator monotone and operator concave on .0; 1/I 7. The entropy function f .t/ D t ln t is operator concave on .0; 1/I 8. The exponential function f .t/ D et is neither operator convex nor operator monotone on any interval of R. The following monotonicity property for the function f .t/ D t r with r 2 Œ0; 1 is well known in the literature as the L¨owner–Heinz inequality and was established essentially in 1934: Theorem 1.32 (L¨owner–Heinz Inequality). Let A and B be positive operators on a Hilbert space H: If A  B  0; then Ar  B r for all r 2 Œ0; 1 : The following characterization of operator convexity holds, see [1, p. 10]. Theorem 1.33 (Jensen’s Operator Inequality). Let H and K be Hilbert spaces. Let f be a real-valued continuous function on an interval J: Let A and Aj be selfadjoint operators on H with spectra contained in J; for each j D 1; 2; : : : ; k: Then the following conditions are mutually equivalent: (i) f is operator convex on J I (ii) f .C  AC /  C  f .A/C for every selfadjoint operator A W H ! H and isometry C W K ! H; i:e: C  C D 1K I

1.5 The Spectral Decomposition of Selfadjoint Operators

13

(iii) f .C  AC /  C  f .A/C for every selfadjoint operator A W H ! H and isometry HI P C W H !    Pk k   (iv) f  C A C j D1 j j j j D1 Cj f Aj Cj for every selfadjoint operator Aj W H ! H and bounded linear operators Cj W K ! H; with Pk Cj Cj D 1K .j D 1; : : : ; k/ I j D1    Pk Pk   (v) f  j D1 Cj Aj Cj j D1 Cj f Aj Cj for every selfadjoint operator A W H ! H and bounded linear operators Cj W H ! H; with Pjk Cj Cj D 1H .j D 1; : : : ; k/ I j D1    Pk Pk (vi) f  P A P j j j j D1 j D1 Pj f Aj Pj for every selfadjoint operator Pk Aj W H ! H and projection Pj W H ! H; with j D1 Pj D 1H .j D 1; : : : ; k/ : The following well-known result due to Hansen and Pedersen also holds: Theorem 1.34 (Hansen–Pedersen–Jensen’s Inequality). Let J be an interval containing 0 and let f be a real-valued continuous function defined on J: Let A and Aj be selfadjoint operators on H with spectra contained in J; for each j D 1; 2; : : : ; k: Then the following conditions are mutually equivalent: (i) f is operator convex on J and f .0/  0I (ii) f .C  AC/  C  f .A/C for every selfadjoint operator A W H ! H and contraction C WH ! i:e: C  C  1H I P  H; P   k k   (iii) f  j D1 Cj Aj Cj j D1 Cj f Aj Cj for every selfadjoint operator A W H ! H and bounded linear operators Cj W H ! H; with Pjk  j D1 Cj Cj  1H .j D 1; : : : ; k/ I (iv) f .PAP/  Pf .A/P for every selfadjoint operator A W H ! H and projection P: The case of continuous and negative functions is as follows [1, p. 13]: Theorem 1.35. Let f be continuous on Œ0; 1/: If f .t/  0 for all t 2 Œ0; 1/; then each of the conditions (i)–(vi) from Theorem 1.33 is equivalent with (vii) f is an operator monotone function. Corollary 1.36. Let f be a real-valued continuous function mapping the positive half line Œ0; 1/ into itself. Then f is operator monotone if and only if f is operator concave. The following result may be stated as well [1, p. 14]: Theorem 1.37. Let f be continuous on the interval Œ0; r/ with r  1: Then the following conditions are mutually equivalent: (i) f is operator convex and f .0/  0I (ii) The function t 7! f .tt / is operator monotone on .0; r/:

14

1 Functions of Selfadjoint Operators on Hilbert Spaces

As a particular case of interest, we can state that [1, p. 15]: Corollary 1.38. Let f be continuous on Œ0; 1/ and taking positive values. The function f is operator monotone if and only if the function t 7! f .tt / is operator monotone. Finally, we recall the following result as well [1, p. 16]: Theorem 1.39. Let f be a real-valued continuous function on the interval J D Œ˛; 1/ and bounded below, i.e. there exists m 2 R such that m  f .t/ for all t 2 J: Then the following conditions are mutually equivalent: (i) f is operator concave on J I (ii) f is operator monotone on J: As a particular case of this result we note that, the function f .t/ D t r is operator monotone on Œ0; 1/ if and only if 0  r  1: The function f .t/ D t r is operator convex on .0; 1/ if either 1  r  2 or 1  r  0 and is operator concave on .0; 1/ if 0  r  1:

References 1. T. Furuta, J. Mi´ci´c, J. Peˇcari´c and Y. Seo, Mond-Peˇcari´c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005. 2. G. Helmberg, Introduction to Spectral Theory in Hilbert Space, John Wiley & Sons, Inc., New York, 1969.

Chapter 2

Inequalities of the Jensen Type

2.1 Introduction Jensen’s type inequalities in their various settings ranging from discrete to continuous case play an important role in different branches of Modern Mathematics. A simple search in the MathSciNet database of the American Mathematical Society with the key words “jensen” and “inequality” in the title reveals that there are more than 300 items intimately devoted to this famous result. However, the number of papers where this inequality is applied is a lot larger and far more difficult to find. It can be a good project in itself for someone to write a monograph devoted to Jensen’s inequality in its different forms and its applications across Mathematics. In the introductory chapter we have recalled a number of Jensen’s type inequalities for convex and operator convex functions of selfadjoint operators in Hilbert spaces. In this chapter we present some recent results obtained by the author that deal with different aspects of this well-researched inequality than those recently reported in the book [19]. They include but are not restricted to the operator version of the Dragomir–Ionescu inequality, Slater’s type inequalities for operators and its inverses, Jensen’s inequality for twice differentiable functions whose second derivatives satisfy some upper and lower bounds conditions, Jensen’s type inequalities for log-convex functions and for differentiable log-convex functions and their applications to Ky Fan’s inequality. Finally, some Hermite–Hadamard’s type inequalities for convex functions and Hermite–Hadamard’s type inequalities for operator convex functions are presented as well. All the above results are exemplified for some classes of elementary functions of interest such as the power function and the logarithmic function.

ˇ sev and Gr¨uss Type, S.S. Dragomir, Operator Inequalities of the Jensen, Cebyˇ SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-1521-3 2, © Silvestru Sever Dragomir 2012

15

16

2 Inequalities of the Jensen Type

2.2 Reverses of the Jensen Inequality 2.2.1 An Operator Version of the Dragomir–Ionescu Inequality The following result holds: Theorem 2.1 (Dragomir, 2008, [8]). Let I be an interval and f W I ! R be a convex and differentiable function on I˚ (the interior of I / whose derivative f 0 ˚ If A is a selfadjoint operators on the Hilbert space H with is continuous on I. ˚ then Sp.A/  Œm; M   I; ˝ ˛ ˝ ˛ .0 / hf .A/x; xi  f .hAx; xi/  f 0 .A/Ax; x  hAx; xi  f 0 .A/x; x

(2.1)

for any x 2 H with kxk D 1: Proof. Since f is convex and differentiable, we have that f .t/  f .s/  f 0 .t/  .t  s/ for any t; s 2 Œm; M : Now, if we chose in this inequality s D hAx; xi 2 Œm; M  for any x 2 H with kxk D 1 since Sp.A/  Œm; M ; then we have f .t/  f .hAx; xi/  f 0 .t/  .t  hAx; xi/

(2.2)

for any t 2 Œm; M  any x 2 H with kxk D 1: If we fix x 2 H with kxk D 1 in (2.2) and apply property (P) then we get ˝ ˛ hŒf .A/  f .hAx; xi/ 1H  x; xi  f 0 .A/  .A  hAx; xi 1H / x; x for each x 2 H with kxk D 1; which is clearly equivalent to the desired inequality (2.1). t u Corollary 2.2 (Dragomir, 2008, [8]).   Assume that f is as in Theorem 2.1. If Aj ˚ j 2 f1; : : : ; ng and xj 2 are selfadjoint operators with Sp Aj  Œm; M   I, Pn  2 xj  D 1, then H; j 2 f1; : : : ; ng with j D1

0 1 n n X X ˝   ˛ ˝ ˛ .0 / f Aj xj ; xj  f @ Aj xj ; xj A j D1



j D1

n n n X ˝ 0  ˛ X ˝ ˛ X ˝ 0  ˛ f Aj Aj xj ; xj  Aj xj ; xj  f Aj xj ; xj : j D1

j D1

j D1

(2.3)

2.2 Reverses of the Jensen Inequality

17

Corollary 2.3 (Dragomir, 2008, [8]). Assume that f is as in Theorem 2.1. If Aj ˚ are selfadjoint operators Pn with Sp Aj  Œm; M   I, j 2 f1; : : : ; ng and pj  0; j 2 f1; : : : ; ng with j D1 pj D 1; then

.0 /

* n X

0* + +1 n X   pj f Aj x; x  f @ pj Aj x; x A

j D1



* n X

j D1

+ + * n n X X     0 pj f Aj Aj x; x  pj Aj x; x  pj f Aj x; x ; (2.4) +

*

0

j D1

j D1

j D1

for each x 2 H with kxk D 1: Remark 2.4. Inequality (2.4), in the scalar case, namely

.0 /

n X

0 1 n X   pj f xj  f @ pj xj A

j D1



n X

j D1

n n X X     pj f 0 xj xj  pj xj  pj f 0 xj ;

j D1

j D1

(2.5)

j D1

where xj 2 ˚I, j 2 f1; : : : ; ng; has been obtained by the first time in 1994 by Dragomir and Ionescu, see [16].

2.2.2 Further Reverses In applications would be perhaps more useful to find upper bounds for the quantity hf .A/x; xi  f .hAx; xi/;

x2H

with

kxk D 1;

that are in terms of the spectrum margins m; M and of the function f . The following result may be stated: Theorem 2.5 (Dragomir, 2008, [8]). Let I be an interval and f W I ! R be a convex and differentiable function on I˚ (the interior of I / whose derivative f 0 is ˚ If A is a selfadjoint operator on the Hilbert space H with Sp.A/  continuous on I: ˚ then Œm; M   I;

18

2 Inequalities of the Jensen Type

.0 / hf .A/x; xi  f .hAx; xi/ 8 h i1=2 1 2 2 ˆ 0 0 ˆ  .M  m/ .A/xk  .A/x; xi kf hf ˆ < 2  ˆ i1=2 h ˆ ˆ : 1  .f 0 .M /  f 0 .m// kAxk2  hAx; xi2 2   1  .M  m/ f 0 .M /  f 0 .m/ 4

(2.6)

for any x 2 H with kxk D 1: We also have the inequality .0 / hf .A/x; xi  f .hAx; xi/ 



  1 .M  m/ f 0 .M /  f 0 .m/ 4 8 1 0 0 0 0 2 ˆ < ŒhM x  Ax; Ax  mxi hf .M /x  f .A/x; f .A/x  f .m/xi ;  ˇ ˇ ˆ : ˇˇhAx; xi  M Cm ˇˇ ˇˇhf 0 .A/x; xi  f 0 .M /Cf 0 .m/ ˇˇ 2 2   1 .M  m/ f 0 .M /  f 0 .m/ 4

(2.7)

for any x 2 H with kxk D 1: Moreover, if m > 0 and f 0 .m/ > 0; then we also have .0 / hf .A/x; xi  f .hAx; xi/ 8 0 .M /f 0 .m// 1 ˆ <  .Mpm/.f hAx; xi hf 0 .A/x; xi; 0 .M /f 0 .m/ M mf 4     ˆ : pM  pm pf 0 .M /  pf 0 .m/ ŒhAx; xi hf 0 .A/x; xi 12

(2.8)

for any x 2 H with kxk D 1: Proof. We use the following Gr¨uss’ type result we obtained in [6]: Let A be a selfadjoint operator on the Hilbert space .H I h:; :i/ and assume that Sp.A/  Œm; M  for some scalars m < M: If h and g are continuous on Œm; M  and  WD mint 2Œm;M  h.t/ and  WD maxt 2Œm;M  h.t/; then jhh.A/g.A/x; xi  hh.A/x; xi  hg.A/x; xij h i1=2 1   .   / kg.A/xk2  hg.A/x; xi2 2   1  .   / .  ı/ 4

(2.9)

for each x 2 H with kxk D1; where ı WD mint 2Œm;M  g.t/ and  WD maxt 2Œm;M  g.t/:

2.2 Reverses of the Jensen Inequality

19

Therefore, we can state that ˝

˛ ˝ ˛ Af 0 .A/x; x  hAx; xi  f 0 .A/x; x h 2 ˝ ˛2 i1=2 1   .M  m/ f 0 .A/x   f 0 .A/x; x 2   1  .M  m/ f 0 .M /  f 0 .m/ 4

(2.10)

and ˝

˛ ˝ ˛ Af 0 .A/x; x  hAx; xi  f 0 .A/x; x 

i1=2 h 1  0  f .M /  f 0 .m/ kAxk2  hAx; xi2 2



  1 .M  m/ f 0 .M /  f 0 .m/ 4

(2.11)

for each x 2 H with kxk D 1; which together with (2.1) provide the desired result (2.6). On making use of the inequality obtained in [7]: jhh.A/g.A/x; xi  hh.A/x; xi hg.A/x; xij 

1  .   / .  ı/ 4 8 1 ˆ < Œh x  h.A/x; f .A/x   xi hx  g.A/x; g.A/x  ıxi 2;  ˇ ˇ ˆ : ˇˇhh.A/x; xi   C ˇˇ ˇˇhg.A/x; xi  Cı ˇˇ 2 2

(2.12)

for each x 2 H with kxk D 1; we can state that ˝ 0 ˛ ˝ ˛ Af .A/x; x  hAx; xi  f 0 .A/x; x 

  1 .M  m/ f 0 .M /  f 0 .m/ 4 8 1 ˆ < ŒhM x  Ax; Ax  mxi hf 0 .M /x  f 0 .A/x; f 0 .A/x  f 0 .m/xi 2 ;  ˇ ˇ ˇˇ 0 0 ˆ : ˇhAx; xi  M Cm ˇ ˇˇhf 0 .A/x; xi  f .M /Cf .m/ ˇˇ 2 2

for each x 2 H with kxk D 1; which together with (2.1) provide the desired result (2.7). Further, in order to prove the third inequality, we make use of the following result of Gr¨uss type obtained in [7]:

20

2 Inequalities of the Jensen Type

If  and ı are positive, then jhh.A/g.A/x; xi  hh.A/x; xi hg.A/x; xij 8 1 /.ı/ ˆ <  . p hh.A/x; xi hg.A/x; xi;   ı  4p   p  p 1 p ˆ :      ı Œhh.A/x; xi hg.A/x; xi 2

(2.13)

for each x 2 H with kxk D 1: Now, on making use of (2.13) we can state that ˝

˛ ˝ ˛ Af 0 .A/x; x  hAx; xi  f 0 .A/x; x 8 1 .M m/.f 0 .M /f 0 .m// ˆ ˆ <  pM mf 0 .M /f 0 .m/ hAx; xi hf 0 .A/x; xi; 4    p p ˆ 1 p  p 0 ˆ : M m f .M /  f 0 .m/ ŒhAx; xi hf 0 .A/x; xi 2

for each x 2 H with kxk D 1; which together with (2.1) provide the desired result (2.8). t u Corollary 2.6 (Dragomir, 2008,[8]). Assume that f is as in Theorem 2.5. If Aj ˚ j 2 f1; : : : ; ng, then are selfadjoint operators with Sp Aj  Œm; M   I, 0 1 n n X X ˝   ˛ ˝ ˛ f Aj xj ; xj  f @ Aj xj ; xj A .0 / j D1

j D1

8

    ˝ 0  ˛2 1=2 Pn  1 ˆ f 0 Aj xj 2  Pn ˆ  .M  m/ f A x ; x ; ˆ j j j j D1 j D1 <2 

 ˆ1  ˝ ˛2 1=2 ˆ Pn  ˆ Aj xj 2  Pn :  .f 0 .M /  f 0 .m// A x ; x ; j j j j D1 j D1 2   1  .M  m/ f 0 .M /  f 0 .m/ (2.14) 4 for any xj 2 H; j 2 f1; : : : ; ng with We also have the inequality

Pn

j D1

 2 xj  D 1:

0 1 n n X X ˝   ˛ ˝ ˛ .0 / f Aj xj ; xj  f @ Aj xj ; xj A j D1



j D1

  1 .M  m/ f 0 .M /  f 0 .m/ 4

2.2 Reverses of the Jensen Inequality

21

8" #1 ˆ n ˝ ˆ ˛ 2 P ˆ ˆ M xj  Aj x; Aj xj  mxj ˆ ˆ ˆ j D1 ˆ ˆ ˆ #1=2 " ˆ < n ˝     ˛ P   f 0 .M /xj  f 0 Aj xj ; f 0 Aj xj  f 0 .m/xj ; ˆ j D1 ˆ ˆ ˆ ˆ ˇ ˇˇ ˇ ˆ ˆ ˆ ˇˇ P n ˝ n ˝ ˛ M Cm ˇˇ ˇˇ P   ˛ f 0 .M /Cf 0 .m/ ˇˇ ˆ ˆ 0 ˆ Aj xj ; xj  2 ˇ ˇ f Aj xj ; xj  ˇ : ˇˇ 2 ˇˇ ˇ j D1

j D1

  1 .M  m/ f 0 .M /  f 0 .m/ 4  2 P for any xj 2 H; j 2 f1; : : : ; ng with nj D1 xj  D 1: Moreover, if m > 0 and f 0 .m/ > 0; then we also have 0 1 n n X X ˝   ˛ ˝ ˛ f Aj xj ; xj  f @ Aj xj ; xj A .0 / 

j D1

(2.15)

j D1

81 ˝ ˛ Pn ˝ 0   ˛ .M m/.f 0 .M /f 0 .m// Pn ˆ Aj xj ; xj ; ˆ j D1 Aj xj ; xj j D1 f ˆ 4  pM mf 0 .M /f 0 .m/ ˆ ˆ < p  p p  p 0  M m f .M /  f 0 .m/ ˆ ˆ ˆ i1 h ˆ ˆ :  Pn ˝A x ; x ˛ Pn ˝f 0 A  x ; x ˛ 2 j D1

j

j

j

j D1

for any xj 2 H; j 2 f1; : : : ; ng with

j

Pn

j D1

j

(2.16)

j

 2 xj  D 1:

The following corollary also holds: Corollary 2.7 (Dragomir, 2008, [8]). Assume that f is as in Theorem 2.1. If Aj ˚ are selfadjoint operators Pn with Sp Aj  Œm; M   I, j 2 f1; : : : ; ng and pj  0; j 2 f1; : : : ; ng with j D1 pj D 1; then 0* + +1 * n n X X   pj f Aj x; x  f @ pj Aj x; x A .0 / j D1





j D1

8 2 +2 31=2 * ˆ n n ˆ  0   2   P P 1 ˆ ˆ ˆ  .M  m/ 4 pj f Aj x   pj f 0 Aj x; x 5 ; ˆ ˆ <2 j D1 j D1 2 * +2 31=2 ˆ ˆ ˆ n n   P P 1 ˆ 2 ˆ ˆ  .f 0 .M /  f 0 .m// 4 pj Aj x   pj Aj x; x 5 ; ˆ :2 j D1 j D1   1 .M  m/ f 0 .M /  f 0 .m/ 4

for any x 2 H with kxk D 1:

(2.17)

22

2 Inequalities of the Jensen Type

We also have the inequality .0 /

* n X

0* + +1 n X   pj f Aj x; x  f @ pj Aj x; x A

j D1



j D1

  1 .M  m/ f 0 .M /  f 0 .m/ 4 8" #1 ˆ n ˆ ˝ ˛ 2 P ˆ ˆ pj M x  Aj x; Aj x  mx ˆ ˆ ˆ j D1 ˆ ˆ " #1=2 ˆ ˆ < n ˝ 0     ˛ P 0 0 0 pj f .M /x  f Aj x; f Aj x  f .m/x ;   ˆ j D1 ˆ ˆ ˆ ˆ ˇ* ˇ ˇ* ˇ + + ˆ ˆ ˇ n ˇˇ n ˇ ˆ   ˆˇ P f 0 .M /Cf 0 .m/ ˇ M Cm ˇ ˇ P 0 ˆ ˆ A x; x  p A x; x  p f ˇ ˇ ˇ ˇ j j j j :ˇ 2 ˇˇ 2 ˇ j D1



j D1

  1 .M  m/ f 0 .M /  f 0 .m/ 4

(2.18)

for any x 2 H with kxk D 1: Moreover, if m > 0 and f 0 .m/ > 0; then we also have 0* * n + +1 n X X   .0 / pj f Aj x; x  f @ pj Aj x; x A j D1

j D1

8 E E DP   1 .M m/.f 0 .M /f 0 .m// DPn n ˆ ˆ  pM mf 0 .M /f 0 .m/ pj Aj x; x pj f 0 Aj x; x ; ˆ j D1 j D1 ˆ 4 ˆ ˆ  < p p p  p 0 M m f .M /  f 0 .m/  ˆ ˆ ˆ ˆ Ei 1 ED hD ˆ ˆ :  Pn p A x; x Pn p f 0 A  x; x 2 j D1

j

j

j D1

j

j

(2.19) for any x 2 H with kxk D 1: Remark 2.8. Some of the inequalities in Corollary 2.7 can be used to produce reverse norm inequalities for the sum of positive operators in the case when the convex function f is non-negative and monotonic non-decreasing on Œ0; M : For instance, if we use inequality (2.17), then we have   1 0  n  n   X X         @ A .0 /  pj f Aj   f  pj Aj   j D1 j D1   

  1 .M  m/ f 0 .M /  f 0 .m/ : 4

(2.20)

2.3 Some Slater Type Inequalities

23

Moreover, if we use inequality (2.19), then we obtain   1 0 X X    n  n       @ A .0 /  pj f Aj   f  pj Aj   j D1 j D1      8 P n n   P 1 .M m/.f 0 .M /f 0 .m//  ˆ     0 ˆ ˆ  pM mf 0 .M /f 0 .m/  pj Aj   pj f Aj  ; ˆ <4  j D1  j D1 # 1   "  2 P ˆ p  P n n p ˆ   p p 0     ˆ ˆ M m f .M / f 0 .m/  pj Aj   p j f 0 Aj  : :   j D1 j D1 (2.21)

2.3 Some Slater Type Inequalities 2.3.1 Slater Type Inequalities for Functions of Real Variables Suppose that I is an interval of real numbers with interior ˚I and f W I ! R is a convex function on I . Then f is continuous on ˚I and has finite left and right derivatives at each point of ˚I. Moreover, if x; y 2˚I and x < y; then f0 .x/  fC0 .x/  f0 .y/  fC0 .y/ which shows that both f0 and fC0 are non-decreasing function on ˚I. It is also known that a convex function must be differentiable except for at most countably many points. For a convex function f W I ! R, the sub-differential of f denoted by @f is the  ˚ set of all functions ' W I ! Œ1; 1 such that ' I  R and f .x/  f .a/ C .x  a/ '.a/

for any x; a 2 I:

It is also well known that if f is convex on I; then @f is non-empty, f0 , fC0 2 @f and if ' 2 @f , then f0 .x/  ' .x/  fC0 .x/

for any x 2 ˚I.

In particular, ' is a non-decreasing function. If f is differentiable and convex on ˚I, then @f D ff 0 g : The following result is well known in the literature as the Slater inequality: Theorem 2.9 (Slater, 1981, [28]). If f W I ! R is a non-increasing (nonPn decreasing) convex function, x 2 I; p  0 with P WD p > 0 and i i n i i D1 Pn i D1 pi ' .xi / ¤ 0; where ' 2 @f; then   Pn n pi xi ' .xi / 1 X Pi D1 : pi f .xi /  f n Pn i D1 i D1 pi ' .xi /

(2.22)

24

2 Inequalities of the Jensen Type

As pointed out in [5, p. 208], the monotonicity assumption for the derivative ' can be replaced with the condition Pn pi xi ' .xi / Pi D1 2 I; n i D1 pi ' .xi /

(2.23)

which is more general and can hold for suitable points in I and for not necessarily monotonic functions.

2.3.2 Some Slater Type Inequalities for Operators The following result holds: Theorem 2.10 (Dragomir, 2008, [9]). Let I be an interval and f W I ! R be a convex and differentiable function on I˚ (the interior of I / whose derivative f 0 is ˚ If A is a selfadjoint operator on the Hilbert space H with Sp.A/  continuous on I: Œm; M   I˚ and f 0 .A/ is a positive definite operator on H then  hAf 0 .A/x; xi  hf .A/x; xi 0f hf 0 .A/x; xi 

 hAf 0 .A/x; xi  hAx; xi hf 0 .A/x; xi hAf 0 .A/x; xi f0 hf 0 .A/x; xi hf 0 .A/x; xi 

(2.24)

for any x 2 H with kxk D 1: Proof. Since f is convex and differentiable on ˚I, then we have that f 0 .s/  .t  s/  f .t/  f .s/  f 0 .t/  .t  s/

(2.25)

for any t; s 2 Œm; M : Now, if we fix t 2 Œm; M  and apply property (P) for the operator A; then for any x 2 H with kxk D 1 we have ˝ 0 ˛ f .A/  .t  1H  A/ x; x  hŒf .t/  1H  f .A/ x; xi ˝ ˛  f 0 .t/  .t  1H  A/ x; x (2.26) for any t 2 Œm; M  and any x 2 H with kxk D 1: Inequality (2.26) is equivalent with ˝ ˛ ˝ ˛ t f 0 .A/x; x  f 0 .A/Ax; x  f .t/  hf .A/x; xi  f 0 .t/t  f 0 .t/ hAx; xi for any t 2 Œm; M  any x 2 H with kxk D 1:

(2.27)

2.3 Some Slater Type Inequalities

25

Now, since A is selfadjoint with mI  A  MI and f 0 .A/ is positive definite, then mf 0 .A/  Af 0 .A/  Mf 0 .A/; i.e. m hf 0 .A/x; xi  hAf 0 .A/x; xi  M hf 0 .A/x; xi for any x 2 H with kxk D 1; which shows that t0 WD

hAf 0 .A/x; xi 2 Œm; M  hf 0 .A/x; xi

for any x 2 H

with

kxk D 1:

Finally, if we put t D t0 in (2.27), then we get the desired result (2.24). Remark 2.11. It is important to observe that, the condition that f 0 .A/ is a positive definite operator on H can be replaced with the more general assumption that hAf 0 .A/x; xi ˚ 2I hf 0 .A/x; xi

for any x 2 H

with

kxk D 1;

(2.28)

which may be easily verified for particular convex functions f: Remark 2.12. Now, if the functions are concave on ˚I and condition (2.28) holds, then we have the inequality 

 hAf 0 .A/x; xi 0  hf .A/x; xi  f hf 0 .A/x; xi 

 hAx; xi hf 0 .A/x; xi  hAf 0 .A/x; xi hAf 0 .A/x; xi f0 hf 0 .A/x; xi hf 0 .A/x; xi

(2.29)

for any x 2 H with kxk D 1:

2.3.3 Further Reverses The following results that provide perhaps more useful upper bounds for the nonnegative quantity:  f

hAf 0 .A/x; xi hf 0 .A/x; xi

  hf .A/x; xi

for x 2 H

with

kxk D 1

can be stated: Theorem 2.13 (Dragomir, 2008, [9]). Let I be an interval and f W I ! R be a convex and differentiable function on I˚ (the interior of I / whose derivative f 0 is ˚ Assume that A is a selfadjoint operator on the Hilbert space H continuous on I: with Sp.A/  Œm; M   I˚ and f 0 .A/ is a positive definite operator on H: If we define   0  0  1 0 hAf .A/x; xi B f ; AI x WD f ; hf 0 .A/x; xi hf 0 .A/x; xi

26

2 Inequalities of the Jensen Type

then  hAf 0 .A/x; xi  hf .A/x; xi .0 /f hf 0 .A/x; xi 8 h i1=2 1 2 2 ˆ 0 0 ˆ  .M  m/ .A/xk  .A/x; xi kf hf   <2  B f 0 ; AI x  i1=2 h ˆ ˆ : 1  .f 0 .M /  f 0 .m// kAxk2  hAx; xi2 2  0    1  .M  m/ f .M /  f 0 .m/ B f 0 ; AI x (2.30) 4 

and  .0 /f

hAf 0 .A/x; xi hf 0 .A/x; xi 2

  hf .A/x; xi

   61   B f 0 ; AI x  4 .M  m/ f 0 .M /  f 0 .m/ 4 8 3 1 ˆ < ŒhM x  Ax; Ax  mxi hf 0 .M /x  f 0 .A/x; f 0 .A/x  f 0 .m/xi 2 7  ˇ 5; ˇ ˇˇ ˆ : ˇhAx; xi  M Cm ˇ ˇˇhf 0 .A/x; xi  f 0 .M /Cf 0 .m/ ˇˇ 2 2 

    1 .M  m/ f 0 .M /  f 0 .m/ B f 0 ; AI x 4

(2.31)

for any x 2 H with kxk D 1; respectively. Moreover, if A is a positive definite operator, then 

 hAf 0 .A/x; xi  hf .A/x; xi hf 0 .A/x; xi    B f 0 ; AI x 8 0 .M /f 0 .m// 1 ˆ <  .Mpm/.f hAx; xi hf 0 .A/x; xi ; M mf 0 .M /f 0 .m/ 4     ˆ : pM pm pf 0 .M /pf 0 .m/ ŒhAx; xihf 0 .A/x; xi 12

.0 /f

(2.32)

for any x 2 H with kxk D 1: Proof. We use the following Gr¨uss’ type result we obtained in [6]: Let A be a selfadjoint operator on the Hilbert space .H I h:; :i/ and assume that Sp.A/  Œm; M  for some scalars m < M: If h and g are continuous on Œm; M  and  WD mint 2Œm;M  h.t/ and  WD maxt 2Œm;M  h.t/; then

2.3 Some Slater Type Inequalities

27

jhh.A/g.A/x; xi  hh.A/x; xi  hg.A/x; xij h i1=2 1   .   / kg.A/xk2  hg.A/x; xi2 2   1  .   / .  ı/ 4

(2.33)

for each x 2 H with kxk D1; where ı WD mint 2Œm;M  g.t/ and  WD maxt 2Œm;M  g.t/: Therefore, we can state that ˝ 0 ˛ ˝ ˛ Af .A/x; x  hAx; xi  f 0 .A/x; x h 2 ˝ ˛2 i1=2 1   .M  m/ f 0 .A/x   f 0 .A/x; x 2   1  .M  m/ f 0 .M /  f 0 .m/ (2.34) 4 and ˝

˛ ˝ ˛ Af 0 .A/x; x  hAx; xi  f 0 .A/x; x i1=2 h 1    f 0 .M /  f 0 .m/ kAxk2  hAx; xi2 2   1  .M  m/ f 0 .M /  f 0 .m/ 4

(2.35)

for each x 2 H with kxk D 1; which together with (2.24) provide the desired result (2.30). On making use of the inequality obtained in [7] jhh.A/g.A/x; xi  hh.A/x; xi hg.A/x; xij 

1  .   / .  ı/ 4 8 1 < Œh x  h.A/x; f .A/x   xi hx  g.A/x; g.A/x  ıxi 2; ˇ  ˇˇ : ˇhh.A/x; xi   C ˇˇ ˇˇhg.A/x; xi  Cı ˇˇ 2

(2.36)

2

for each x 2 H with kxk D 1; we can state that ˛ ˝ ˛ ˝ 0 Af .A/x; x  hAx; xi  f 0 .A/x; x 

  1 .M  m/ f 0 .M /  f 0 .m/ 4 8 1 < ŒhM x  Ax; Ax  mxi hf 0 .M /x  f 0 .A/x; f 0 .A/x  f 0 .m/xi 2; ˇ ˇ  ˇ : ˇhAx; xi  M Cm ˇˇ ˇˇhf 0 .A/x; xi  f 0 .M /Cf 0 .m/ ˇˇ 2 2

28

2 Inequalities of the Jensen Type

for each x 2 H with kxk D 1; which together with (2.24) provide the desired result (2.31). Further, in order to prove the third inequality, we make use of the following result of Gr¨uss type obtained in [7]: If  and ı are positive, then jhh.A/g.A/x; xi  hh.A/x; xi hg.A/x; xij 8 1 /.ı/ ˆ <  . p hh.A/x; xi hg.A/x; xi;   ı 4  p   p  p 1 ˆ p :      ı Œhh.A/x; xi hg.A/x; xi 2 for each x 2 H with kxk D 1: Now, on making use of (2.37) we can state that ˝ 0 ˛ ˝ ˛ Af .A/x; x  hAx; xi  f 0 .A/x; x 8 1 .M m/.f 0 .M /f 0 .m// ˆ ˆ <  pM mf 0 .M /f 0 .m/ hAx; xi hf 0 .A/x; xi ; 4    p p ˆ 1 p  p 0 ˆ : M m f .M /  f 0 .m/ ŒhAx; xi hf 0 .A/x; xi 2

(2.37)

(2.38)

for each x 2 H with kxk D 1; which together with (2.24) provide the desired result (2.32). t u Remark 2.14. We observe, from the first inequality in (2.32), that .1 /

1 .M  m/ .f 0 .M /  f 0 .m// hAf 0 .A/x; xi   C1 p 4 hAx; xi hf 0 .A/x; xi M mf 0 .M /f 0 .m/

which implies that f

0



hAf 0 .A/x; xi hf 0 .A/x; xi

"

 f

0

# ! 1 .M  m/ .f 0 .M /  f 0 .m//  p C 1 hAx; xi 4 M mf 0 .M /f 0 .m/

for each x 2 H with kxk D 1; since f 0 is monotonic non-decreasing and A is positive definite. Now, the first inequality in (2.32) implies the following result:   hAf 0 .A/x; xi .0 /f  hf .A/x; xi hf 0 .A/x; xi 

1 .M  m/ .f 0 .M /  f 0 .m//  p 4 M mf 0 .M /f 0 .m/ " # ! 1 .M  m/ .f 0 .M /  f 0 .m// 0  f p C 1 hAx; xi hAx; xi 4 M mf 0 .M /f 0 .m/

for each x 2 H with kxk D 1:

(2.39)

2.4 Other Inequalities for Convex Functions

29

From the second inequality in (2.32) we also have  .0 /f 

hAf 0 .A/x; xi hf 0 .A/x; xi

  hf .A/x; xi

p  p p  p 0 M m f .M /  f 0 .m/ " f

0

# !

12 1 .M  m/ .f 0 .M /  f 0 .m// hAx; xi  p C 1 hAx; xi 4 hf 0 .A/x; xi M mf 0 .M /f 0 .m/ (2.40)

for each x 2 H with kxk D 1: Remark 2.15. If the condition that f 0 .A/ is a positive definite operator on H from Theorem 2.13 is replaced by condition (2.28), then inequalities (2.30) and (2.33) will still hold. Similar inequalities for concave functions can be stated. However, the details are not provided here.

2.4 Other Inequalities for Convex Functions 2.4.1 Some Inequalities for Two Operators The following result holds: Theorem 2.16 (Dragomir, 2008, [10]). Let I be an interval and f W I ! R be a convex and differentiable function on I˚ (the interior of I / whose derivative f 0 is ˚ If A and B are selfadjoint operators on the Hilbert space H with continuous on I: ˚ then Sp.A/; Sp.B/  Œm; M   I; ˝

˛ ˝ ˛ f 0 .A/x; x hBy; yi  f 0 .A/Ax; x  hf .B/y; yi  hf .A/x; xi ˝ ˛ ˝ ˛  f 0 .B/By; y  hAx; xi f 0 .B/y; y

(2.41)

for any x; y 2 H with kxk D kyk D 1: In particular, we have ˝

˛ ˝ ˛ f 0 .A/x; x hAy; yi  f 0 .A/Ax; x  hf .A/y; yi  hf .A/x; xi ˝ ˛ ˝ ˛  f 0 .A/Ay; y  hAx; xi f 0 .A/y; y

for any x; y 2 H with kxk D kyk D 1 and

(2.42)

30

2 Inequalities of the Jensen Type

˝ 0 ˛ ˝ ˛ f .A/x; x hBx; xi  f 0 .A/Ax; x  hf .B/x; xi  hf .A/x; xi ˛ ˝ ˛ ˝  f 0 .B/Bx; x  hAx; xi f 0 .B/x; x

(2.43)

for any x 2 H with kxk D 1: Proof. Since f is convex and differentiable on ˚I, then we have that f 0 .s/  .t  s/  f .t/  f .s/  f 0 .t/  .t  s/

(2.44)

for any t; s 2 Œm; M : Now, if we fix t 2 Œm; M  and apply property (P) for the operator A; then for any x 2 H with kxk D 1 we have ˝ 0 ˛ f .A/  .t  1H  A/ x; x  hŒf .t/  1H  f .A/ x; xi ˝ ˛  f 0 .t/  .t  1H  A/ x; x (2.45) for any t 2 Œm; M  and any x 2 H with kxk D 1: Inequality (2.45) is equivalent with ˝ ˛ ˝ ˛ t f 0 .A/x; x  f 0 .A/Ax; x  f .t/  hf .A/x; xi  f 0 .t/t  f 0 .t/ hAx; xi

(2.46)

for any t 2 Œm; M  and any x 2 H with kxk D 1: If we fix x 2 H with kxk D 1 in (2.46) and apply property (P) for the operator B; then we get ˝˝ 0 ˛ ˝ ˛ ˛ f .A/x; x B  f 0 .A/Ax; x 1H y; y  hŒf .B/  hf .A/x; xi 1H  y; yi ˝ ˛  f 0 .B/B  hAx; xi f 0 .B/ y; y for each y 2 H with kyk D 1; which is clearly equivalent to the desired inequality (2.41). t u Remark 2.17. If we fix x 2 H with kxk D 1 and choose B D hAx; xi  1H ; then we obtain from the first inequality in (2.41) the reverse of the Mond–Peˇcari´c inequality obtained by the author in [8]. The second inequality will provide the Mond–Peˇcari´c inequality for convex functions whose derivatives are continuous. The following corollary is of interest: Corollary 2.18. Let I be an interval and f W I ! R be a convex and differentiable ˚ Also, suppose that A is a function on I˚ whose derivative f 0 is continuous on I: ˚ If g is selfadjoint operator on the Hilbert space H with Sp.A/  Œm; M   I: non-increasing and continuous on Œm; M  and f 0 .A/ Œg.A/  A  0

(2.47)

2.4 Other Inequalities for Convex Functions

31

in the operator order of B .H /; then .f ı g/ .A/  f .A/

(2.48)

in the operator order of B .H /: The following result may be stated as well: Theorem 2.19 (Dragomir, 2008, [10]). Let I be an interval and f W I ! R be a convex and differentiable function on I˚ (the interior of I / whose derivative f 0 is ˚ If A and B are selfadjoint operators on the Hilbert space H with continuous on I: ˚ then Sp.A/; Sp.B/  Œm; M   I; f 0 .hAx; xi/ .hBy; yi  hAx; xi/  hf .B/y; yi  f .hAx; xi/ ˛ ˝ ˛ ˝  f 0 .B/By; y  hAx; xi f 0 .B/y; y

(2.49)

for any x; y 2 H with kxk D kyk D 1: In particular, we have f 0 .hAx; xi/ .hAy; yi  hAx; xi/  hf .A/y; yi  f .hAx; xi/ ˛ ˝ ˛ ˝  f 0 .A/Ay; y  hAx; xi f 0 .A/y; y

(2.50)

for any x; y 2 H with kxk D kyk D 1 and f 0 .hAx; xi/ .hBx; xi  hAx; xi/  hf .B/x; xi  f .hAx; xi/ ˝ ˛ ˝ ˛  f 0 .B/Bx; x  hAx; xi f 0 .B/x; x

(2.51)

for any x 2 H with kxk D 1: Proof. Since f is convex and differentiable on ˚I, then we have that f 0 .s/  .t  s/  f .t/  f .s/  f 0 .t/  .t  s/

(2.52)

for any t; s 2 Œm; M : If we choose s D hAx; xi 2 Œm; M ; with a fix x 2 H with kxk D 1; then we have f 0 .hAx; xi/  .t  hAx; xi/  f .t/  f .hAx; xi/  f 0 .t/  .t  hAx; xi/ for any t 2 Œm; M :

(2.53)

32

2 Inequalities of the Jensen Type

Now, if we apply property (P) to inequality (2.53) and the operator B, then we get ˝

f 0 .hAx; xi/  .B  hAx; xi  1H / y; yi  hŒf .B/  f .hAx; xi/  1H  y; yi ˝ ˛  f 0 .B/  .B  hAx; xi  1H / y; y

(2.54)

for any x; y 2 H with kxk D kyk D 1; which is equivalent with the desired result (2.49). Remark 2.20. We observe that if we choose B D A in (2.51) or y D x in (2.50) then we recapture the Mond–Peˇcari´c inequality and its reverse from (2.1). The following particular case of interest follows from Theorem 2.19: Corollary 2.21 (Dragomir, 2008, [10]). Assume that f; A and B are as in Theorem 2.19. If, either f is increasing on Œm; M  and B  A in the operator order of B .H / or f is decreasing and B  A; then we have the Jensen’s type inequality hf .B/x; xi  f .hAx; xi/

(2.55)

for any x 2 H with kxk D 1: The proof is obvious by the first inequality in (2.51) and the details are omitted.

2.5 Some Jensen Type Inequalities for Twice Differentiable Functions 2.5.1 Jensen’s Inequality for Twice Differentiable Functions The following result may be stated: Theorem 2.22 (Dragomir, 2008, [11]). Let A be a positive definite operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with 0 < m < M: If f is a twice differentiable function on .m; M / and for p 2 .1; 0/ [ .1; 1/ we have for some  <  that 

t 2p  f 00 .t/   p .p  1/

t 2 .m; M / ;

(2.56)

   hAp x; xi  hAx; xip  hf .A/x; xi  f .hAx; xi/     hAp x; xi  hAx; xip

(2.57)

for any

then

for each x 2 H with kxk D 1:

2.5 Some Jensen Type Inequalities for Twice Differentiable Functions

33

If ı

t 2p  f 00 .t/   p .1  p/

t 2 .m; M /

(2.58)

  ı hAx; xip  hAp x; xi  hf .A/x; xi  f .hAx; xi/     hAx; xip  hAp x; xi

(2.59)

for any

and for some ı < ; where p 2 .0; 1/, then

for each x 2 H with kxk D 1: Proof. Consider the function g ;p W .m; M / ! R given by g ;p .t/ D f .t/   t p where p 2 .1; 0/ [ .1; 1/ : The function g ;p is twice differentiable, g00;p .t/ D f 00 .t/   p .p  1/ t p2 for any t 2 .m; M / and by (2.56) we deduce that g ;p is convex on .m; M /: Now, applying the Mond and Peˇcari´c inequality for g ;p we have  0  h.f .A/   Ap / x; xi  f .hAx; xi/   hAx; xip  D hf .A/x; xi  f .hAx; xi/   hAp x; xi  hAx; xip which is equivalent with the first inequality in (2.57). By defining the function g;p W .m; M / ! R given by g;p .t/ D  t p  f .t/ and applying the same argument we deduce the second part of (2.57). The rest goes likewise and the details are omitted. t u Remark 2.23. We observe that if f is a twice differentiable function on .m; M / and ' WD inft 2.m;M / f 00 .t/; Φ WD supt 2.m;M / f 00 .t/; then by (2.57) we get the inequality i ˛ 1 h˝ 2 ' A x; x  hAx; xi2  hf .A/x; xi  f .hAx; xi/ 2 i ˛ 1 h˝  Φ A2 x; x  hAx; xi2 2

(2.60)

for each x 2 H with kxk D 1: We observe that inequality (2.60) holds for selfadjoint operators that are not necessarily positive. The next result provides some inequalities for the function f which replace the cases p D 0 and p D 1 that were not allowed in Theorem 2.22.

34

2 Inequalities of the Jensen Type

Theorem 2.24 (Dragomir, 2008, [10]). Let A be a positive definite operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with 0 < m < M: If f is a twice differentiable function on .m; M / and we have for some  <  that   t 2  f 00 .t/  

for any

t 2 .m; M /;

(2.61)

then  .ln .hAx; xi/  hln Ax; xi/  hf .A/x; xi  f .hAx; xi/   .ln .hAx; xi/  hln Ax; xi/

(2.62)

for each x 2 H with kxk D 1: If ı  t  f 00 .t/  

for any

t 2 .m; M /

(2.63)

for some ı < , then ı .hA ln Ax; xi  hAx; xi ln .hAx; xi//  hf .A/x; xi  f .hAx; xi/   .hA ln Ax; xi  hAx; xi ln .hAx; xi//

(2.64)

for each x 2 H with kxk D 1: Proof. Consider the function g ;0 W .m; M / ! R given by g ;0 .t/ D f .t/ C  ln t: The function g ;0 is twice differentiable, g00;p .t/ D f 00 .t/   t 2 for any t 2 .m; M / and by (2.61) we deduce that g ;0 is convex on .m; M /: Now, applying the Mond and Peˇcari´c inequality for g ;0 we have 0  h.f .A/ C  ln A/ x; xi  Œf .hAx; xi/ C  ln .hAx; xi/ D hf .A/x; xi  f .hAx; xi/   Œln .hAx; xi/  hln Ax; xi which is equivalent with the first inequality in (2.62). By defining the function g;0 W .m; M / ! R given by g;0 .t/ D  ln t  f .t/ and applying the same argument we deduce the second part of (2.62). The rest goes likewise for the functions gı;1 .t/ D f .t/  ıt ln t and the details are omitted.

and

g;0 .t/ D t ln t  f .t/

2.6 Some Jensen’s Type Inequalities for Log-Convex Functions

35

2.6 Some Jensen’s Type Inequalities for Log-Convex Functions 2.6.1 Preliminary Results The following result that provides an operator version for the Jensen inequality for convex functions is due to Mond and Peˇcari´c [25] (see also [19, p. 5]): Theorem 2.25 (Mond–Peˇcari´c, 1993, [25]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with m < M: If f is a convex function on Œm; M ; then f .hAx; xi/  hf .A/x; xi

(MP)

for each x 2 H with kxk D 1: Taking into account the above result and its applications for various concrete examples of convex functions, it is therefore natural to investigate the corresponding results for the case of log-convex functions, namely functions f W I ! .0; 1/ for which ln f is convex. We observe that such functions satisfy the elementary inequality f ..1  t/ a C tb/  Œf .a/1t Œf .b/t

(2.65)

for any a; b 2 I and t 2 Œ0; 1 : Also, due to the fact that the weighted geometric mean is less than the weighted arithmetic mean, it follows that any log-convex function is a convex function. However, obviously, there are functions that are convex but not log-convex. As an immediate consequence of the Mond–Peˇcari´c inequality above, we can provide the following result: Theorem 2.26 (Dragomir, 2010, [14]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with m < M: If g W Œm; M  ! .0; 1/ is log-convex, then g .hAx; xi/  exp hln g.A/x; xi  hg.A/x; xi

(2.66)

for each x 2 H with kxk D 1: Proof. Consider the function f WD ln g; which is convex on Œm; M : Writing (MP) for f we get ln Œg .hAx; xi/  hln g.A/x; xi; for each x 2 H with kxk D 1; which, by taking the exponential, produces the first inequality in (2.66). If we also use (MP) for the exponential function, we get exp hln g.A/x; xi  hexp Œln g.A/ x; xi D hg.A/x; xi for each x 2 H with kxk D 1 and the proof is complete.

36

2 Inequalities of the Jensen Type

The case of sequences of operators may be of interest and is embodied in the following corollary: Corollary 2.27 (Dragomir, 2010, [14]).  Assume that g is as in Theorem 2.26. If Aj are selfadjoint operators with Sp Aj  Œm; M , j 2 f1; : : : ; ng and xj 2  2 P H; j 2 f1; : : : ; ng with nj D1 xj  D 1, then 0 g@

n X ˝

˛

1

Aj xj ; xj A  exp

j D1

* n X

  ln g Aj xj ; xj

+

j D1

+ n X    g Aj xj ; xj : *

(2.67)

j D1

t u

Proof. Follows from Theorem 2.26 and we omit the details. In particular we have:

Corollary 2.28 (Dragomir, 2010, [14]).  Assume that g is as in Theorem 2.26. If Aj ˚ j 2 f1; : : : ; ng and pj  0; are selfadjoint operators with Sp A  Œm; M   I, j Pn j 2 f1; : : : ; ng with j D1 pj D 1; then 0* g@

n X

+1

*

n Y    pj g Aj pj Aj x; x A  x; x

j D1

* 

j D1 n X





+

+

pj g Aj x; x

(2.68)

j D1

for each x 2 H with kxk D 1: Proof. Follows from Corollary 2.27 by choosing xj D where x 2 H with kxk D 1:

p pj  x; j 2 f1; : : : ; ng 

The following reverse for the Mond–Peˇcari´c inequality that generalizes the scalar Lah–Ribari´c inequality for convex functions is well known, see for instance [19, p.57]: Theorem 2.29. Let A be a selfadjoint operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with m < M: If f is a convex function on Œm; M ; then hf .A/x; xi 

M  hAx; xi hAx; xi  m  f .m/ C  f .M / M m M m

for each x 2 H with kxk D 1:

(2.69)

2.6 Some Jensen’s Type Inequalities for Log-Convex Functions

37

This result can be improved for log-convex functions as follows: Theorem 2.30 (Dragomir, 2010, [14]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with m < M: If g W Œm; M  ! .0; 1/ is log-convex, then Dh E M1H A Am1H i hg.A/x; xi  Œg.m/ M m Œg.M / M m x; x 

M  hAx; xi hAx; xi  m  g.m/ C  g.M / M m M m

(2.70)

and M hAx;xi

hAx;xim

g .hAx; xi/  Œg.m/ M m Œg.M / M m E Dh M1H A Am1H i  Œg.m/ M m Œg.M / M m x; x

(2.71)

for each x 2 H with kxk D 1: Proof. Observe that, by the log-convexity of g; we have   t m M t mC M g.t/ D g M m M m M t

t m

 Œg.m/ M m Œg.M / M m

(2.72)

for any t 2 Œm; M : Applying property (P) for the operator A, we have that hg.A/x; xi  hΨ.A/x; xi M t

t m

for each x 2 H with kxk D 1; where Ψ.t/ WD Œg.m/ M m Œg.M / M m ; t 2 Œm; M : This proves the first inequality in (2.70). Now, observe that, by the weighted arithmetic mean–geometric mean inequality we have M t

t m

Œg.m/ M m Œg.M / M m 

t m M t  g.m/ C  g.M / M m M m

for any t 2 Œm; M : Applying property (P) for the operator A, we deduce the second inequality in (2.70). Further on, if we use inequality (2.72) for t D hAx; xi 2 Œm; M , then we deduce the first part of (2.71). Now, observe that the function Ψ introduced above can be rearranged to read as

g.M / Ψ.t/ D g.m/ g.m/

Mt m m

showing that Ψ is a convex function on Œm; M :

; t 2 Œm; M 

38

2 Inequalities of the Jensen Type

Applying Mond–Peˇcari´c’s inequality for Ψ we deduce the second part of (2.71) and the proof is complete.

2.6.2 Jensen’s Inequality for Differentiable Log-Convex Functions The following result provides a reverse for the Jensen type inequality (MP): Theorem 2.31 (Dragomir, 2008, [8]). Let J be an interval and f W J ! R be a convex and differentiable function on J˚ (the interior of J / whose derivative f 0 is ˚ If A is a selfadjoint operator on the Hilbert space H with Sp.A/  continuous on J: ˚ then Œm; M   J; .0 / hf .A/x; xi  f .hAx; xi/ ˝ ˛ ˝ ˛  f 0 .A/Ax; x  hAx; xi  f 0 .A/x; x

(2.73)

for any x 2 H with kxk D 1: The following result may be stated: Proposition 2.32 (Dragomir, 2010, [14]). Let J be an interval and g W J ! R be ˚ If a differentiable log-convex function on J˚ whose derivative g0 is continuous on J. ˚ A is a selfadjoint operator on the Hilbert space H with Sp.A/  Œm; M   J; then .1 /

hD E exp hln g.A/x; xi  exp g0 .A/ Œg.A/1 Ax; x g .hAx; xi/ Ei D  hAx; xi  g 0 .A/ Œg.A/1 x; x

(2.74)

for each x 2 H with kxk D 1: Proof. It follows by inequality (2.73) written for the convex function f D ln g that hln g.A/x; xi  ln g .hAx; xi/ E D E D C g 0 .A/ Œg.A/1 Ax; x  hAx; xi  g 0 .A/ Œg.A/1 x; x for each x 2 H with kxk D 1: Now, taking the exponential and dividing by g .hAx; xi/ > 0 for each x 2 H with kxk D 1; we deduce the desired result (2.74). u t The following result that provides both a refinement and a reverse of the multiplicative version of Jensen’s inequality can be stated as well: Theorem 2.33 (Dragomir, 2010, [14]). Let J be an interval and g W J ! R be a ˚ If A log-convex differentiable function on J˚ whose derivative g0 is continuous on J. ˚ then is a selfadjoint operator on the Hilbert space H with Sp.A/  Œm; M   J;

2.6 Some Jensen’s Type Inequalities for Log-Convex Functions

39



 g 0 .hAx; xi/ .A  hAx; xi 1H / x; x 1  exp g .hAx; xi/ h i E hg.A/x; xi D  exp g 0 .A/ Œg.A/1 .A  hAx; xi 1H / x; x  g .hAx; xi/

(2.75)

for each x 2 H with kxk D 1; where 1H denotes the identity operator on H: ˚ Proof. It is well known that if h W J ! R is a convex differentiable function on J, then the following gradient inequality holds: h.t/  h.s/  h0 .s/.t  s/ ˚ for any t; s 2J. Now, if we write this inequality for the convex function h D ln g; then we get ln g.t/  ln g.s/ 

g 0 .s/ .t  s/ g.s/

(2.76)

which is equivalent with

g0 .s/ .t  s/ g.t/  g.s/ exp g.s/

(2.77)

˚ for any t; s 2J. ˚ for a fixed x 2 H with kxk D 1; Further, if we take s WD hAx; xi 2 Œm; M   J; in inequality (2.77), then we get g.t/  g .hAx; xi/ exp



g 0 .hAx; xi/ .t  hAx; xi/ g .hAx; xi/

˚ for any t 2J. Utilizing property (P) for the operator A and the Mond–Peˇcari´c inequality for the exponential function, we can state the following inequality that is of interest in itself as well: hg.A/y; yi

0

 g .hAx; xi/  g .hAx; xi/ exp .A  hAx; xi 1H / y; y g .hAx; xi/

0 g .hAx; xi/ .hAy; yi  hAx; xi/  g .hAx; xi/ exp g .hAx; xi/

(2.78)

for each x; y 2 H with kxk D kyk D 1: Further, if we put y D x in (2.78), then we deduce the first and the second inequality in (2.75).

40

2 Inequalities of the Jensen Type

Now, if we replace s with t in (2.77) we can also write the inequality g.t/ exp

g 0 .t/ .s  t/  g.s/ g.t/

which is equivalent with

g 0 .t/ g.t/  g.s/ exp .t  s/ g.t/

(2.79)

˚ for any t; s 2J. ˚ for a fixed x 2 H with kxk D 1; Further, if we take s WD hAx; xi 2 Œm; M   J; in inequality (2.79), then we get g.t/  g .hAx; xi/ exp

g 0 .t/ .t  hAx; xi/ g.t/

˚ for any t 2J. Utilizing property (P) for the operator A, then we can state the following inequality that is of interest in itself as well: hg.A/y; yi

D h i E  g .hAx; xi/ exp g 0 .A/ Œg.A/1 .A  hAx; xi 1H / y; y

(2.80)

for each x; y 2 H with kxk D kyk D 1: Finally, if we put y D x in (2.80), then we deduce the last inequality in (2.75). u t The following reverse inequality may be proven as well: Theorem 2.34 (Dragomir, 2010, [14]). Let J be an interval and g W J ! R be a ˚ If A log-convex differentiable function on J˚ whose derivative g0 is continuous on J. ˚ then is a selfadjoint operators on the Hilbert space H with Sp.A/  Œm; M   J;

.1 /



D E Am1H M1H A Œg.M / M m Œg.m/ M m x; x hg.A/x; xi  0 D h g .M / H/  g.A/ exp .M1H A/.Am1 M m g.M /



hg.A/x; xi

g0 .M / g 0 .m/ 1 .M  m/   exp 4 g.M / g.m/ for each x 2 H with kxk D 1:

g 0 .m/ g.m/

i

E x; x



(2.81)

2.6 Some Jensen’s Type Inequalities for Log-Convex Functions

41

Proof. Utilizing inequality (2.76) we have successively

and



g0 .s/ g ..1  / t C s/  exp .1  / .t  s/ g.s/ g.s/

(2.82)



g0 .t/ g ..1  / t C s/  exp  .t  s/ g.t/ g.t/

(2.83)

for any t; s 2J˚ and any  2 Œ0; 1 : Now, if we take the power  in inequality (2.82) and the power 1   in (2.83) and multiply the obtained inequalities, we deduce Œg.t/1 Œg.s/ g ..1  / t C s/ 

 0 g .t/ g 0 .s/  .t  s/  exp .1  /  g.t/ g.s/ for any t; s 2J˚ and any  2 Œ0; 1 : Further on, if we choose in (2.84) t D M; s D m and  D (2.84) we get the inequality

(2.84)

M u ; M m

then, from

M u

um

Œg.M / M m Œg.m/ M m g .u/  

.M  u/ .u  m/ g0 .M / g 0 .m/   exp M m g.M / g.m/

(2.85)

which, together with the inequality 1 .M  u/ .u  m/  .M  m/ M m 4 produce um

M u

Œg.M / M m Œg.m/ M m  

.M  u/ .u  m/ g0 .M / g 0 .m/   g .u/ exp M m g.M / g.m/  0 

0 g .M / g .m/ 1 .M  m/   g .u/ exp 4 g.M / g.m/

(2.86)

for any u 2 Œm; M : If we apply property (P) to inequality (2.86) and for the operator A we deduce the desired result. t u

42

2 Inequalities of the Jensen Type

2.6.3 More Inequalities for Differentiable Log-Convex Functions The following results providing companion inequalities for the Jensen inequality for differentiable log-convex functions obtained above hold: Theorem 2.35 (Dragomir, 2010, [15]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with m < M: If g W J ! .0; 1/ is a differentiable log-convex function with the derivative continuous on JV and Œm; M   JV , then

hg 0 .A/Ax; xi hg.A/Ax; xi hg 0 .A/x; xi   hg.A/x; xi hg.A/x; xi hg.A/x; xi i h ln g.A/x;xi exp hg.A/ hg.A/x;xi   1  g hg.A/Ax;xi hg.A/x;xi



exp

for each x 2 H with kxk D 1: If hg 0 .A/Ax; xi 2 JV for each x 2 H with kxk D 1; hg 0 .A/x; xi

(2.87)

(C)

then 2 exp 4

g0 g

 

hg 0 .A/Ax;xi hg 0 .A/x;xi

hg 0 .A/Ax;xi hg 0 .A/x;xi







3  hg 0 .A/Ax; xi hAg.A/x; xi 5  hg 0 .A/x; xi hg.A/x; xi

  0 .A/Ax;xi g hg 0 hg .A/x;xi  1   hg.A/ ln g.A/x;xi exp hg.A/x;xi

(2.88)

for each x 2 H with kxk D 1: Proof. By the gradient inequality for the convex function ln g we have g 0 .s/ g 0 .t/ .t  s/  ln g.t/  ln g.s/  .t  s/ g.t/ g.s/

(2.89)

for any t; s 2 JV , which by multiplication with g.t/ > 0 is equivalent with g0 .t/.t  s/  g.t/ ln g.t/  g.t/ ln g.s/  for any t; s 2 JV :

g0 .s/ .tg.t/  sg.t// g.s/

(2.90)

2.6 Some Jensen’s Type Inequalities for Log-Convex Functions

43

Fix s 2 JV and apply property (P) to get that ˝

˛ ˝ ˛ g0 .A/Ax; x  s g 0 .A/x; x  hg.A/ ln g.A/x; xi  hg.A/x; xi ln g.s/ 

g 0 .s/ .hAg.A/x; xi  s hg.A/x; xi/ g.s/

(2.91)

for any x 2 H with kxk D 1; which is an inequality of interest in itself as well. Since hg.A/Ax; xi 2 Œm; M  for any x 2 H with kxk D 1 hg.A/x; xi then on choosing s WD

hg.A/Ax;xi hg.A/x;xi

in (2.91) we get

˛ hg.A/Ax; xi ˝ 0 ˛ ˝ 0 g .A/x; x g .A/Ax; x  hg.A/x; xi  hg.A/ ln g.A/x; xi  hg.A/x; xi ln g



hg.A/Ax; xi hg.A/x; xi

  0;

which, by division with hg.A/x; xi > 0; produces hg 0 .A/Ax; xi hg.A/Ax; xi hg 0 .A/x; xi   hg.A/x; xi hg.A/x; xi hg.A/x; xi   hg.A/ ln g.A/x; xi hg.A/Ax; xi  ln g 0  hg.A/x; xi hg.A/x; xi

(2.92)

for any x 2 H with kxk D 1: Taking the exponential in (2.92) we deduce the desired inequality (2.87). 0 .A/Ax;xi Now, assuming that condition (C) holds, then by choosing s WD hg hg 0 .A/x;xi in (2.91) we get  0  hg .A/Ax; xi 0  hg.A/ ln g.A/x; xi  hg.A/x; xi ln g hg 0 .A/x; xi  0  .A/Ax;xi   g0 hg hg 0 .A/x;xi hg 0 .A/Ax; xi  hAg.A/x; xi  xi   0 hg.A/x; hg 0 .A/x; xi g hg .A/Ax;xi hg 0 .A/x;xi

which, by dividing with hg.A/x; xi > 0 and rearranging, is equivalent with g0 g

 

hg 0 .A/Ax;xi hg 0 .A/x;xi

hg 0 .A/Ax;xi hg 0 .A/x;xi







hg 0 .A/Ax; xi hAg.A/x; xi  hg 0 .A/x; xi hg.A/x; xi



44

2 Inequalities of the Jensen Type



hg0 .A/Ax; xi  ln g hg 0 .A/x; xi

 

hg.A/ ln g.A/x; xi 0 hg.A/x; xi

(2.93)

for any x 2 H with kxk D 1: Finally, on taking the exponential in (2.93) we deduce the desired inequality (2.88). t u Remark 2.36. We observe that a sufficient condition for (C) to hold is that either g 0 .A/ or g 0 .A/ is a positive definite operator on H: Corollary 2.37 (Dragomir, 2010, [15]). Assume that A and g are as in Theorem 2.35. If condition (C) holds, then we have the double inequality   0 hg.A/ ln g.A/x; xi hg .A/Ax; xi  ln g hg 0 .A/x; xi hg.A/x; xi   hg.A/Ax; xi (2.94)  ln g hg.A/x; xi for any x 2 H with kxk D 1: The following result providing different inequalities also holds: Theorem 2.38 (Dragomir, 2010, [15]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with m < M: If g W J ! .0; 1/ is a differentiable log-convex function with the derivative continuous on JV and Œm; M   JV , then



  hg.A/Ax; xi exp g0 .A/ A  1H x; x hg.A/x; xi 1g.A/ 0 + * g.A/ A @   x; x  g hg.A/Ax;xi hg.A/x;xi 2  hg.A/Ax;xi  3 * +  g 0 hg.A/x;xi  xi hg.A/Ax;  Ag.A/  g.A/ 5 x; x  1 (2.95)  exp 4  hg.A/x; xi g hg.A/Ax;xi hg.A/x;xi

for each x 2 H with kxk D 1: If condition (C) from Theorem 2.35 holds, then 2  hg0 .A/Ax;xi  3 * +  g 0 hg0 .A/x;xi  hg 0 .A/Ax; xi  exp 4  0 g.A/  Ag.A/ 5 x; x hg 0 .A/x; xi g hg .A/Ax;xi hg 0 .A/x;xi

+ *   g.A/ hg 0 .A/Ax; xi 1 Œg.A/ x; x  g hg 0 .A/x; xi

2.6 Some Jensen’s Type Inequalities for Log-Convex Functions







hg 0 .A/Ax; xi 1H  A  exp g .A/ hg 0 .A/x; xi 0

45

 x; x  1

(2.96)

for each x 2 H with kxk D 1: Proof. By taking the exponential in (2.90) we have the following inequality: 

 0  0 g.t/ g.t / g .s/ .tg.t/  sg.t// (2.97)  exp exp g .t/.t  s/  g.s/ g.s/ for any t; s 2 JV : If we fix s 2 JV and apply property (P) to inequality (2.97), we deduce * +  ˝  0 ˛ g.A/ g.A/ exp g .A/ .A  s1H / x; x  x; x g.s/



0 g .s/ .Ag.A/  sg.A// x; x  exp g.s/

(2.98)

for each x 2 H with kxk D 1; where 1H is the identity operator on H: By Mond–Peˇcari´c’s inequality applied for the convex function exp, we also have

0

 g .s/ exp .Ag.A/  sg.A// x; x g.s/   0 g .s/ .hAg.A/x; xi  s hg.A/x; xi/ (2.99)  exp g.s/ for each s 2 JV and x 2 H with kxk D 1: Now, if we choose s WD hg.A/Ax;xi 2 Œm; M  in (2.98) and (2.99) we deduce the hg.A/x;xi desired result (2.95). Observe that, inequality (2.97) is equivalent with 0

   g .s/ g.s/ g.t / exp .sg.t/  tg.t//   exp g 0 .t/ .s  t/ (2.100) g.s/ g.t/ for any t; s 2 JV : If we fix s 2 JV and apply property (P) to inequality (2.100) we deduce

0

 g .s/ exp .sg.A/  Ag.A// x; x g.s/ 

 g.A/ 1 x; x  g.s/ Œg.A/ ˛ ˝   exp g 0 .A/ .s1H  A/ x; x for each x 2 H with kxk D 1:

(2.101)

46

2 Inequalities of the Jensen Type

By Mond–Peˇcari´c’s inequality we also have ˝

 ˛  ˝ ˛ ˝ ˛ exp g0 .A/ .s1H  A/ x; x  exp s g 0 .A/x; x  g0 .A/Ax; x

(2.102)

for each s 2 JV and x 2 H with kxk D 1: Taking into account that condition (C) is valid, then we can choose in (2.101) 0 .A/Ax;xi and (2.102) s WD hg t u hg 0 .A/x;xi to get the desired result (2.96).

2.6.4 A Reverse Inequality The following reverse inequality is also of interest: Theorem 2.39 (Dragomir, 2010, [15]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with m < M: If g W J ! .0; 1/ is a differentiable log-convex function with the derivative continuous on JV and Œm; M   JV , then M hAx;xi

hAx;xim

Œg.m/ M m Œg.M / M m .1 / exp hln g.A/x; xi  

h.M1H  A/ .A  m1H / x; xi g 0 .M / g 0 .m/   exp M m g.M / g.m/  0 

0 .M  hAx; xi/ .hAx; xi  m/ g .M / g .m/   exp M m g.M / g.m/  0 

g .M / g 0 .m/ 1 .M  m/  (2.103)  exp 4 g.M / g.m/ for each x 2 H with kxk D 1: Proof. Utilizing inequality (2.89) we have successively ln g ..1  / t C s/  ln g.s/  .1  /

g 0 .s/ .t  s/ g.s/

and ln g ..1  / t C s/  ln g.t/  

g0 .t/ .t  s/ g.t/

(2.104)

(2.105)

for any t; s 2J˚ and any  2 Œ0; 1 : Now, if we multiply (2.104) by  and (2.105) by 1   and sum the obtained inequalities, we deduce

2.7 Hermite–Hadamard’s Type Inequalities

47

.1  / ln g.t/ C  ln g.s/  ln g ..1  / t C s/  0 

g .t/ g 0 .s/  .1  /   .t  s/ g.t/ g.s/

(2.106)

for any t; s 2J˚ and any  2 Œ0; 1 : M u Now, if we choose  WD M ; s WD m and t WD M in (2.106) then we get the m inequality M u um ln g.M / C ln g.m/  ln g .u/ M m M m  

.M  u/ .u  m/ g0 .M / g 0 .m/   M m g.M / g.m/

(2.107)

for any u 2 Œm; M : If we use property (P) for the operator A we get M  hAx; xi hAx; xi  m ln g.M / C ln g.m/  hln g.A/x; xi M m M m  

h.M1H  A/ .A  m1H / x; xi g 0 .M / g 0 .m/   M m g.M / g.m/

(2.108)

for each x 2 H with kxk D 1: Taking the exponential in (2.108) we deduce the first inequality in (2.103). Now, consider the function h W Œm; M  ! R, h.t/ D .M  t/ .t  m/ : This function is concave in Œm; M  and by Mond–Peˇcari´c’s inequality we have h.M1H  A/ .A  m1H / x; xi  .M  hAx; xi/ .hAx; xi  m/ for each x 2 H with kxk D 1; which proves the second inequality in (2.103). For the last inequality, we observe that .M  hAx; xi/ .hAx; xi  m/ 

1 .M  m/2 ; 4

and the proof is complete.

t u

2.7 Hermite–Hadamard’s Type Inequalities 2.7.1 Scalar Case If f W I ! R is a convex function on the interval I; then for any a; b 2 I with a ¤ b we have the following double inequality:

48

2 Inequalities of the Jensen Type

 f

aCb 2

 

1 ba

Z a

b

f .t/dt 

f .a/ C f .b/ : 2

(HH)

This remarkable result is well known in the literature as the Hermite–Hadamard inequality [24]. For various generalizations, extensions, reverses and related inequalities, see [1, 2, 18, 20–24] the monograph [17] and the references therein.

2.7.2 Some Inequalities for Convex Functions The following inequality related to the Mond–Peˇcari´c result also holds: Theorem 2.40 (Dragomir, 2010, [13]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with m < M: If f is a convex function on Œm; M ; then

 f .m/ C f .M / f .A/ C f ..m C M / 1H  A/  x; x 2 2 f .hAx; xi/ C f .m C M  hAx; xi/ 2   mCM f 2



for each x 2 H with kxk D 1: In addition, if x 2 H with kxk D 1 and hAx; xi ¤

mCM 2 ;

(2.109)

then also

f .hAx; xi/ C f .m C M  hAx; xi/ 2   Z mCM hAx;xi 2 mCM : f .u/ du  f  mCM 2  hAx; xi hAx;xi 2

(2.110)

Proof. Since f is convex on Œm; M  then for each u 2 Œm; M  we have the inequalities M u um f .m/ C f .M / M m M m   um M u mC M D f .u/ f M m M m

(2.111)

2.7 Hermite–Hadamard’s Type Inequalities

49

and um M u f .M / C f .m/  f M m M m



um M u MC m M m M m

D f .M C m  u/ :



(2.112)

If we add these two inequalities we get f .m/ C f .M /  f .u/ C f .M C m  u/ for any u 2 Œm; M ; which, by property (P) applied for the operator A; produces the first inequality in (2.109). By the Mond–Peˇcari´c inequality we have hf ..m C M / 1H  A/ x; xi  f .m C M  hAx; xi/; which together with the same inequality produces the second inequality in (2.109). The third part follows by the convexity of f: In order to prove (2.110), we use the Hermite–Hadamard inequality (HH) for the convex functions f and the choices a D hAx; xi and b D m C M  hAx; xi: The proof is complete. t u Remark 2.41. We observe that, from inequality (2.109) we have the following inequality in the operator order of B .H /:

f .m/ C f .M / f .A/ C f ..m C M / 1H  A/ 1H  2 2   mCM 1H ; f 2

(2.113)

where f is a convex function on Œm; M  and A a selfadjoint operator on the Hilbert space H with Sp.A/  Œm; M  for some scalars m; M with m < M: The case of log-convex functions may be of interest for applications and therefore is stated in: Corollary 2.42 (Dragomir, 2010, [13]). If g is a log-convex function on Œm; M ; then D E p g.m/g.M /  exp ln Œg.A/g ..m C M / 1H  A/1=2 x; x p  g .hAx; xi/ g .m C M  hAx; xi/   mCM g (2.114) 2 for each x 2 H with kxk D 1:

50

2 Inequalities of the Jensen Type

In addition, if x 2 H with kxk D 1 and hAx; xi ¤ p

mCM 2 ;

then also

g .hAx; xi/ g .m C M  hAx; xi/ " # Z mCM hAx;xi 2  exp mCM ln g .u/ du  hAx; xi hAx;xi 2   mCM : g 2

(2.115)

The following result also holds: Theorem 2.43 (Dragomir, 2010, [13]). Let A and B selfadjoint operators on the Hilbert space H and assume that Sp.A/; Sp.B/  Œm; M  for some scalars m; M with m < M: If f is a convex function on Œm; M ; then  f

ACB x; x 2



1 Œf ..1  t/ hAx; xi C t hBx; xi/ C f .t hAx; xi C .1  t/ hBx; xi/ 2 

1 Œf ..1  t/ A C tB/ C f .tA C .1  t/ B/ x; x  2 ˝ ˝ ACB ˛ ˛ M  ACB x; x x; x  m 2 2  f .m/ C f .M / (2.116) M m M m 

for any t 2 Œ0; 1 and each x 2 H with kxk D 1: Moreover, we have the Hermite–Hadamard’s type inequalities: 

 ACB f x; x 2 Z 1  f ..1  t/ hAx; xi C t hBx; xi/ dt 0

Z

1



 f ..1  t/ A C tB/ dt x; x

0

˝ ACB ˝ ˛ ˛ M  ACB x; x x; x  m 2 2 f .m/ C f .M /  M m M m each x 2 H with kxk D 1:

(2.117)

2.7 Hermite–Hadamard’s Type Inequalities

51

In addition, if we assume that B  A is a positive definite operator, then  ACB x; x h.B  A/ x; xi f 2

Z Z hBx;xi  f .u/ du  h.B  A/ x; xi 

hAx;xi

"

˝ ACB

˛

1

 f ..1  t/ A C tB/ dt x; x

0

M  2 x; x f .m/ C  h.B  A/ x; xi M m

# ˛ x; x  m 2 f .M / : (2.118) M m

˝ ACB

Proof. It is obvious that for any t 2 Œ0; 1 we have Sp ..1  t/ A C tB/; Sp .tA C .1  t/ B/  Œm; M : On making use of the Mond–Peˇcari´c inequality we have f ..1  t/ hAx; xi C t hBx; xi/  hf ..1  t/ A C tB/ x; xi

(2.119)

f .t hAx; xi C .1  t/ hBx; xi/  hf .tA C .1  t/ B/ x; xi

(2.120)

and

for any t 2 Œ0; 1 and each x 2 H with kxk D 1: Adding (2.119) with (2.120) and utilizing the convexity of f we deduce the first two inequalities in (2.116). By the Lah–Ribari´c inequality (2.69) we also have hf ..1  t/ A C tB/ x; xi 

M  .1  t/ hAx; xi  t hBx; xi  f .m/ M m .1  t/ hAx; xi C t hBx; xi  m  f .M / (2.121) C M m

and hf .tA C .1  t/ B/ x; xi 

M  t hAx; xi  .1  t/ hBx; xi  f .m/ M m t hAx; xi C .1  t/ hBx; xi  m  f .M / (2.122) C M m

for any t 2 Œ0; 1 and each x 2 H with kxk D 1: Now, if we add inequalities (2.121) with (2.122) and divide by two, we deduce the last part in (2.116).

52

2 Inequalities of the Jensen Type

Integrating the inequality over t 2 Œ0; 1, utilizing the continuity property of the inner product and the properties of the integral of operator-valued functions we have  f

ACB x; x 2



1  2

Z

1

f ..1  t/ hAx; xi C t hBx; xi/ dt

0

Z

1

C

f .t hAx; xi C .1  t/ hBx; xi/ dt

0

Z 1 1  f ..1  t/ A C tB/ dt 2 0

 Z 1 f .tA C .1  t/ B/ dt x; x C 0

˝ ACB ˛ ˛ M  2 x; x 2 x; x  m f .m/ C f .M /: (2.123)  M m M m ˝ ACB

Since Z

Z

1

1

f ..1  t/ hAx; xi C t hBx; xi/ dt D

f .t hAx; xi C .1  t/ hBx; xi/ dt

0

0

and

Z

Z

1

f ..1  t/ A C tB/ dt D 0

1

f .tA C .1  t/ B/ dt 0

then, by (2.123), we deduce inequality (2.117). Inequality (2.118) follows from (2.117) by observing that for hBx; xi > hAx; xi we have Z

1 0

1 f ..1  t/ hAx; xi C t hBx; xi/ dt D hBx; xi  hAx; xi

Z

hBx;xi

f .u/ du hAx;xi

for each x 2 H with kxk D 1:



Remark 2.44. We observe that, from inequalities (2.116) and (2.117) we have the following inequalities in the operator order of B .H /: 1 Œf ..1  t/ A C tB/ C f .tA C .1  t/ B/ 2  f .m/

ACB M1H  ACB  m1H 2 C f .M / 2 ; M m M m

(2.124)

where f is a convex function on Œm; M  and A; B are selfadjoint operator on the Hilbert space H with Sp.A/; Sp.B/  Œm; M  for some scalars m; M with m < M:

2.7 Hermite–Hadamard’s Type Inequalities

53

The case of log-convex functions is as follows: Corollary 2.45 (Dragomir, 2010, [13]). If g is a log-convex function on Œm; M ; then   ACB g x; x 2 p  g ..1  t/ hAx; xi C t hBx; xi/ g .t hAx; xi C .1  t/ hBx; xi/

 1  exp Œln g ..1  t/ A C tB/ C ln g .tA C .1  t/ B/ x; x 2  g.m/

h ACB 2 x;x i

M

M m

g.M /

h ACB 2 x;x im

(2.125)

M m

for any t 2 Œ0; 1 and each x 2 H with kxk D 1: Moreover, we have the Hermite–Hadamard’s type inequalities:  g

 ACB x; x 2 Z 1

 exp ln g ..1  t/ hAx; xi C t hBx; xi/ dt 0

Z

1

 exp

 ln g ..1  t/ A C tB/ dt x; x

0

 g.m/

h ACB 2 x;x i

M

M m

g.M /

h ACB 2 x;x im M m

(2.126)

for each x 2 H with kxk D 1: In addition, if we assume that B  A is a positive definite operator, then 

h.BA/x;xi ACB g x; x 2 "Z # hBx;xi  exp ln g .u/ du hAx;xi



Z

1

 exp h.B  A/ x; xi "  g.m/



ln g ..1  t/ A C tB/ dt x; x

0

h ACB 2 x;x i

M

M m

for each x 2 H with kxk D 1:

g.M /

h ACB 2 x;x im M m

#h.BA/x;xi (2.127)

54

2 Inequalities of the Jensen Type

From a different perspective, we have the following result as well: Theorem 2.46 (Dragomir, 2010, [13]). Let A and B selfadjoint operators on the Hilbert space H and assume that Sp.A/; Sp.B/  Œm; M  for some scalars m; M with m < M: If f is a convex function on Œm; M ; then  f

 hAx; xi C hBy; yi 2 Z 1  f ..1  t/ hAx; xi C t hBy; yi/ dt 0

Z

1



 f ..1  t/ A C t hBy; yi 1H / dt x; x

0

1 Œhf .A/x; xi C f .hBy; yi/ 2 1  Œhf .A/x; xi C hf .B/y; yi 2 

(2.128)

and  f

hAx; xi C hBy; yi 2



   A C hBy; yi 1H  f x; x 2

Z 1

  f ..1  t/ A C t hBy; yi 1H / dt x; x (2.129) 0

for each x; y 2 H with kxk D kyk D 1: Proof. For a convex function f and any u; v 2 Œm; M  and t 2 Œ0; 1, we have the double inequality:  f

uCv 2

 

1 Œf ..1  t/ u C tv/ C f .tu C .1  t/ v/ 2



1 Œf .u/ C f .v/ : 2

(2.130)

Utilizing the second inequality in (2.130) we have 1 f ..1  t/ u C t hBy; yi/ C f .tu C .1  t/ hBy; yi/ 2 1  Œf .u/ C f .hBy; yi/ 2 for any u 2 Œm; M , t 2 Œ0; 1 and y 2 H with kyk D 1:

(2.131)

2.7 Hermite–Hadamard’s Type Inequalities

55

Now, on applying property (P) to inequality (2.131) for the operator A we have 1 Œhf ..1  t/ A C t hBy; yi/ x; xi C hf .tA C .1  t/ hBy; yi/ x; xi 2 1  Œhf .A/x; xi C f .hBy; yi/ (2.132) 2 for any t 2 Œ0; 1 and x; y 2 H with kxk D kyk D 1: On applying the Mond–Peˇcari´c inequality we also have 1 f ..1  t/ hAx; xi C t hBy; yi/ C f .t hAx; xi C .1  t/ hBy; yi/ 2 1  Œhf ..1  t/ A C t hBy; yi 1H / x; xi C hf .tA C .1  t/ hBy; yi 1H / x; xi 2 (2.133) for any t 2 Œ0; 1 and x; y 2 H with kxk D kyk D 1: Now, integrating over t on Œ0; 1 inequalities (2.132) and (2.133) and taking into account that Z 1 hf ..1  t/ A C t hBy; yi 1H / x; xi dt 0

Z

1

hf .tA C .1  t/ hBy; yi 1H / x; xi dt

D 0

Z



1



f ..1  t/ A C t hBy; yi 1H / dt x; x

D 0

and Z

Z

1

1

f ..1  t/ hAx; xi C t hBy; yi/ dt D 0

f .t hAx; xi C .1  t/ hBy; yi/ dt; 0

we obtain the second and the third inequality in (2.128). Further, on applying the Jensen integral inequality for the convex function f we also have Z 1 f ..1  t/ hAx; xi C t hBy; yi/ dt 0

Z

1

f  Df

 Œ.1  t/ hAx; xi C t hBy; yi dt

0

hAx; xi C hBy; yi 2



for each x; y 2 H with kxk D kyk D 1, proving the first part of (2.128).

56

2 Inequalities of the Jensen Type

Now, on utilizing the first part of (2.130) we can also state that  f

u C hBy; yi 2





1 Œf ..1  t/ u C t hBy; yi/ C f .tu C .1  t/ hBy; yi/ 2

(2.134)

for any u 2 Œm; M , t 2 Œ0; 1 and y 2 H with kyk D 1: Further, on applying property (P) to inequality (2.134) and for the operator A we get

   A C hBy; yi 1H f x; x 2 

1 Œhf ..1  t/ A C t hBy; yi 1H / x; xi C hf .tA C .1  t/ hBy; yi 1H / x; xi 2

for each x; y 2 H with kxk D kyk D 1; which, by integration over t in Œ0; 1 produces the second inequality in (2.129). The first inequality is obvious. u t Remark 2.47. It is important to remark that, from inequalities (2.128) and (2.129) we have the following Hermite–Hadamard’s type results in the operator order of B .H / and for the convex function f W Œm; M  ! R:   Z 1 A C hBy; yi 1H f  f ..1  t/ A C t hBy; yi 1H / dt 2 0 

1 Œf .A/ C f .hBy; yi/ 1H  2

(2.135)

for any y 2 H with kyk D 1 and any selfadjoint operators A; B with spectra in Œm; M : In particular, we have from (2.135)  f

A C hAy; yi 1H 2



Z

1



f ..1  t/ A C t hAy; yi 1H / dt 0



1 Œf .A/ C f .hAy; yi/ 1H  2

(2.136)

for any y 2 H with kyk D 1 and   Z 1 A C s1H f  f ..1  t/ A C ts1H / dt 2 0  for any s 2 Œm; M :

1 Œf .A/ C f .s/1H  2

(2.137)

2.7 Hermite–Hadamard’s Type Inequalities

57

As a particular case of the above theorem, we have the following refinement of the Mond–Peˇcari´c inequality: Corollary 2.48 (Dragomir, 2010, [13]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp.A/  Œm; M  for some scalars m; M with m < M: If f is a convex function on Œm; M ; then

   A C hAx; xi 1H f .hAx; xi/  f x; x 2

Z 1

  f ..1  t/ A C t hAx; xi 1H / dt x; x 0

1  Œhf .A/x; xi C f .hAx; xi/  hf .A/x; xi : 2

(2.138)

Finally, the case of log-convex functions is as follows: Corollary 2.49 (Dragomir, 2010, [13]). If g is a log-convex function on Œm; M ; then 

 hAx; xi C hBy; yi g 2 Z 1

 exp ln g ..1  t/ hAx; xi C t hBy; yi/ dt 0

Z



1



ln g ..1  t/ A C t hBy; yi 1H / dt x; x

 exp 0

1 Œhln g.A/x; xi C ln g .hBy; yi/  exp 2

1 Œhln g.A/x; xi C hln g.B/y; yi  exp 2

(2.139)

and 

hAx; xi C hBy; yi g 2





  A C hBy; yi 1H  exp ln g x; x 2

Z 1

  exp ln g ..1  t/ A C t hBy; yi 1H / dt x; x 0

(2.140)

58

2 Inequalities of the Jensen Type

and   A C hAx; xi 1H x; x g .hAx; xi/  exp ln g 2

Z 1

  exp ln g ..1  t/ A C t hAx; xi 1H / dt x; x



 exp

0

1 Œhln g.A/x; xi C ln g .hAx; xi/ 2

 exp hln g.A/x; xi

(2.141)

respectively, for each x 2 H with kxk D 1 and A; B selfadjoint operators with spectra in Œm; M : It is obvious that all the above inequalities can be applied for particular convex or log-convex functions of interest. The details are left to the interested reader.

2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions 2.8.1 Introduction The following inequality holds for any convex function f defined on R:  .b  a/f

aCb 2



Z

b

f .x/dx < .b  a/

< a

f .a/ C f .b/ ; 2

a; b 2 R: (2.142)

It was first discovered by Ch. Hermite in 1881 in the journal Mathesis (see [24]). But this result was nowhere mentioned in the mathematical literature and was not widely known as Hermite’s result [27]. E.F. Beckenbach, a leading expert on the history and the theory of convex functions, wrote that this inequality was proven by J. Hadamard in 1893 [3]. In 1974, D.S. Mitrinovi´c found Hermite’s note in Mathesis [24]. Since (2.142) was known as Hadamard’s inequality, the inequality is now commonly referred as the Hermite–Hadamard inequality [27]. Let X be a vector space, x; y 2 X; x ¤ y. Define the segment Œx; y WD f.1  t/x C ty; t 2 Œ0; 1g:

2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions

59

We consider the function f W Œx; y ! R and the associated function g.x; y/ W Œ0; 1 ! R; g.x; y/.t/ WD f Œ.1  t/x C ty; t 2 Œ0; 1: Note that f is convex on Œx; y if and only if g.x; y/ is convex on Œ0; 1. For any convex function defined on a segment Œx:y  X , we have the Hermite– Hadamard integral inequality (see [4, p. 2])  f

xCy 2



Z

1

 0

f Œ.1  t/x C tydt 

f .x/ C f .y/ ; 2

(2.143)

which can be derived from the classical Hermite–Hadamard inequality (2.142) for the convex function g.x; y/ W Œ0; 1 ! R. Since f .x/ D kxkp (x 2 X and 1  p < 1) is a convex function, we have the following norm inequality from (2.143) (see [26, p. 106]):   Z 1  x C y p kxkp C kykp p    k.1  t/x C tyk dt   2  2 0

(2.144)

for any x; y 2 X . Motivated by the above results, we investigate in this section the operator version of the Hermite–Hadamard inequality for operator convex functions. The operator quasi-linearity of some associated functionals are also provided. A real-valued continuous function f on an interval I is said to be operator convex (operator concave) if f ..1  / A C B/  ./ .1  / f .A/ C f .B/

(OC)

in the operator order, for all  2 Œ0; 1 and for every selfadjoint operator A and B on a Hilbert space H whose spectra are contained in I: Notice that a function f is operator concave if f is operator convex. A real-valued continuous function f on an interval I is said to be operator monotone if it is monotone with respect to the operator order, i.e. A  B with Sp.A/; Sp.B/  I imply f .A/  f .B/: For some fundamental results on operator convex (operator concave) and operator monotone functions, see [19] and the references therein. As examples of such functions, we note that f .t/ D t r is operator monotone on Œ0; 1/ if and only if 0  r  1: The function f .t/ D t r is operator convex on .0; 1/ if either 1  r  2 or 1  r  0 and is operator concave on .0; 1/ if 0  r  1: The logarithmic function f .t/ D ln t is operator monotone and operator concave on .0; 1/: The entropy function f .t/ D t ln t is operator concave on .0; 1/: The exponential function f .t/ D et is neither operator convex nor operator monotone.

60

2 Inequalities of the Jensen Type

2.8.2 Some Hermite–Hadamard’s Type Inequalities We start with the following result: Theorem 2.50 (Dragomir, 2010, [12]). Let f W I ! R be an operator convex function on the interval I: Then for any selfadjoint operators A and B with spectra in I we have the inequality        

ACB 1 3A C B A C 3B f  f Cf 2 2 4 4 Z 1  f ..1  t/A C tB/dt 0

 

  f .A/ C f .B/ f .A/ C f .B/ 1 ACB f C  :  2 2 2 2

(2.145)

Proof. First of all, since the function f is continuous, the operator-valued integral R1 0 f ..1 t/ACtB/dt exists for any selfadjoint operators A and B with spectra in I: We give here two proofs, the first using only the definition of operator convex functions and the second using the classical Hermite–Hadamard inequality for realvalued functions. 1. By the definition of operator convex functions we have the double inequality:  f

C CD 2

 

1 Œf ..1  t/ C C tD/ C f ..1  t/ D C tC / 2



1 Œf .C / C f .D/ 2

(2.146)

for any t 2 Œ0; 1 and any selfadjoint operators C and D with the spectra in I: Integrating inequality (2.146) over t 2 Œ0; 1 and taking into account that Z

1

Z

1

f ..1  t/ C C tD/ dt D

0

f ..1  t/ D C tC / dt

0

then we deduce the Hermite–Hadamard inequality for operator convex functions  f

C CD 2



Z

1



f ..1  t/ C C tD/ dt  0

1 Œf .C / C f .D/ 2

that holds for any selfadjoint operators C and D with the spectra in I:

(HHO)

2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions

61

Now, on making use of the change of variable u D 2t we have Z

1=2

f ..1  t/A C tB/dt D 0

1 2

Z

1

f 0

  ACB du .1  u/ A C u 2

and by the change of variable u D 2t  1 we have Z

1

1 f ..1  t/A C tB/dt D 2 1=2

Z

1 0

  ACB C uB du: f .1  u/ 2

Utilizing the Hermite–Hadamard inequality (HHO) we can write  f

3A C B 4



  ACB du .1  u/ A C u 2 0  

1 ACB f .A/ C f  2 2 Z

1



f

and  f

A C 3B 4



  ACB C uB du .1  u/ 2 0  

1 ACB f .A/ C f ;  2 2 Z



1

f

which by summation and division by two produces the desired result (2.145). 2. Consider now x 2 H; kxk D 1 and two selfadjoint operators A and B with spectra in I . Define the real-valued function ' x;A;B W Œ0; 1 ! R given by ' x;A;B .t/ D hf ..1  t/A C tB/x; xi: Since f is operator convex, then for any t1 ; t2 2 Œ0; 1 and ˛; ˇ  0 with ˛ C ˇ D 1 we have ' x;A;B .˛t1 C ˇt2 / D hf ..1  .˛t1 C ˇt2 // A C .˛t1 C ˇt2 / B/ x; xi D hf .˛ Œ.1  t1 / A C t1 B C ˇ Œ.1  t2 / A C t2 B/ x; xi  ˛ hf .Œ.1  t1 / A C t1 B/ x; xi C ˇ hf .Œ.1  t2 / A C t2 B/ x; xi D ˛' x;A;B .t1 / C ˇ' x;A;B .t2 / showing that ' x;A;B is a convex function on Œ0; 1 :

62

2 Inequalities of the Jensen Type

Now we use the Hermite–Hadamard inequality for real-valued convex functions   Z b 1 g.a/ C g.b/ aCb g  g.s/ds  2 ba a 2 to get that ' x;A;B and

    Z 1=2 ' x;A;B .0/ C ' x;A;B 12 1 2 ' x;A;B .t/dt  4 2 0

    Z 1 ' x;A;B 12 C ' x;A;B .1/ 3 2 ' x;A;B .t/dt  ' x;A;B 4 2 1=2

which by summation and division by two produces 1 2

   

 3A C B A C 3B f Cf x; x 4 4 Z 1  hf ..1  t/A C tB/x; xi dt 0

1 2



 

 f .A/ C f .B/ ACB f C x; x : 2 2

(2.147)

Finally, since by the continuity of the function f we have Z

Z

1 0



1

f ..1  t/A C tB/dtx; x

hf ..1  t/A C tB/x; xi dt D 0

for any x 2 H; kxk D 1 and any two selfadjoint operators A and B with spectra in I; we deduce from (2.147) the desired result (2.145). t u A simple consequence of the above theorem is that the integral is closer to the left bound than to the right, namely we can state: Corollary 2.51 (Dragomir, 2010, [12]). With the assumptions in Theorem 2.50 we have the inequality Z



1

.0 /

f ..1  t/A C tB/dt  f 0



f .A/ C f .B/  2

Z

ACB 2



1

f ..1  t/A C tB/dt:

(2.148)

0

Remark 2.52. Utilizing different examples of operator convex or concave functions, we can provide inequalities of interest.

2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions

63

If r 2 Œ1; 0 [ Œ1; 2 then we have the inequalities for powers of operators     

 3A C B r ACB r 1 A C 3B r  C 2 2 4 4 Z 1  ..1  t/A C tB/r dt



0

1  2



ACB 2

r C

Ar C B r 2

  Ar C B r  2

(2.149)

for any two selfadjoint operators A and B with spectra in .0; 1/ : If r 2 .0; 1/ the inequalities in (2.149) hold with “  ” instead of “  ”: We also have the following inequalities for logarithm:        

ACB 1 3A C B A C 3B ln  ln C ln 2 2 4 4 Z 1  ln..1  t/A C tB/dt 0



 

  1 ln.A/ C ln.B/ ln.A/ C ln.B/ ACB ln C  2 2 2 2

(2.150)

for any two selfadjoint operators A and B with spectra in .0; 1/:

2.8.3 Some Operator Quasi-linearity Properties Consider an operator convex function f W I  R ! R defined on the interval I and two distinct selfadjoint operators A; B with the spectra in I . We denote by ŒA; B the closed operator segment defined by the family of operators f.1  t/A C tB, t 2 Œ0; 1g : We also define the operator-valued functional f .A; BI t/ WD .1  t/f .A/ C tf .B/  f ..1  t/A C tB/  0

(2.151)

in the operator order, for any t 2 Œ0; 1 : The following result concerning an operator quasi-linearity property for the functional f .; I t/ may be stated: Theorem 2.53 (Dragomir, 2010, [12]). Let f W I  R ! R be an operator convex function on the interval I . Then for each A; B two distinct selfadjoint operators with the spectra in I and C 2 ŒA; B we have .0 / f .A; C I t/ C f .C; BI t/  f .A; BI t/

(2.152)

64

2 Inequalities of the Jensen Type

for each t 2 Œ0; 1 ; i.e. the functional f .; I t/ is operator super-additive as a function of interval. If ŒC; D  ŒA; B; then .0 / f .C; DI t/  f .A; BI t/

(2.153)

for each t 2 Œ0; 1 ; i.e. the functional f .; I t/ is operator non-decreasing as a function of interval. Proof. Let C D .1  s/ A C sB with s 2 .0; 1/ : For t 2 .0; 1/ we have f .C; BI t/ D .1  t/f ..1  s/ A C sB/ C tf .B/  f ..1  t/ Œ.1  s/ A C sB C tB/ and f .A; C I t/ D .1  t/f .A/ C tf ..1  s/ A C sB/  f ..1  t/A C t Œ.1  s/ A C sB/ giving that f .A; C I t/ C f .C; BI t/  f .A; BI t/ D f ..1  s/ A C sB/ C f ..1  t/A C tB/  f ..1  t/ .1  s/ A C Œ.1  t/s C t B/  f ..1  ts/ A C tsB/ : (2.154) Now, for a convex function ' W I  R ! R, where I is an interval, and any real numbers t1 ; t2 ; s1 and s2 from I and with the properties that t1  s1 and t2  s2 we have that ' .s1 /  ' .s2 / ' .t1 /  ' .t2 /  : t1  t2 s1  s2 Indeed, since ' is convex on I then for any a 2 I the function .t/ WD

(2.155) W I n fag ! R

'.t/  '.a/ t a

is monotonic non-decreasing where is defined. Utilizing this property repeatedly we have ' .t1 /  ' .t2 / ' .s1 /  ' .t2 / ' .t2 /  ' .s1 /  D t1  t2 s1  t2 t2  s1  which proves inequality (2.155).

' .s2 /  ' .s1 / ' .s1 /  ' .s2 / D ; s2  s1 s1  s2

2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions

65

For a vector x 2 H , with kxk D 1; consider the function ' x W Œ0; 1 ! R given by ' x .t/ WD hf ..1  t/A C tB/x; xi: Since f is operator convex on I it follows that ' x is convex on Œ0; 1 : Now, if we consider, for given t; s 2 .0; 1/; t1 WD ts < s DW s1 and t2 WD t < t C .1  t/s DW s2 ; then we have ' x .t1 / D hf ..1  ts/ A C tsB/ x; xi and ' x .t2 / D hf ..1  t/A C tB/x; xi giving that ' x .t1 /  ' x .t2 / D t1  t2



 f ..1  ts/ A C tsB/  f ..1  t/A C tB/ x; x : t .s  1/

Also ' x .s1 / D hf ..1  s/ A C sB/ x; xi and ' x .s2 / D hf ..1  t/ .1  s/ A C Œ.1  t/s C t B/ x; xi giving that ' x .s1 /  ' x .s2 / s1  s2 

f ..1  s/ A C sB/  f ..1  t/ .1  s/ A C Œ.1  t/s C t B/ x; x : D t .s  1/ Utilizing inequality (2.155) and multiplying with t .s  1/ < 0, we deduce the following inequality in the operator order: f ..1  ts/ A C tsB/  f ..1  t/A C tB/  f ..1  s/ A C sB/  f ..1  t/ .1  s/ A C Œ.1  t/s C t B/ :

(2.156)

Finally, by (2.154) and (2.156) we get the desired result (2.152). Applying repeatedly the superadditivity property we have for ŒC; D  ŒA; B that f .A; C I t/ C f .C; DI t/ C f .D; BI t/  f .A; BI t/ giving that 0  f .A; C I t/ C f .D; BI t/  f .A; BI t/  f .C; DI t/ which proves (2.153).

t u

66

2 Inequalities of the Jensen Type

For t D

1 2

we consider the functional

    f .A/ C f .B/ 1 ACB D f ; f .A; B/ WD f A; BI 2 2 2 which obviously inherits the superadditivity and monotonicity properties of the functional f .; I t/ : We are able then to state the following: Corollary 2.54 (Dragomir, 2010, [12]). Let f W I  R ! R be an operator convex function on the interval I . Then for each A; B two distinct selfadjoint operators with the spectra in I we have the following bounds in the operator order:   

   C CB ACB ACC Cf  f .C / D f f C 2ŒA;B 2 2 2 inf

(2.157)

and  

C CD f .C / C f .D/ f 2 2 C;D2ŒA;B   f .A/ C f .B/ ACB f : D 2 2

sup

(2.158)

Proof. By the superadditivity of the functional f .; / we have for each C 2 ŒA; B that  ACB 2     f .C / C f .B/ f .A/ C f .C / ACC C CB f C f  2 2 2 2

f .A/ C f .B/ f 2



which is equivalent with  f

ACC 2



 Cf

C CB 2



  f .C /  f

 ACB : 2

(2.159)

Since the equality case in (2.159) is realized for either C D A or C D B we get the desired bound (2.157). The bound (2.158) is obvious by the monotonicity of the functional f .; / as a function of interval. t u Consider now the following functional: f .A; BI t/ WD f .A/ C f .B/  f ..1  t/A C tB/  f ..1  t/B C tA/;

2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions

67

where, as above, f W I  R ! R is a convex function on the convex set I and A; B two distinct selfadjoint operators with the spectra in I while t 2 Œ0; 1 : We notice that f .A; BI t/ D f .B; AI t/ D f .A; BI 1  t/ and f .A; BI t/ D f .A; BI t/ C f .A; BI 1  t/  0 for any A; B and t 2 Œ0; 1 : Therefore, we can state the following result as well: Corollary 2.55 (Dragomir, 2010, [12]). Let f W I  R ! R be an operator convex function on the interval I . Then for each A; B two distinct selfadjoint operators with the spectra in I; the functional f .; I t/ is operator superadditive and operator non-decreasing as a function of interval. In particular, if C 2 ŒA; B then we have the inequality 1 Œf ..1  t/A C tB/ C f ..1  t/B C tA/ 2 1  Œf ..1  t/A C tC / C f ..1  t/C C tA/ 2 1 C Œf ..1  t/C C tB/ C f ..1  t/B C tC /  f .C /: 2

(2.160)

Also, if C; D 2 ŒA; B then we have the inequality f .A/ C f .B/  f ..1  t/A C tB/  f ..1  t/B C tA/  f .C / C f .D/  f ..1  t/C C tD/  f ..1  t/C C tD/

(2.161)

for any t 2 Œ0; 1 : Perhaps the most interesting functional we can consider is the following one: f .A/ C f .B/ Θf .A; B/ D  2

Z

1

f ..1  t/A C tB/dt:

(2.162)

0

Notice that, by the second Hermite–Hadamard inequality for operator convex functions we have that Θf .A; B/  0 in the operator order. We also observe that Z Θf .A; B/ D 0

1

Z

1

f .A; BI t/ dt D 0

f .A; BI 1  t/ dt:

(2.163)

68

2 Inequalities of the Jensen Type

Utilizing this representation, we can state the following result as well: Corollary 2.56 (Dragomir, 2010, [12]). Let f W I  R ! R be an operator convex function on the interval I . Then for each A; B two distinct selfadjoint operators with the spectra in I; the functional Θf .; / is operator superadditive and operator non-decreasing as a function of interval. Moreover, we have the bounds in the operator order Z

1

inf

C 2ŒA;B

Z

Œf ..1  t/A C tC / C f ..1  t/C C tB/ P dt  f .C /

0 1

f ..1  t/A C tB/dt

D

(2.164)

0

and

Z 1 f .C / C f .D/  f ..1  t/C C tD/dt 2 0 C;D2ŒA;B Z 1 f .A/ C f .B/  f ..1  t/A C tB/dt: D 2 0 sup

(2.165)

Remark 2.57. The above inequalities can be applied to various concrete operator convex function of interest. If we choose for instance inequality (2.165), then we get the following bounds in the operator order:

Z 1 C r C Dr  ..1  t/C C tD/r dt 2 0 C;D2ŒA;B Z 1 r r A CB  ..1  t/A C tB/r dt; D 2 0 sup

(2.166)

where r 2 Œ1; 0 [ Œ1; 2 and A; B are selfadjoint operators with spectra in .0; 1/ : If r 2 .0; 1/ then Z

1

..1  t/C C tD/r dt 

sup C;D2ŒA;B

Z

0

1

D

..1  t/A C tB/r dt  0

C r C Dr 2

Ar C B r 2

and A; B are selfadjoint operators with spectra in .0; 1/ :



(2.167)

References

69

We also have the operator bound for the logarithm Z

1

ln..1  t/C C tD/dt 

sup C;D2ŒA;B

Z

0

1

D

ln..1  t/A C tB/dt  0

ln.C / C ln.D/ 2

ln.A/ C ln.B/ ; 2



(2.168)

where A; B are selfadjoint operators with spectra in .0; 1/ :

References 1. G. Allasia, C. Giordano, J. Peˇcari´c, Hadamard-type inequalities for (2r)-convex functions with applications, Atti Acad. Sci. Torino-Cl. Sc. Fis., 133 (1999), 1–14. 2. A.G. Azpeitia, Convex functions and the Hadamard inequality, Rev.-Colombiana-Mat., 28(1) (1994), 7–12. 3. E.F. Beckenbach and R. Bellman, Inequalities, 4th Edition, Springer-Verlag, Berlin, 1983. 4. S.S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3 (2002), No.3, Article 35. 5. S.S. Dragomir, Discrete Inequalities of the Cauchy-Bunyakovsky-Schwarz Type, Nova Science Publishers, NY, 2004. 6. S.S. Dragomir, Gr¨uss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, Preprint, RGMIA Res. Rep. Coll., 11(e) (2008), Art. 11. 7. S.S. Dragomir, Some new Gr¨uss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, Sarajevo J. Math. 6(18)(2010), 89–107. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 12. 8. S.S. Dragomir, Some reverses of the Jensen inequality for functions of selfadjoint operators in Hilbert spaces, J. Ineq. & Appl. Vol 2010, Article ID 496821. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 15. 9. S.S. Dragomir, Some Slater’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces, Rev. Un. Mat. Argentina, 52(2011), No.1, 109–120. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 7. 10. S.S. Dragomir, Some inequalities for convex functions of selfadjoint operators in Hilbert spaces, Filomat 23(2009), No. 3, 81–92. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 10. 11. S.S. Dragomir, Some Jensen’s type inequalities for twice differentiable functions of selfadjoint operators in Hilbert spaces, Filomat 23(2009), No. 3, 211–222. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 13. 12. S.S. Dragomir, Hermite-Hadamard’s type inequalities for operator convex functions, Appl. Math. Comp. 218(2011), No. 3, 766–772. Preprint RGMIA Res. Rep. Coll., 13(2010), No. 1, Art. 7. 13. S.S. Dragomir, Hermite-Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., 13(2010), No. 2, Art 1. 14. S.S. Dragomir, Some Jensen’s type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces, Bull. Malays. Math. Sci. Soc. 34(2011), No. 3, 445–454. Preprint RGMIA Res. Rep. Coll., 13(2010), Sup. Art. 2. 15. S.S. Dragomir, New Jensen’s type inequalities for differentiable log-convex functions of selfadjoint operators in Hilbert spaces, Sarajevo J. Math. 19(2011), No. 1, 67–80. Preprint RGMIA Res. Rep. Coll., 13(2010), Sup. Art. 2.

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2 Inequalities of the Jensen Type

16. S.S. Dragomir and N.M. Ionescu, Some converse of Jensen’s inequality and applications. Rev. Anal. Num´er. Th´eor. Approx. 23 (1994), no. 1, 71–78. MR:1325895 (96c:26012). 17. S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Type Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. [http://rgmia.org/ monographs.php]. 18. A.M. Fink, Toward a theory of best possible inequalities, Nieuw Archief von Wiskunde, 12 (1994), 19–29. 19. T. Furuta, J. Mi´ci´c, J. Peˇcari´c and Y. Seo, Mond-Peˇcari´c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005. 20. B. Gavrea, On Hadamard’s inequality for the convex mappings defined on a convex domain in the space, Journal of Ineq. in Pure & Appl. Math., 1 (2000), No. 1, Article 9, http://jipam.vu. edu.au/. 21. K.-C. Lee and K.-L. Tseng, On a weighted generalisation of Hadamard’s inequality for G-convex functions, Tamsui Oxford Journal of Math. Sci., 16(1) (2000), 91–104. 22. A. Lupas¸, A generalisation of Hadamard’s inequality for convex functions, Univ. Beograd. Publ. Elek. Fak. Ser. Mat. Fiz., No. 544–576, (1976), 115–121 23. D. M. Maksimovi´c, A short proof of generalized Hadamard’s inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., (1979), No. 634–677 126–128. 24. D.S. Mitrinovi´c and I.B. Lackovi´c, Hermite and convexity, Aequationes Math. 28 (1985), 229–232. 25. B. Mond and J. Peˇcari´c, Convex inequalities in Hilbert space, Houston J. Math., 19(1993), 405–420. 26. J.E. Peˇcari´c and S.S. Dragomir, A generalization of Hadamard’s inequality for isotonic linear functionals, Radovi Mat. (Sarajevo) 7 (1991), 103–107. 27. J.E. Peˇcari´c, F. Proschan, and Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., San Diego, 1992.. 28. M.S. Slater, A companion inequality to Jensen’s inequality, J. Approx. Theory, 32(1981), 160–166.

Chapter 3

ˇ ¨ Type Inequalities of the Cebyˇ sev and Gruss

3.1 Introduction ˇ sev, or in a different spelling, Chebyshev inequality which compares The Cebyˇ the integral/discrete mean of the product with the product of the integral/discrete means is famous in the literature devoted to Mathematical Inequalities. It has been extended, generalized, refined, etc. by many authors during the last century. A simple search utilizing either spellings and the key word “inequality” in the title in the comprehensive MathSciNet database of the American Mathematical Society produces more than 200 research articles devoted to this result. The sister result due to Gr¨uss which provides error bounds for the magnitude of the difference between the integral mean of the product and the product of the integral means has also attracted much interest since it has been discovered in 1935 with more than 180 papers published, as a simple search in the same database reveals. Far more publications have been devoted to the applications of these inequalities and an accurate picture of the impacted results in various fields of Modern Mathematics is difficult to provide. In this chapter, however, we present only some recent results due to the author for the corresponding operator versions of these two famous inequalities. Applications for particular functions of selfadjoint operators such as the power, logarithmic and exponential functions are provided as well.

ˇ 3.2 Cebyˇ sev’s Inequality ˇ sev’s Inequality for Real Numbers 3.2.1 Cebyˇ First of all, let us recall a number of classical results for sequences of real numbers ˇ sev inequality. concerning the celebrated Cebyˇ ˇ sev and Gr¨uss Type, S.S. Dragomir, Operator Inequalities of the Jensen, Cebyˇ SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-1521-3 3, © Silvestru Sever Dragomir 2012

71

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

72

Consider the real sequences .n  tuples/ a D .a1 ; : : : ; anP / ; b D .b1 ; : : : ; bn / and the non-negative sequence p D .p1 ; : : : ; pn / with Pn WD niD1 pi > 0: Define ˇ sev’s functional the weighted Cebyˇ Tn .pI a; b/ WD

n n n 1 X 1 X 1 X pi ai bi  pi ai  pi b i : Pn i D1 Pn i D1 Pn i D1

(3.1)

ˇ In 1882–1883, Cebyˇ sev [7, 8] proved that if a and b are monotonic in the same (opposite) sense, then Tn .pI a; b/  ./ 0:

(3.2)

In the special case p D a  0, it appears that inequality (3.2) has been obtained ˇ by Laplace long before Cebyˇ sev (see for example [48, p. 240]). Inequality (3.2) was mentioned by Hardy, Littlewood and P´olya in their book [44] in 1934 in the more general setting of synchronous sequences, i.e. if a; b are synchronous (asynchronous), this means that    ai  aj bi  bj  ./ 0 for any i; j 2 f1; : : : ; ng;

(3.3)

then (3.2) holds true as well. A relaxation of the synchronicity condition was provided by M. Biernacki in 1951, [5], P which showed that, if a; b are monotonic in mean in the same sense, i.e. for Pk WD kiD1 pi ; k D 1; : : : ; n  1I

and

k kC1 1 X 1 X pi ai  ./ pi ai ; k 2 f1; : : : ; n  1g Pk i D1 PkC1 i D1

(3.4)

k kC1 1 X 1 X pi bi  ./ pi bi ; k 2 f1; : : : ; n  1g; Pk i D1 PkC1 i D1

(3.5)

then (3.2) holds with “  ”. If a; b are monotonic in mean in the opposite sense then (3.2) holds with “  ”. If one would like to drop the assumption of non-negativity for the components of p; then one may state the following inequality obtained by Mitrinovi´c and Peˇcari´c in 1991, [47]: If 0  Pi  Pn for each i 2 f1; : : : ; n  1g, then: Tn .pI a; b/  0;

(3.6)

provided a and b are sequences with the same monotonicity. If a and b are monotonic in the opposite sense, the sign of inequality (3.6) reverses. Similar integral inequalities may be stated, however, we do not present them here.

ˇ 3.2 Cebyˇ sev’s Inequality

73

ˇ For other recent results on the Cebyˇ sev inequality in either discrete or integral form see [6, 18, 19, 25, 37, 38, 46, 48, 49, 53–55] and the references therein. The main aim of the present section is to provide operator versions for the ˇ Cebyˇ sev inequality in different settings. Related results and some particular cases of interest are also given.

ˇ sev Inequality for One Operator 3.2.2 A Version of the Cebyˇ We say that the functions f; g W Œa; b ! R are synchronous (asynchronous) on the interval Œa; b if they satisfy the following condition: .f .t/  f .s// .g .t/  g .s//  ./ 0

for each t; s 2 Œa; b :

It is obvious that, if f; g are monotonic and have the same monotonicity on the interval Œa; b ; then they are synchronous on Œa; b while if they have opposite monotonicity, they are asynchronous. ˇ sev inequality for synchronous (asynFor some extensions of the discrete Cebyˇ chronous) sequences of vectors in an inner product space, see [39, 40]. ˇ The following result provides an inequality of Cebyˇ sev type for functions of selfadjoint operators: Theorem 3.1 (Dragomir, 2008, [29]). Let A be a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M: If f; g W Œm; M  ! R are continuous and synchronous (asynchronous) on Œm; M  ; then hf .A/ g .A/ x; xi  ./ hf .A/ x; xi  hg .A/ x; xi

(3.7)

for any x 2 H with kxk D 1: Proof. We consider only the case of synchronous functions. In this case, we have then f .t/ g .t/ C f .s/ g .s/  f .t/ g .s/ C f .s/ g .t/ (3.8) for each t; s 2 Œa; b : If we fix s 2 Œa; b and apply property (P) for inequality (3.8), then we have for each x 2 H with kxk D 1 that h.f .A/ g .A/ C f .s/ g .s/ 1H / x; xi  h.g .s/ f .A/ C f .s/ g .A// x; xi; which is clearly equivalent with hf .A/ g .A/ x; xi C f .s/ g .s/  g .s/ hf .A/ x; xi C f .s/ hg .A/ x; xi (3.9) for each s 2 Œa; b :

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74

Now, if we apply again property (P) for inequality (3.9), then we have for any y 2 H with kyk D 1 that h.hf .A/ g .A/ x; xi 1H C f .A/ g .A// y; yi  h.hf .A/ x; xi g .A/ C hg .A/ x; xi f .A// y; yi ; which is clearly equivalent with hf .A/ g .A/ x; xi C hf .A/ g .A/ y; yi  hf .A/ x; xi hg .A/ y; yi C hf .A/ y; yi hg .A/ x; xi

(3.10)

for each x; y 2 H with kxk D kyk D 1: This is an inequality of interest in itself. Finally, on making y D x in (3.10), we deduce the desired result (3.7).  Some particular cases are of interest for applications. In the first instance we consider the case of power functions. Remark 3.2. We observe, from the proof of the above theorem that, if A and B are selfadjoint operators and Sp .A/ ; Sp .B/  Œm; M  ; then for any continuous synchronous (asynchronous) functions f; g W Œm; M  ! R we have the more general result hf .A/ g .A/ x; xi C hf .B/ g .B/ y; yi  ./ hf .A/ x; xi hg .B/ y; yi C hf .B/ y; yi hg .A/ x; xi

(3.11)

for each x; y 2 H with kxk D kyk D 1: If f W Œm; M  ! .0; 1/ is continuous then the functions f p ; f q are synchronous in the case when p; q > 0 or p; q < 0 and asynchronous when either p > 0; q < 0 or p < 0; q > 0: In this situation if A and B are positive definite operators, then we have the inequality ˝ pCq ˛ ˝ ˛ f .A/ x; x C f pCq .B/ y; y  hf p .A/ x; xi hf q .B/ y; yi C hf p .B/ y; yi hf q .A/ x; xi

(3.12)

for each x; y 2 H with kxk D kyk D 1, where either p; q > 0 or p; q < 0: If p > 0; q < 0 or p < 0; q > 0 then the reverse inequality also holds in (3.12). As particular cases, we should observe that for p D q D 1 and f .t/ D t; we get from (3.12) the inequality ˝ 2 ˛ ˝ ˛ A x; x C B 2 y; y  2  hAx; xi hBy; yi for each x; y 2 H with kxk D kyk D 1.

(3.13)

ˇ 3.2 Cebyˇ sev’s Inequality

75

For p D 1 and q D 1 we have from (3.12) ˝ ˛ ˝ ˛ hAx; xi B 1 y; y C hBy; yi A1 x; x  2

(3.14)

for each x; y 2 H with kxk D kyk D 1:

3.2.3 Related Results for One Operator ˇ The following result that is related to the Cebyˇ sev inequality may be stated: Theorem 3.3 (Dragomir, 2008, [29]). Let A be a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M: If f; g W Œm; M  ! R are continuous and synchronous on Œm; M  ; then hf .A/ g .A/ x; xi  hf .A/ x; xi  hg .A/ x; xi  Œhf .A/ x; xi  f .hAx; xi/  Œg .hAx; xi/  hg .A/ x; xi

(3.15)

for any x 2 H with kxk D 1: If f; g are asynchronous, then hf .A/ x; xi  hg .A/ x; xi  hf .A/ g .A/ x; xi  Œhf .A/ x; xi  f .hAx; xi/  Œhg .A/ x; xi  g .hAx; xi/

(3.16)

for any x 2 H with kxk D 1: Proof. Since f; g are synchronous and m  hAx; xi  M for any x 2 H with kxk D 1; then we have Œf .t/  f .hAx; xi/ Œg .t/  g .hAx; xi/  0

(3.17)

for any t 2 Œa; b and x 2 H with kxk D 1: On utilizing property (P) for inequality (3.17) we have that hŒf .B/  f .hAx; xi/ Œg .B/  g .hAx; xi/ y; yi  0

(3.18)

for any B a bounded linear operator with Sp .B/  Œm; M  and y 2 H with kyk D 1: Since hŒf .B/  f .hAx; xi/ Œg .B/  g .hAx; xi/ y; yi D hf .B/ g .B/ y; yi C f .hAx; xi/ g .hAx; xi/  hf .B/ y; yi g .hAx; xi/  f .hAx; xi/ hg .B/ y; yi;

(3.19)

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76

then from (3.18) we get hf .B/ g .B/ y; yi C f .hAx; xi/ g .hAx; xi/  hf .B/ y; yi g .hAx; xi/ C f .hAx; xi/ hg .B/ y; yi ; which is clearly equivalent with hf .B/ g .B/ y; yi  hf .A/ y; yi  hg .A/ y; yi  Œhf .B/ y; yi  f .hAx; xi/  Œg .hAx; xi/  hg .B/ y; yi

(3.20)

for each x; y 2 H with kxk D kyk D 1: This inequality is of interest in its own right. Now, if we choose B D A and y D x in (3.20), then we deduce the desired result (3.15).  ˇ The following result which improves the Cebyˇ sev inequality may be stated: Corollary 3.4 (Dragomir, 2008, [29]). Let A be a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M: If f; g W Œm; M  ! R are continuous, synchronous and one is convex while the other is concave on Œm; M  ; then hf .A/ g .A/ x; xi  hf .A/ x; xi  hg .A/ x; xi  Œhf .A/ x; xi  f .hAx; xi/  Œg .hAx; xi/  hg .A/ x; xi  0

(3.21)

for any x 2 H with kxk D 1: If f; g are asynchronous and either both of them are convex or both of them concave on Œm; M , then hf .A/ x; xi  hg .A/ x; xi  hf .A/ g .A/ x; xi  Œhf .A/ x; xi  f .hAx; xi/  Œhg .A/ x; xi  g .hAx; xi/  0

(3.22)

for any x 2 H with kxk D 1: Proof. The second inequality follows by making use of the result due to Mond and Peˇcari´c, see [50, 51] or [42, p. 5]: hh .A/ x; xi  ./ h .hAx; xi/

(MP)

for any x 2 H with kxk D 1 provided that A is a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M and h is convex (concave) on the given interval Œm; M . 

3.3 Gr¨uss Inequality

77

¨ Inequality 3.3 Gruss 3.3.1 Some Elementary Inequalities of Gruss ¨ Type In 1935, G. Gr¨uss [43] proved the following integral inequality which gives an approximation of the integral of the product in terms of the product of the integrals as follows: ˇ ˇ Z b Z b ˇ 1 Z b ˇ 1 1 ˇ ˇ f .x/ g .x/ dx  f .x/ dx  g .x/ dx ˇ ˇ ˇb  a a ˇ ba a ba a 

1 .ˆ  / .   / ; 4

(3.23)

where f , g W Œa; b ! R are integrable on Œa; b and satisfy the condition   f .x/  ˆ,

  g .x/  

(3.24)

for each x 2 Œa; b ; where ; ˆ;  ;  are given real constants. Moreover, the constant 14 is sharp in the sense that it cannot be replaced by a smaller one. In 1950, M. Biernacki, H. Pidek and C. Ryll-Nardjewski [48, Chap. X] established the following discrete version of Gr¨uss’ inequality: Let a D .a1 ; : : : ; an / ; b D .b1 ; : : : ; bn / be two n-tuples of real numbers such that r  ai  R and s  bi  S for i D 1; : : : ; n: Then one has ˇ n ˇ n n ˇ1 X 1X 1 X ˇˇ ˇ ai bi  ai  bi ˇ ˇ ˇn n i D1 n i D1 ˇ i D1   1 hni 1 hni  1 .R  r/ .S  s/; (3.25) n 2 n 2 where Œx denotes the integer part of x; x 2 R: For a simple proof of (3.23) as well as for some other integral inequalities of Gr¨uss type, see Chap. X of the recent book [48]. For other related results see the papers [1–4, 9–36, 41, 52, 56] and the references therein.

3.3.2 An Inequality of Gruss’ ¨ Type for One Operator The following result may be stated: Theorem 3.5 (Dragomir, 2008, [30]). Let A be a selfadjoint operator on the Hilbert space .H I h:; :i/ and assume that Sp .A/  Œm; M  for some scalars

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78

m < M: If f and g are continuous on Œm; M  and  WD mint 2Œm;M  f .t/ and  WD maxt 2Œm;M  f .t/ then ˇ ˇ ˇ hf .A/ g .A/ y; yi  hf .A/ y; yi  hg .A/ x; xi ˇ ˇ ˇ  C Œhg .A/ y; yi  hg .A/ x; xiˇˇ  2 h i1=2 1   .   / kg .A/ yk2 C hg .A/ x; xi2  2 hg .A/ x; xi hg .A/ y; yi 2 (3.26) for any x; y 2 H with kxk D kyk D 1: Proof. First of all, observe that, for each  2 R and x; y 2 H; kxk D kyk D 1 we have the identity h.f .A/    1H / .g .A/  hg .A/ x; xi  1H / y; yi D hf .A/ g .A/ y; yi    Œhg .A/ y; yi  hg .A/ x; xi  hg .A/ x; xi hf .A/ y; yi:

(3.27)

Taking the modulus in (3.27) we have jhf .A/ g .A/ y; yi    Œhg .A/ y; yi  hg .A/ x; xi  hg .A/ x; xi hf .A/ y; yij D jh.g .A/  hg .A/ x; xi  1H / y; .f .A/    1H / yij  kg .A/ y  hg .A/ x; xi yk kf .A/ y  yk i1=2 h D kg .A/ yk2 C hg .A/ x; xi2  2 hg .A/ x; xi hg .A/ y; yi  kf .A/ y  yk h i1=2  kg .A/ yk2 C hg .A/ x; xi2  2 hg .A/ x; xi hg .A/ y; yi  kf .A/    1H k

(3.28)

for any x; y 2 H; kxk D kyk D 1: Now, since  D mint 2Œm;M  f .t/ and  D maxt 2Œm;M  f .t/ ; then by property (P) we have that   hf .A/ y; yi   for each y 2 H with kyk D 1 which is clearly equivalent with ˇ ˇ ˇ ˇ ˇhf .A/ y; yi   C  kyk2 ˇ  1 .   / ˇ ˇ 2 2

3.3 Gr¨uss Inequality

or with

79

ˇ  ˇ ˇ ˇ ˇ f .A/   C  1H y; y ˇ  1 .   / ˇ 2 ˇ 2

for each y 2 H with kyk D 1: Taking the supremum in this inequality we get     f .A/   C   1H   1 .   /   2 2 which together with inequality (3.28) applied for  D result (3.26).

 C 2

produces the desired 

As a particular case of interest, we can derive from the above theorem the following result of Gr¨uss’ type: Corollary 3.6 (Dragomir, 2008, [30]). With the assumptions in Theorem 3.5 we have jhf .A/ g .A/ x; xi  hf .A/ x; xi  hg .A/ x; xij  h i1=2  1 1 2 2   .   / kg .A/ xk  hg .A/ x; xi  .   / .  ı/ 2 4 (3.29) for each x 2 H with kxk D 1; where ı WD mint 2Œm;M  g .t/ and  WD maxt 2Œm;M  g .t/ : Proof. The first inequality follows from (3.26) by putting y D x: Now, if we write the first inequality in (3.29) for f D g we get ˝ ˛ 0  kg .A/ xk2  hg .A/ x; xi2 D g 2 .A/ x; x  hg .A/ x; xi2 h i1=2 1  .  ı/ kg .A/ xk2  hg .A/ x; xi2 2 which implies that h

kg .A/ xk2  hg .A/ x; xi2

i1=2



1 .  ı/ 2

for each x 2 H with kxk D 1: This together with the first part of (3.29) proves the desired bound.



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80

¨ Type 3.4 More Inequalities of Gruss 3.4.1 Some Vectorial Gruss’ ¨ Type Inequalities The following lemmas, that are of interest in their own right, collect some Gr¨uss type inequalities for vectors in inner product spaces obtained earlier by the author: Lemma 3.7 (Dragomir, 2003 and 2004, [22, 27]). Let .H; h; i/ be an inner product space over the real or complex number field K, u; v; e 2 H; kek D 1; and ˛; ˇ;  ; ı 2 K such that Re hˇe  u; u  ˛ei  0;

Re hıe  v; v   ei  0

(3.30)

or equivalently,     u  ˛ C ˇ e   1 jˇ  ˛j ;   2 2

    v   C ı e   1 jı  j :   2 2

(3.31)

Then jhu; vi  hu; ei he; vij 1  jˇ  ˛j jı   j 4 8 1 < ŒRe hˇe  u; u  ˛ei Re hıe  v; v   ei 2 ; ˇˇ ˇ  ˇˇ : ˇhu; ei  ˛Cˇ ˇˇ ˇˇhv; ei   Cı ˇˇ : 2 2 

(3.32)

The first inequality has been obtained in [22] (see also [26, p. 44]) while the second result was established in [27] (see also [26, p. 90]). They provide refinements of the earlier result from [15] where only the first part of the bound, i.e. 14 jˇ  ˛j jı   j has been given. Notice that, as pointed out in [27], the upper bounds for the Gr¨uss functional incorporated in (3.32) cannot be compared in general, meaning that one is better than the other depending on appropriate choices of the vectors and scalars involved. Another result of this type is the following one: Lemma 3.8 (Dragomir, 2004 & 2006, [23], [28]). With the assumptions in Lemma 3.7 and if Re .ˇ˛/ > 0; Re .ı / > 0 then

3.4 More Inequalities of Gr¨uss Type

jhu; vi  hu; ei he; vij 8 1 ˆ ˆ  jˇ˛jjı j 1 jhu; ei he; vij; ˆ ˆ < 4 ŒRe.ˇ˛/ Re.ı/ 2



i 12 1 1  h j˛ C ˇj  2 ŒRe .ˇ˛/ 2 jı C  j  2 ŒRe .ı / 2 ˆ ˆ ˆ ˆ 1 :  Œjhu; ei he; vij 2 :

81

(3.33)

The first inequality has been established in [23] (see [26, p. 62]) while the second one can be obtained in a canonical manner from the reverse of the Schwarz inequality given in [28]. The details are omitted. Finally, another inequality of Gr¨uss type that has been obtained in [24] (see also [26, p. 65]) can be stated as: Lemma 3.9 (Dragomir, 2004, [24]). With the assumptions in Lemma 3.7 and if ˇ ¤ ˛; ı ¤  then jhu; vi  hu; ei he; vij 

1 1 jˇ  ˛j jı   j  Œ.kuk C jhu; eij/ .kvk C jhv; eij/ 2 : 4 Œjˇ C ˛j jı C  j 12

(3.34)

3.4.2 Some Inequalities of Gruss’ ¨ Type for One Operator The following results incorporates some new inequalities of Gr¨uss’ type for two functions of a selfadjoint operator. Theorem 3.10 (Dragomir, 2008, [31]). Let A be a selfadjoint operator on the Hilbert space .H I h:; :i/ and assume that Sp .A/  Œm; M  for some scalars m < M: If f and g are continuous on Œm; M  and  WD mint 2Œm;M  f .t/,  WD maxt 2Œm;M  f .t/, ı WD mint 2Œm;M  g .t/ and  WD maxt 2Œm;M  g .t/ then jhf .A/ g .A/ x; xi  hf .A/ x; xi hg .A/ x; xij 

1 .   / .  ı/ 4 8 1 ˆ < Œh x  f .A/ x; f .A/ x   xi hx  g .A/ x; g .A/ x  ıxi 2 ;  ˇ ˇ ˆ : ˇˇhf .A/ x; xi   C ˇˇ ˇˇhg .A/ x; xi  Cı ˇˇ 2 2 (3.35)

for each x 2 H with kxk D 1: Moreover if  and ı are positive, then we also have

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82

jhf .A/ g .A/ x; xi  hf .A/ x; xi hg .A/ x; xij 81 . /.ı/ ˆ ˆ < 4  p  ı hf .A/ x; xi hg .A/ x; xi; 



ˆ ˆ : p  p p  pı Œhf .A/ x; xi hg .A/ x; xi 12 ;

(3.36)

while for  C  ;  C ı ¤ 0 we have jhf .A/ g .A/ x; xi  hf .A/ x; xi hg .A/ x; xij 

1 .   / .  ı/  4 Œj C  j j C ıj 12 1

 Œ.kf .A/ xk C jhf .A/ x; xij/ .kg .A/ xk C jhg .A/ x; xij/ 2

(3.37)

for each x 2 H with kxk D 1: Proof. Since  WD mint 2Œm;M  f .t/,  WD maxt 2Œm;M  f .t/, ı WD mint 2Œm;M  g .t/ and  WD maxt 2Œm;M  g .t/, the by property (P) we have that   1H  f .A/    1H

and

ı  1H  g .A/    1H

in the operator order, which imply that Œf .A/    1 Œ  1H  f .A/  0 and Œ  1H  g .A/ Œg .A/  ı  1H   0

(3.38)

in the operator order. We then have from (3.38) hŒf .A/    1 Œ  1H  f .A/ x; xi  0 and hŒ  1H  g .A/ Œg .A/  ı  1H  x; xi  0 for each x 2 H with kxk D 1; which, by the fact that the involved operators are selfadjoint, are equivalent with the inequalities h x  f .A/ x; f .A/ x   xi  0, hx  g .A/ x; g .A/ x  ıxi  0

(3.39)

for each x 2 H with kxk D 1: Now, if we apply Lemma 3.7 for u D f .A/ x; v D g .A/ x, e D x; and the real scalars ; ;  and ı defined in the statement of the theorem, then we can state the inequality

3.4 More Inequalities of Gr¨uss Type

83

jhf .A/ x; g .A/ xi  hf .A/ x; xi hx; g .A/ xij 

1  .   / .  ı/ 4 8 1 ˆ < ŒRe h x  f .A/ x; f .A/ x   xi Re hx  g .A/ x; g .A/ x  ıxi 2 ;  ˇ ˇ ˆ : ˇˇhf .A/ x; xi   C ˇˇ ˇˇhg .A/ x; xi  Cı ˇˇ 2 2 (3.40)

for each x 2 H with kxk D 1; which is clearly equivalent with inequality (3.35). Inequalities (3.36) and (3.37) follow by Lemmas 3.8 and 3.9 respectively and the details are omitted.  Remark 3.11. The first inequality in (3.36) can be written in a more convenient way as ˇ ˇ ˇ hf .A/ g .A/ x; xi ˇ 1 .   / .  ı/ ˇ ˇ p (3.41) ˇ hf .A/ x; xi hg .A/ x; xi  1ˇ  4    ı for each x 2 H with kxk D 1; while the second inequality has the following equivalent form: ˇ ˇ ˇ ˇ hf .A/ g .A/ x; xi ˇ 1=2 ˇ  Œhf .A/ x; xi hg .A/ x; xi ˇ ˇ 1=2 ˇ Œhf .A/ x; xi hg .A/ x; xi ˇ p



p p p      ı

(3.42)

for each x 2 H with kxk D 1: We know, from [29] that if f; g are synchronous (asynchronous) functions on the interval Œm; M  ; i.e. we recall that Œf .t/  f .s/ Œg .t/  g .s/ ./  0

for each t; s 2 Œm; M  ;

then we have the inequality hf .A/ g .A/ x; xi  ./ hf .A/ x; xi hg .A/ x; xi

(3.43)

for each x 2 H with kxk D 1; provided f; g are continuous on Œm; M  and A is a selfadjoint operator with Sp .A/  Œm; M . Therefore, if f; g are synchronous, then we have from (3.41) and from (3.42) the following results: 0

1 .   / .  ı/ hf .A/ g .A/ x; xi 1  p 4 hf .A/ x; xi hg .A/ x; xi   ı

(3.44)

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84

and 0

hf .A/ g .A/ x; xi

Œhf .A/ x; xi hg .A/ x; xi1=2 p p

p p      ı

 Œhf .A/ x; xi hg .A/ x; xi1=2 (3.45)

for each x 2 H with kxk D 1; respectively. If f; g are asynchronous then 01

1 .   / .  ı/ hf .A/ g .A/ x; xi   p 4 hf .A/ x; xi hg .A/ x; xi   ı

(3.46)

and 0  Œhf .A/ x; xi hg .A/ x; xi1=2  

p p

p p     ı

hf .A/ g .A/ x; xi Œhf .A/ x; xi hg .A/ x; xi1=2 (3.47)

for each x 2 H with kxk D 1; respectively.

ˇ 3.5 More Inequalities for the Cebyˇ sev Functional 3.5.1 A Refinement and Some Related Results The following result can be stated: Theorem 3.12 (Dragomir, 2008, [32]). Let A be a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M: If f; g W Œm; M  ! R are continuous with ı WD mint 2Œm;M  g .t/ and  WD maxt 2Œm;M  g .t/, then 1 .  ı/ hjf .A/  hf .A/ x; xi  1H j x; xi 2 1  .  ı/ C 1=2 .f; f I AI x/ 2

jC .f; gI AI x/j 

(3.48)

for any x 2 H with kxk D 1: Proof. Since ı WD mint 2Œm;M  g .t/ and  WD maxt 2Œm;M  g .t/ ; we have ˇ ˇ ˇ ˇ ˇg .t/   C ı ˇ  1 .  ı/ ˇ 2 ˇ 2 for any t 2 Œm; M  and for any x 2 H with kxk D 1:

(3.49)

ˇ 3.5 More Inequalities for the Cebyˇ sev Functional

85

If we multiply inequality (3.49) with jf .t/  hf .A/ x; xij we get ˇ ˇ ˇ ˇ ˇf .t/ g .t/  hf .A/ x; xi g .t/   C ı f .t/ C  C ı hf .A/ x; xiˇ ˇ ˇ 2 2 

1 .  ı/ jf .t/  hf .A/ x; xij 2

(3.50)

for any t 2 Œm; M  and for any x 2 H with kxk D 1: Now, if we apply property (P) for inequality (3.50) and a selfadjoint operator B with Sp .B/  Œm; M  ; then we get the following inequality of interest in itself: jhf .B/ g .B/ y; yi  hf .A/ x; xi hg .B/ y; yi  

ˇ ˇ Cı Cı hf .B/ y; yi C hf .A/ x; xiˇˇ 2 2

1 .  ı/ hjf .B/  hf .A/ x; xi  1H j y; yi 2

(3.51)

for any x; y 2 H with kxk D kyk D 1: If we choose in (3.51) y D x and B D A; then we deduce the first inequality in (3.48). Now, by the Schwarz inequality in H we have hjf .A/  hf .A/ x; xi  1H j x; xi  kjf .A/  hf .A/ x; xi  1H j xk D kf .A/ x  hf .A/ x; xi  xk i1=2 h D kf .A/ xk2  hf .A/ x; xi2 D C 1=2 .f; f I AI x/ for any x 2 H with kxk D 1; and the second part of (3.48) is also proved.



Let U be a selfadjoint operator on the Hilbert space .H; h:; :i/ with the spectrum Sp .U / included in the interval Œm; M  for some real numbers m < M and let fE g2R be its spectral family. Then for any continuous function f W Œm; M  ! R, it is well known that we have the following representation in terms of the Riemann– Stieltjes integral: Z

M

hf .U / x; xi D

f ./ d .hE x; xi/

(3.52)

m0

for any x 2 H with kxk D 1: The function gx ./ WD hE x; xi is monotonic non-decreasing on the interval Œm; M  and gx .m  0/ D 0 for any x 2 H with kxk D 1:

and

gx .M / D 1

(3.53)

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

86

The following result is of interest: Theorem 3.13 (Dragomir, 2008, [32]). Let A and B be selfadjoint operators with Sp .A/ ; Sp .B/  Œm; M  for some real numbers m < M: If f W Œm; M  ! R is of r  L-H¨older type, i.e. for a given r 2 .0; 1 and L > 0 we have jf .s/  f .t/j  L js  tjr

for any s; t 2 Œm; M ;

then we have the Ostrowski type inequality for selfadjoint operators: ˇ ˇ ˇ m C M ˇˇ r 1 ˇ .M  m/ C ˇs  jf .s/  hf .A/ x; xij  L 2 2 ˇ 

(3.54)

for any s 2 Œm; M  and any x 2 H with kxk D 1. Moreover, we have jhf .B/ y; yi  hf .A/ x; xij  hjf .B/  hf .A/ x; xi  1H j y; yi ˇ  r ˇ  ˇ ˇ mCM 1 ˇ ˇ .M  m/ C ˇB   1H ˇ y; y L 2 2 (3.55) for any x; y 2 H with kxk D kyk D 1: Proof. We use the following Ostrowski type inequality for the Riemann–Stieltjes integral obtained by the author in [21]: ˇ ˇ Z b ˇ ˇ ˇ ˇ f .t/ du .t/ˇ ˇf .s/ Œu .b/  u .a/  ˇ ˇ a  L

ˇ ˇ b ˇ a C b ˇˇ r _ 1 .b  a/ C ˇˇs  .u/ 2 2 ˇ a

(3.56)

for any s 2 Œa; b ; provided that f is of r L-H¨older type on Œa; b ; u is of bounded _b variation on Œa; b and .u/ denotes the total variation of u on Œa; b : a Now, applying this inequality for u ./ D gx ./ WD hE x; xi, where x 2 H with kxk D 1 we get ˇ Z ˇ ˇf .s/  ˇ  L

M m0

ˇ ˇ f ./ d .hE x; xi/ˇˇ

ˇ ˇ M ˇ m C M ˇˇ r _ 1 .M  m/ C ˇˇs  .gx / 2 2 ˇ m0

which, by (3.52) and (3.53) is equivalent with (3.54).

(3.57)

ˇ 3.5 More Inequalities for the Cebyˇ sev Functional

87

By applying property (P) for inequality (3.54) and the operator B we have hjf .B/  hf .A/ x; xi  1H j y; yi ˇ r ˇ   ˇ ˇ mCM 1 .M  m/ C ˇˇB   1H ˇˇ y; y L 2 2 ˇ ˇ r  ˇ m C M ˇˇ 1 ˇ .M  m/ C ˇB   1 y; y L H 2 2 ˇ ˇ  r ˇ  ˇ ˇ mCM 1 ˇ ˇ .M  m/ C ˇB   1H ˇ y; y DL 2 2 for any x; y 2 H with kxk D kyk D 1; which proves the second inequality in (3.55). Further, by the Jensen inequality for convex functions of selfadjoint operators (see for instance [42, p. 5]) applied for the modulus, we can state that jhh .A/ x; xij  hjh .A/j x; xi

(M)

for any x 2 H with kxk D 1; where h is a continuous function on Œm; M  : Now, if we apply inequality (M), then we have jhŒf .B/  hf .A/ x; xi  1H  y; yij  hjf .B/  hf .A/ x; xi  1H j y; yi which shows the first part of (3.55), and the proof is complete.



Remark 3.14. With the above assumptions for f; A and B we have the following particular inequalities of interest: ˇ ˇ   ˇ ˇ ˇf m C M  hf .A/ x; xiˇ  1 L .M  m/r ˇ 2r ˇ 2

(3.58)

and jf .hAx; xi/  hf .A/ x; xij ˇ ˇ  ˇ m C M ˇˇ r 1 ˇ .M  m/ C ˇhAx; xi  L 2 2 ˇ

(3.59)

for any x 2 H with kxk D 1. We also have the inequalities: jhf .A/ y; yi  hf .A/ x; xij  hjf .A/  hf .A/ x; xi  1H j y; yi ˇ  r ˇ  ˇ ˇ mCM 1 ˇ .M  m/ C ˇA   1H ˇˇ y; y L 2 2

(3.60)

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

88

for any x; y 2 H with kxk D kyk D 1; jhŒf .B/  f .A/ x; xij  hjf .B/  hf .A/ x; xi  1H j x; xi ˇ  r ˇ  ˇ ˇ mCM 1 .M  m/ C ˇˇB   1H ˇˇ x; x L 2 2 (3.61) and, more particularly, hjf .A/  hf .A/ x; xi  1H j x; xi ˇ  r ˇ  ˇ ˇ mCM 1 .M  m/ C ˇˇA   1H ˇˇ x; x L 2 2

(3.62)

for any x 2 H with kxk D 1: We also have the norm inequality  r    mCM 1  .M  m/ C B   1H  kf .B/  f .A/k  L  : 2 2 

(3.63)

The following corollary of the above Theorem 3.13 can be useful for applications: Corollary 3.15 (Dragomir, 2008, [32]). Let A and B be selfadjoint operators with Sp .A/ ; Sp .B/  Œm; M  for some real numbers m < M: If f W Œm; M  ! R is absolutely continuous, then we have the Ostrowski type inequality for selfadjoint operators: jf .s/  hf .A/ x; xij 8 ˇ 1 ˆ .M  m/ C ˇs  ˆ 2 ˆ <  ˇ

1 ˆ ˆ ˆ : 2 .M  m/ C ˇs 

ˇ

mCM ˇ 2

kf 0 k1;Œm;M 

ˇ mCM ˇ 1=q 2

if f 0 2 L1 Œm; M  I

kf 0 kp;Œm;M 

if f 0 2 Lp Œm; M  ; p; q > 1; p1 C q1 D 1 (3.64)

for any s 2 Œm; M  and any x 2 H with kxk D 1, where kkp;Œm;M  are the Lebesgue norms, i.e. khk1;Œm;M  WD ess sup kh .t/k t 2Œm;M 

and

Z khkp;Œm;M  WD

1=p

M

jh .t/j m

p

;

p  1:

ˇ 3.5 More Inequalities for the Cebyˇ sev Functional

89

Moreover, we have jhf .B/ y; yi  hf .A/ x; xij  hjf .B/  hf .A/ x; xi  1H j y; yi 8 ˇ ˛ ˝ˇ M m ˆ C ˇB  mCM  1H ˇ y; y kf 0 k1;Œm;M  if f 0 2 L1 Œm; M  I ˆ 2 2 ˆ ˆ <  ˆ ˇ ˛ q1

M m ˝ˇ if f 0 2 Lp Œm; M  ; ˆ mCM ˆ 0 ˇ ˇ ˆ y; y C B   1 kf k H : 2 p;Œm;M  2 p; q > 1; p1 C q1 D 1 (3.65) for any x; y 2 H with kxk D kyk D 1: Now, on utilizing Theorem 3.12 we can provide the following upper bound for ˇ the Cebyˇ sev functional that may be more useful in applications: Corollary 3.16 (Dragomir, 2008, [32]). Let A be a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M: If g W Œm; M  ! R is continuous with ı WD mint 2Œm;M  g .t/ and  WD maxt 2Œm;M  g .t/ ; then for any f W Œm; M  ! R of r  L-H¨older type we have the inequality: jC .f; gI AI x/j

ˇ  r  ˇ ˇ ˇ mCM 1 1 ˇ ˇ .M  m/ C ˇA   1H ˇ x; x  .  ı/ L 2 2 2

(3.66)

for any x 2 H with kxk D 1: Remark 3.17. With the assumptions from Corollary 3.16 for g and A and if f is absolutely continuos on Œm; M  ; then we have the inequalities: 1 jC .f; gI AI x/j  .  ı/ 2 8 ˇ ˛ ˝ˇ 1 mCM ˇ ˆ  1H ˇ x; x kf 0 k1;Œm;M  if f 0 2 L1 Œm; M  I ˆ 2 .M  m/ C A  2 ˆ <  ˇ

1 ˛1=q ˝ˇ if f 0 2 L1 Œm; M  ; ˆ mCM ˆ ˆ kf 0 kp;Œm;M  : 2 .M  m/ C ˇA  2  1H ˇ x; x p; q > 1; p1 C q1 D 1 (3.67) for any x 2 H with kxk D 1:

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

90

ˇ 3.6 Bounds for the Cebyˇ sev Functional of Lipschitzian Functions 3.6.1 The Case of Lipschitzian Functions The following result can be stated: Theorem 3.18 (Dragomir, 2008, [33]). Let A be a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M: If f W Œm; M  ! R is Lipschitzian with the constant L > 0 and g W Œm; M  ! R is continuous with ı WD mint 2Œm;M  g .t/ and  WD maxt 2Œm;M  g .t/ ; then jC .f; gI AI x/j 

(3.68)

1 .  ı/ L h`A;x .A/ x; xi  .  ı/ LC .e; eI AI x/ 2

for any x 2 H with kxk D 1; where `A;x .t/ WD hjt  1H  Aj x; xi is a continuous function on Œm; M  ; e .t/ D t and C .e; eI AI x/ D kAxk2  hAx; xi2 . 0/ :

(3.69)

Proof. First of all, by the Jensen inequality for convex functions of selfadjoint operators (see for instance [42, p. 5]) applied for the modulus, we can state that jhh .A/ x; xij  hjh .A/j x; xi

(M)

for any x 2 H with kxk D 1; where h is a continuous function on Œm; M  : Since f is Lipschitzian with the constant L > 0; then for any t; s 2 Œm; M  we have jf .t/  f .s/j  L jt  sj :

(3.70)

Now, if we fix t 2 Œm; M  and apply property (P) for inequality (3.70) and the operator A we get hjf .t/  1H  f .A/j x; xi  L hjt  1H  Aj x; xi

(3.71)

for any x 2 H with kxk D 1: Utilizing property (M) we get jf .t/  hf .A/ x; xij D jhf .t/  1H  f .A/ x; xij  hjf .t/  1H  f .A/j x; xi

ˇ 3.6 Bounds for the Cebyˇ sev Functional of Lipschitzian Functions

91

which together with (3.71) gives jf .t/  hf .A/ x; xij  L`A;x .t/

(3.72)

for any t 2 Œm; M  and for any x 2 H with kxk D 1: Since ı WD mint 2Œm;M  g .t/ and  WD maxt 2Œm;M  g .t/ ; we also have ˇ ˇ ˇ ˇ ˇg .t/   C ı ˇ  1 .  ı/ ˇ 2 ˇ 2

(3.73)

for any t 2 Œm; M  and for any x 2 H with kxk D 1: If we multiply inequality (3.72) with (3.73) we get ˇ ˇ ˇ ˇ ˇf .t/ g .t/  hf .A/ x; xi g .t/   C ı f .t/ C  C ı hf .A/ x; xiˇ ˇ ˇ 2 2 1 1 .  ı/ L`A;x .t/ D .  ı/ L hjt  1H  Aj x; xi 2 2 D E1=2 1  .  ı/ L jt  1H  Aj2 x; x 2 ˝ 1=2 ˛ 1 D .  ı/ L A2 x; x  2 hAx; xi t C t 2 2 

(3.74)

for any t 2 Œm; M  and for any x 2 H with kxk D 1: Now, if we apply property (P) for inequality (3.74) and a selfadjoint operator B with Sp .B/  Œm; M  ; then we get the following inequality of interest in itself: hf .B/ g .B/ y; yi  hf .A/ x; xi hg .B/ y; yi

ˇ ˇ Cı Cı  hf .B/ y; yi C hf .A/ x; xiˇˇ 2 2

1 .  ı/ L h`A;x .B/ y; yi 2 D˝ E 1=2 ˛ 1  .  ı/ L A2 x; x 1H  2 hAx; xi B C B 2 y; y 2 ˝ ˛ ˝ ˛ 1  .  ı/ L A2 x; x  2 hAx; xi hBy; yi C B 2 y; y 2



(3.75)

for any x; y 2 H with kxk D kyk D 1: Finally, if we choose in (3.75) y D x and B D A; then we deduce the desired result (3.68). 

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

92

In the case of two Lipschitzian functions, the following result may be stated as well: Theorem 3.19 (Dragomir, 2008, [33]). Let A be a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M: If f; g W Œm; M  ! R are Lipschitzian with the constants L; K > 0; then jC .f; gI AI x/j  LKC .e; eI AI x/

(3.76)

for any x 2 H with kxk D 1: Proof. Since f; g W Œm; M  ! R are Lipschitzian, then jf .t/  f .s/j  L jt  sj and jg .t/  g .s/j  K jt  sj for any t; s 2 Œm; M  ; which gives the inequality   jf .t/ g .t/  f .t/ g .s/  f .s/ g .t/ C f .s/ g .s/j  KL t 2  2ts C s 2 for any t; s 2 Œm; M  : Now, fix t 2 Œm; M  and if we apply properties (P) and (M) for the operator A, we get successively jf .t/ g .t/  hg .A/ x; xi f .t/  hf .A/ x; xi g .t/ C hf .A/ g .A/ x; xij D jhŒf .t/ g .t/  1H  f .t/ g .A/  f .A/ g .t/ C f .A/ g .A/ x; xij  hjf .t/ g .t/  1H  f .t/ g .A/  f .A/ g .t/ C f .A/ g .A/j x; xi ˝  ˛  ˝ ˛  KL t 2  1H  2tA C A2 x; x D KL t 2  2t hAx; xi C A2 x; x (3.77) for any x 2 H with kxk D 1: Further, fix x 2 H with kxk D 1: On applying the same properties for inequality (3.77) and another selfadjoint operator B with Sp .B/  Œm; M  ; we have jhf .B/ g .B/ y; yi  hg .A/ x; xi hf .B/ y; yi  hf .A/ x; xi hg .B/ y; yi C hf .A/ g .A/ x; xij ˝ ˛ ˝ ˛   KL B 2  2 hAx; xi B C A2 x; x 1H y; y ˝ ˛ ˝ ˛ D KL B 2 y; y  2 hAx; xi hBy; yi C A2 x; x

(3.78)

for any x; y 2 H with kxk D kyk D 1; which is an inequality of interest in its own right. Finally, on making B D A and y D x in (3.78) we deduce the desired result (3.76). 

ˇ 3.6 Bounds for the Cebyˇ sev Functional of Lipschitzian Functions

93

3.6.2 The Case of .'; ˆ/-Lipschitzian Functions The following lemma may be stated: Lemma 3.20. Let u W Œa; b ! R and '; ˆ 2 R with ˆ > ': The following statements are equivalent: 1 (i) The function u 'Cˆ 2 e; where e .t/ D t; t 2 Œa; b ; is 2 .ˆ  '/-Lipschitzian; (ii) We have the inequality:

'

u .t/  u .s/  ˆ for each t; s 2 Œa; b with t ¤ sI t s

(3.79)

(iii) We have the inequality: ' .t  s/  u .t/  u .s/  ˆ .t  s/ for each t; s with t > s:

(3.80)

Following [45], we can introduce the concept: Definition 3.21. The function u W Œa; b ! R which satisfies one of the equivalent conditions (i)–(iii) is said to be .'; ˆ/-Lipschitzian on Œa; b : Notice that in [45], the definition was introduced on utilizing the statement (iii) and only the equivalence (i) , (iii) was considered. Utilizing Lagrange’s mean value theorem, we can state the following result that provides practical examples of .'; ˆ/-Lipschitzian functions: Proposition 3.22. Let u W Œa; b ! R be continuous on Œa; b and differentiable on .a; b/ : If  1 <  WD inf u0 .t/ ; t 2.a;b/

sup u0 .t/ DW  < 1;

(3.81)

t 2.a;b/

then u is . ;  /-Lipschitzian on Œa; b : The following result can be stated: Theorem 3.23 (Dragomir, 2008, [33]). Let A be a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M: If f W Œm; M  ! R is .'; ˆ/-Lipschitzian on Œa; b and g W Œm; M  ! R is continuous with ı WD mint 2Œm;M  g .t/ and  WD maxt 2Œm;M  g .t/ ; then ˇ ˇ ˇ ˇ ˇC .f; gI AI x/  ' C ˆ C .e; gI AI x/ˇ ˇ ˇ 2 1 .  ı/ .ˆ  '/ h`A;x .A/ x; xi 4 1  .  ı/ .ˆ  '/ C .e; eI AI x/ 2



for any x 2 H with kxk D 1:

(3.82)

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

94

The proof follows by Theorem 3.18 applied for the 12 .ˆ  '/-Lipschitzian function f  'Cˆ 2  e (see Lemma 3.20) and the details are omitted. Theorem 3.24 (Dragomir, 2008, [33]). Let A be a selfadjoint operator with Sp .A/  Œm; M  for some real numbers m < M and f; g W Œm; M  ! R. If f is .'; ˆ/-Lipschitzian and g is . ; ‰/-Lipschitzian on Œa; b ; then ˇ ˇ ˇC .f; gI AI x/  ˆ C ' C .e; gI AI x/ ˇ 2 

‰C 2



1 .ˆ  '/ .‰  4

C .f; eI AI x/ C

ˆC' ‰C  2 2

ˇ ˇ C .e; eI AI x/ˇˇ

/ C .e; eI AI x/

(3.83)

for any x 2 H with kxk D 1: The proof follows by Theorem 3.19 applied for the 12 .ˆ  '/-Lipschitzian function f  'Cˆ  e and the 12 .‰  /-Lipschitzian function g  ‰C  e: The 2 2 details are omitted. Similar results can be derived for sequences of operators, however, they will not be presented here.

¨ Type Inequalities 3.7 Quasi-Gruss’ 3.7.1 Introduction In [15], in order to generalize the above result in abstract structures the author has proved the following Gr¨uss’ type inequality in real or complex inner product spaces: Theorem 3.25 (Dragomir, 1999, [15]). Let .H; h:; :i/ be an inner product space over K .K D R,C/ and e 2 H; kek D 1: If ';  ; ˆ;  are real or complex numbers and x; y are vectors in H such that the conditions Re hˆe  x; x  'ei  0 and Re h e  y; y   ei  0

(3.84)

hold, then we have the inequality jhx; yi  hx; ei he; yij  The constant constant.

1 4

1 jˆ  'j  j   j : 4

(3.85)

is best possible in the sense that it can not be replaced by a smaller

3.7 Quasi-Gr¨uss’ Type Inequalities

95

For other results of this type, see the recent monograph [26] and the references therein. Let U be a selfadjoint operator on the complex Hilbert space .H; h:; :i/ with the spectrum Sp .U / included in the interval Œm; M  for some real numbers m < M and let fE g be its spectral family. Then for any continuous function f W Œm; M  ! C, it is well known that we have the following spectral representation theorem in terms of the Riemann–Stieltjes integral: Z

M

f .U / D

f ./ dE ;

(3.86)

f ./ d hE x; yi

(3.87)

m0

which in terms of vectors can be written as Z

M

hf .U / x; yi D m0

for any x; y 2 H: The function gx;y ./ WD hE x; yi is of bounded variation on the interval Œm; M  and gx;y .m  0/ D 0 and gx;y .M / D hx; yi for any x; y 2 H: It is also well known that gx ./ WD hE x; xi is monotonic non-decreasing and right continuous on Œm; M .

3.7.2 Vector Inequalities In this section, we provide various bounds for the magnitude of the difference hf .A/ x; yi  hx; yi hf .A/ x; xi under different assumptions on the continuous function, the selfadjoint operator A W H ! H and the vectors x; y 2 H with kxk D 1: Theorem 3.26 (Dragomir, 2010, [34]). Let A be a selfadjoint operator in the Hilbert space H with the spectrum Sp .A/  Œm; M  for some real numbers m < M and let fE g be its spectral family. Assume that x; y 2 H; kxk D 1 are such that there exists  ;  2 C with either Re h x  y; y   xi  0 or, equivalently

  y   C   2

  1 x   2 j  j :

(3.88)

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

96

1. If f W Œm; M  ! C is a continuous function of bounded variation on Œm; M , then we have the inequality jhf .A/ x; yi  hx; yi hf .A/ x; xij  max jhE x; yi  hE x; xi hx; yij 2Œm;M 

M _

1=2

 max .hE x; xi h.1H  E / x; xi/ 2Œm;M 

M

1=2 _ 2 2 .f / kyk  jhy; xij m

1=2 _ _ 1 1 .f /  j   j .f / : kyk2  jhy; xij2 2 4 m m M



.f /

m

M

(3.89)

2. If f W Œm; M  ! C is a Lipschitzian function with the constant L > 0 on Œm; M , then we have the inequality jhf .A/ x; yi  hx; yi hf .A/ x; xij Z M L jhE x; yi  hE x; xi hx; yij d m0

1=2 Z 2 2  L kyk  jhy; xij

M

.hE x; xi h.1H  E / x; xi/1=2 d

m0

1=2  L kyk2  jhy; xij2 h.M1H  A/ x; xi1=2 h.A  m1H / x; xi1=2 



1=2 1 1 .M  m/ L kyk2  jhy; xij2  j   j .M  m/ L: 2 4

(3.90)

3. If f W Œm; M  ! R is a continuous monotonic non-decreasing function on Œm; M , then we have the inequality jhf .A/ x; yi  hx; yi hf .A/ x; xij Z M  jhE x; yi  hE x; xi hx; yij df ./ m0



 kyk2  jhy; xij2

1=2 Z

1=2  kyk2  jhy; xij2

M

.hE x; xi h.1H  E / x; xi/1=2 df ./

m0

 h.f .M / 1H  f .A// x; xi1=2 h.f .A/  f .m/ 1H / x; xi1=2

1=2 1 1  Œf .M /  f .m/ kyk2  jhy; xij2  j   j Œf .M /  f .m/: 2 4 (3.91)

3.7 Quasi-Gr¨uss’ Type Inequalities

97

Proof. First of all, we notice that by the Schwarz inequality in H we have for any u; v; e 2 H with kek D 1 that

1=2

1=2 : jhu; vi  hu; ei he; vij  kuk2  jhu; eij2 kvk2  jhv; eij2

(3.92)

Now on utilizing (3.92), we can state that jhE x; yi  hE x; xi hx; yij

1=2

1=2  kE xk2  jhE x; xij2 kyk2  jhy; xij2

(3.93)

for any  2 Œm; M  : Since E are projections and E  0, then kE xk2  jhE x; xij2 D hE x; xi  hE x; xi2 D hE x; xi h.1H  E / x; xi 

1 4

(3.94)

for any  2 Œm; M  and x 2 H with kxk D 1: Also, by making use of the Gr¨uss’ type inequality in inner product spaces obtained by the author in [15] we have

1=2 1 (3.95)  j  j : kyk2  jhy; xij2 2 Combining the relations (3.93)–(3.95) we deduce the following inequality that is of interest in itself: jhE x; yi  hE x; xi hx; yij

1=2  .hE x; xi h.1H  E / x; xi/1=2 kyk2  jhy; xij2 

1=2 1 1  j   j kyk2  jhy; xij2 2 4

(3.96)

for any  2 Œm; M  : It is well known that if p W Œa; b ! C is a continuous function, v W Œa; b ! C is Rb of bounded variation then the Riemann–Stieltjes integral a p .t/ dv .t/ exists and the following inequality holds: ˇZ ˇ b ˇ b ˇ _ ˇ ˇ p .t/ dv .t/  max .v/ ; .t/j jp ˇ ˇ ˇ a ˇ t 2Œa;b a where

_b a

.v/ denotes the total variation of v on Œa; b :

(3.97)

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

98

Utilizing this property of the Riemann–Stieltjes integral and inequality (3.96) we have ˇZ ˇ ˇ ˇ

M m0

ˇ ˇ ŒhE x; yi  hE x; xi hx; yi df ./ˇˇ

 max jhE x; yi  hE x; xi hx; yij 2Œm;M 

M _

.f /

m

M

1=2 _  max .hE x; xi h.1H  E / x; xi/1=2 kyk2  jhy; xij2 .f / 2Œm;M 



1 2

m

kyk2  jhy; xij2

M

1=2 _ m

.f / 

1 j   j 4

M _

.f /

(3.98)

m

for x and y as in the assumptions of the theorem. Now, integrating by parts in the Riemann–Stieltjes integral and making use of the spectral representation theorem we have Z

M

ŒhE x; yi  hE x; xi hx; yi df ./ m0

D ŒhE x; yi  hE x; xi hx; yi f ./jM m0 Z M  f ./ d ŒhE x; yi  hE x; xi hx; yi m0

Z

D hx; yi

M m0

Z f ./ d hE x; xi 

M

f ./ d hE x; yi m0

D hx; yi hf .A/ x; xi  hf .A/ x; yi

(3.99)

which together with (3.98) produces the desired result (3.89). Now, recall that if p W Œa; b ! C is a Riemann integrable function and v W Œa; b ! C is Lipschitzian with the constant L > 0, i.e. jf .s/  f .t/j  L js  tj for any t; s 2 Œa; b ; Rb then the Riemann–Stieltjes integral a p .t/ dv .t/ exists and the following inequality holds: ˇZ ˇ Z b ˇ b ˇ ˇ ˇ p .t/ dv .t/ˇ  L jp .t/j dt: ˇ ˇ a ˇ a

3.7 Quasi-Gr¨uss’ Type Inequalities

99

Now, on applying this property of the Riemann–Stieltjes integral we have from (3.96) that ˇZ ˇ ˇ ˇ

ˇ ˇ ŒhE x; yi  hE x; xi hx; yi df ./ˇˇ m0 Z M L jhE x; yi  hE x; xi hx; yij d M

m0

1=2 Z  L kyk2  jhy; xij2

M

.hE x; xi h.1H  E / x; xi/1=2 d: (3.100)

m0

If we use the Cauchy–Bunyakovsky–Schwarz integral inequality and the spectral representation theorem, we have successively Z

M

.hE x; xi h.1H  E / x; xi/1=2 d

m0

Z

M



1=2 Z hE x; xi d

m0

1=2

M

h.1H  E / x; xi d m0

 Z D hE x; xi jM  m0 

1=2

M

d hE x; xi m0

 h.1H  E / x; xi jM m0  D h.M1H  A/ x; xi

1=2

Z



M

d h.1H  E / x; xi m0

h.A  m1H / x; xi1=2 :

(3.101)

On utilizing (3.101), (3.100) and (3.99) we deduce the first three inequalities in (3.90). The fourth inequality follows from the fact that h.M1H  A/ x; xi h.A  m1H / x; xi 

1 1 Œh.M1H  A/ x; xi C h.A  m1H / x; xi2 D .M  m/2 : 4 4

The last part follows from (3.95). Further, from the theory of Riemann–Stieltjes integral it is also well known that if p W Œa; b ! C is of bounded variation and v W Œa; b ! R is continuous and Rb monotonic non-decreasing, then the Riemann–Stieltjes integrals a p .t/ dv .t/ and Rb a jp .t/j dv .t/ exist and ˇ Z ˇZ ˇ ˇ b b ˇ ˇ p .t/ dv .t/ˇ  jp .t/j dv .t/ : ˇ ˇ ˇ a a

(3.102)

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

100

Utilizing this property and inequality (3.96), we have successively ˇZ ˇ ˇ ˇ

ˇ ˇ ŒhE x; yi  hE x; xi hx; yi df ./ˇˇ m0 Z M  jhE x; yi  hE x; xi hx; yij df ./ M

m0

1=2 Z  kyk2  jhy; xij2

M

.hE x; xi h.1H  E / x; xi/1=2 df ./ :

m0

(3.103) Applying the Cauchy–Bunyakovsky–Schwarz integral inequality for the Riemann– Stieltjes integral with monotonic integrators and the spectral representation theorem, we have Z

M

.hE x; xi h.1H  E / x; xi/1=2 df ./

m0

Z

1=2 Z

M



M

hE x; xi df ./ m0

1=2 h.1H  E / x; xi df ./

m0



Z

D hE x; xi f

./jM m0

1=2

M



f ./ d hE x; xi m0



Z

 h.1H  E / x; xi f

./jM m0

1=2

M



f ./ d h.1H  E / x; xi m0

D h.f .M / 1H  f .A// x; xi1=2 h.f .A/  f .m/ 1H / x; xi1=2 

1 Œf .M /  f .m/ 2

(3.104) 

and the proof is complete.

Remark 3.27. If we drop the conditions on x; y; we can obtain from inequalities (3.89)–(3.90) the following results that can be easily applied for particular functions: 1. If f W Œm; M  ! C is a continuous function of bounded variation on Œm; M , then we have the inequality ˇ ˇ ˇ ˇ ˇhf .A/ x; yi kxk2  hx; yi hf .A/ x; xiˇ

1=2 _ 1 .f / kxk2 kyk2 kxk2  jhy; xij2 2 m M



for any x; y 2 H; x ¤ 0:

(3.105)

3.7 Quasi-Gr¨uss’ Type Inequalities

101

2. If f W Œm; M  ! C is a Lipschitzian function with the constant L > 0 on Œm; M , then we have the inequality ˇ ˇ ˇ ˇ 2 ˇhf .A/ x; yi kxk  hx; yi hf .A/ x; xiˇ

1=2  L kyk2 kxk2  jhy; xij2  Œh.M1H  A/ x; xi h.A  m1H / x; xi1=2

1=2 1  .M  m/ L kxk2 kyk2 kxk2  jhy; xij2 2

(3.106)

for any x; y 2 H; x ¤ 0: 3. If f W Œm; M  ! R is a continuous monotonic non-decreasing function on Œm; M , then we have the inequality ˇ ˇ ˇ ˇ 2 ˇhf .A/ x; yi kxk  hx; yi hf .A/ x; xiˇ

1=2  kyk2 kxk2  jhy; xij2  Œh.f .M / 1H  f .A// x; xi h.f .A/  f .m/ 1H / x; xi1=2

1=2 1  Œf .M /  f .m/ kxk2 kyk2 kxk2  jhy; xij2 (3.107) 2 for any x; y 2 H; x ¤ 0: We are able now to provide the following corollary: Corollary 3.28 (Dragomir, 2010, [34]). With the assumptions of Theorem 3.26 and if f W Œm; M  ! R is a .'; ˆ/-Lipschitzian function, then we have jhf .A/ x; yi  hx; yi hf .A/ x; xij Z M 1  .ˆ  '/ jhE x; yi  hE x; xi hx; yij d 2 m0

1=2 Z M 1 2 2 .hE x; xi h.1H  E / x; xi/1=2 d  .ˆ  '/ kyk  jhy; xij 2 m0

1=2 1  .ˆ  '/ kyk2  jhy; xij2 2  h.M1H  A/ x; xi1=2 h.A  m1H / x; xi1=2

1=2 1  .M  m/ .ˆ  '/ kyk2  jhy; xij2 4 1  j  j .M  m/ .ˆ  '/ : 8

(3.108)

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

102

The proof follows from the second part of Theorem 3.26 applied for the .ˆ  '/-Lipschitzian function f  ˆC' 2 e by performing the required calculations in the first term of the inequality. The details are omitted. 1 2

3.7.3 Applications for Gruss’ ¨ Type Inequalities The following result provides some Gr¨uss’ type inequalities for two functions of two selfadjoint operators: Proposition 3.29 (Dragomir, 2010, [34]). Let A; B be two selfadjoint operators in the Hilbert space H with the spectra Sp .A/ ; Sp .B/  Œm; M  for some real numbers m < M and let fE g be the spectral family of A. Assume that g W Œm; M  ! R is a continuous function and denote n WD mint 2Œm;M  g .t/ and N WD maxt 2Œm;M  g .t/ : 1. If f W Œm; M  ! C is a continuous function of bounded variation on Œm; M , then we have the inequality jhf .A/ x; g .B/ xi  hf .A/ x; xi hg .B/ x; xij  max jhE x; g .B/ xi  hE x; xi hx; g .B/ xij

M _

2Œm;M 

.f /

m

 max .hE x; xi h.1H  E / x; xi/1=2 2Œm;M 

M

1=2 _  kg .B/ xk2  jhg .B/ x; xij2 .f / m

1=2 _ _ 1 1 .f /  .N  n/ .f / kg .B/ xk2  jhg .B/ x; xij2 2 4 m m (3.109) M

M



for any x 2 H; kxk D 1: 2. If f W Œm; M  ! C is a Lipschitzian function with the constant L > 0 on Œm; M , then we have the inequality jhf .A/ x; g .B/ xi  hf .A/ x; xi hg .B/ x; xij Z M L jhE x; g .B/ xi  hE x; xi hx; g .B/ xij d m0



 L kg .B/ xk2  jhg .B/ x; xij2 Z 

M

m0

1=2

.hE x; xi h.1H  E / x; xi/1=2 d

3.7 Quasi-Gr¨uss’ Type Inequalities

103

1=2  L kg .B/ xk2  jhg .B/ x; xij2  h.M1H  A/ x; xi1=2 h.A  m1H / x; xi1=2

1=2 1  .M  m/ L kg .B/ xk2  jhg .B/ x; xij2 2 1  .N  n/ .M  m/ L 4

(3.110)

for any x 2 H; kxk D 1: 3. If f W Œm; M  ! R is a continuous monotonic non-decreasing function on Œm; M , then we have the inequality jhf .A/ x; g .B/ xi  hf .A/ x; xi hg .B/ x; xij Z M  jhE x; g .B/ xi  hE x; xi hx; g .B/ xij df ./ m0



 kg .B/ xk2  jhg .B/ x; xij2 Z 

M

1=2

.hE x; xi h.1H  E / x; xi/1=2 df ./

m0

 kg .B/ xk2  jhg .B/ x; xij2

1=2

 h.f .M / 1H  f .A// x; xi1=2 h.f .A/  f .m/ 1H / x; xi1=2

1=2 1  Œf .M /  f .m/ kg .B/ xk2  jhg .B/ x; xij2 2 1 (3.111)  .N  n/ Œf .M /  f .m/ 4 for any x 2 H; kxk D 1: Proof. We notice that, since n WD mint 2Œm;M  g .t/ and N WD maxt 2Œm;M  g .t/ ; then n  hg .B/ x; xi  N which implies that hg .B/ x  nx; M x  g .B/ xi  0 for any x 2 H; kxk D 1: On applying Theorem 3.26 for y D Bx;  D N and  D n we deduce the desired result.  Remark 3.30. We observe that if the function f takes real values and is a .'; ˆ/Lipschitzian function on Œm; M , then inequality (3.110) can be improved as follows: jhf .A/ x; g .B/ xi  hf .A/ x; xi hg .B/ x; xij Z M 1  .ˆ  '/ jhE x; g .B/ xi  hE x; xi hx; g .B/ xij d 2 m0

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

104





1=2 1 .ˆ  '/ kg .B/ xk2  jhg .B/ x; xij2 2 Z M  .hE x; xi h.1H  E / x; xi/1=2 d m0





1=2 1 .ˆ  '/ kg .B/ xk2  jhg .B/ x; xij2 2

 h.M1H  A/ x; xi1=2 h.A  m1H / x; xi1=2

1=2 1  .M  m/ .ˆ  '/ kg .B/ xk2  jhg .B/ x; xij2 4 1  .N  n/ .M  m/ .ˆ  '/ 8

(3.112)

for any x 2 H; kxk D 1:

¨ Type Inequalities 3.8 Two Operators Gruss’ 3.8.1 Some Representation Results We start with the following representation result that will play a key role in obtaining various bounds for different choices of functions including continuous functions of bounded variation, Lipschitzian functions or monotonic and continuous functions. Theorem 3.31 (Dragomir, 2010, [35]). Let A; B be two selfadjoint operators in the Hilbert space H with the spectra Sp .A/ ; Sp .B/  Œm;˚M  for some real numbers m < M and let fE g be the spectral family of A and F  the spectral family of B: If f; g W Œm; M  ! C are continuous, then we have the representation hf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xi  Z M Z M

˝ ˛ ˝ ˛ D hE x; xi x; F x  E x; F x d .g .// d .f .// m0

m0

(3.113) for any x 2 H with kxk D 1: Proof. Integrating by parts in the Riemann–Stieltjes integral and making use of the spectral representation theorem we have Z

M

ŒhE x; yi  hE x; xi hx; yi df ./ m0

3.8 Two Operators Gr¨uss’ Type Inequalities

105

D ŒhE x; yi  hE x; xi hx; yi f ./jM m0 Z M  f ./ d ŒhE x; yi  hE x; xi hx; yi m0

Z

D hx; yi

M

Z

M

f ./ d hE x; xi 

f ./ d hE x; yi

m0

m0

D hx; yi hf .A/ x; xi  hf .A/ x; yi

(3.114)

for any x; y 2 H with kxk D 1: Now, if we chose y D g .B/ x in (3.114) then we get that Z

M

ŒhE x; g .B/ xi  hE x; xi hx; g .B/ xi df ./ m0

D hx; g .B/ xi hf .A/ x; xi  hf .A/ x; g .B/ xi

(3.115)

for any x 2 H with kxk D 1: Utilizing the spectral representation theorem for B we also have for each fixed  2 Œm; M  that hE x; g .B/ xi  hE x; xi hx; g .B/ xi   Z  Z M g ./ dF x  hE x; xi x; D E x; Z

m0

˝ ˛ g ./ d E x; F x  hE x; xi

M

D

g ./ dF x m0

Z

m0



M

M

˝ ˛ g ./ d x; F x

(3.116)

m0

for any x 2 H with kxk D 1: Integrating by parts in the Riemann–Stieltjes integral we have Z

M

˝ ˛ g ./ d E x; F x

m0

˛M ˝ D g ./ E x; F x m0  Z

˝

M

D g .M / hE x; xi 

Z

M

˝

˛ E x; F x dg ./

m0

˛ E x; F x d .g .//

m0

and Z

M

˝

g ./ d x; F x

˛

˝

D g ./ x; F x

m0

Z D g .M / 

˛M

M m0

Z

 m0

˝

M

˝ ˛ x; F x d .g .//

m0

˛

x; F x d .g .// ;

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

106

therefore Z

M

˝ ˛ g ./ d E x; F x  hE x; xi

m0

Z

˝ ˛ E x; F x d .g .//

m0

 Z  hE x; xi g .M /  D hE x; xi Z D

˝

 x; F x d .g .//

˝ ˛ x; F x d .g .// 

m0 M

M m0

M

˝ ˛ g ./ d x; F x

M m0

M

D g .M / hE x; xi 

Z

Z

˛

Z

M

˝ ˛ E x; F x d .g .//

m0

˝ ˛ ˝ ˛ hE x; xi x; F x  E x; F x d .g .//

(3.117)

m0

for any x 2 H with kxk D 1 and  2 Œm; M  : Utilizing (3.115)–(3.117) we deduce the desired result (3.113).



Remark 3.32. In particular, if we take B D A; then we get from (3.113) the equality hf .A/ x; g .A/ xi  hf .A/ x; xi hx; g .A/ xi  Z M Z M

˝ ˛ ˝ ˛ D hE x; xi x; E x  E x; E x d .g .// d .f .// m0

m0

(3.118) for any x 2 H with kxk D 1; which for g D f produces the representation result for the variance of the selfadjoint operator f .A/ ; kf .A/ xk2  hf .A/ x; xi2  Z M Z M

˝ ˛ ˝ ˛ D hE x; xi x; E x  E x; E x d .f .// d .f .// m0

m0

(3.119) for any x 2 H with kxk D 1:

3.8.2 Bounds for f of Bounded Variation The first vectorial Gr¨uss’ type inequality when one of the functions is of bounded variation is as follows: Theorem 3.33 (Dragomir, 2010, [35]). Let A; B be two selfadjoint operators in the Hilbert space H with the spectra Sp .A/ ; Sp .B/  Œm; M  for some real

3.8 Two Operators Gr¨uss’ Type Inequalities

107

˚  numbers m < M and let fE g be the spectral family of A and F  the spectral family of B: Also, assume that f W Œm; M  ! C is continuous and of bounded variation on Œm; M . 1. If g W Œm; M  ! C is continuous and of bounded variation on Œm; M  ; then we have the inequality jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij 

max

.;/2Œm;M 2

M M _ ˇ ˝ ˛ ˝ ˛ˇ _ ˇhE x; xi x; F x  E x; F x ˇ .g/ .f / m

m

 max ŒhE x; xi h.1H  E / x; xi1=2 2Œm;M 

 max

2Œm;M 

m

m

_ 1_ .g/ .f / 4 m m M



M M _

˝  ˛1=2 _ ˛ ˝ F x; x 1H  F x; x .g/ .f /

M

(3.120)

for any x 2 H with kxk D 1: 2. If g W Œm; M  ! C is Lipschitzian with the constant K > 0 on Œm; M  ; then we have the inequality jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij Z M _ M ˇ ˝ ˛ ˝ ˛ˇ ˇ ˇ  K max .f / hE x; xi x; F x  E x; F x d 2Œm;M 

K

M _

Z

m0

m

.f / max ŒhE x; xi h.1H  E / x; xi1=2 2Œm;M 

m M



˝

F x; x

˛ ˝

 ˛1=2 1H  F x; x d

m0



1 _ K .f / h.M1H  B/ x; xi1=2 h.B  m1H / x; xi1=2 2 m



_ 1 K .M  m/ .f / 4 m

M

M

(3.121)

for any x 2 H with kxk D 1: 3. If g W Œm; M  ! R is continuous and monotonic non-decreasing on Œm; M  ; then we have the inequality jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

108

Z

_ M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dg ./ .f /

M

 max

2Œm;M 



M _

m0

m

.f / max ŒhE x; xi h.1H  E / x; xi1=2 2Œm;M 

m

Z

M



˝  ˛1=2 ˛ ˝ F x; x 1H  F x; x dg ./

m0



1 2

M _

.f / h.g .M / 1H  g .B// x; xi1=2 h.g .B/  g .m/ 1H / x; xi1=2

m

_ 1 Œg .M /  g .m/ .f / 4 m M



(3.122)

for any x 2 H with kxk D 1: Proof. 1. It is well known that if p W Œa; b ! C is a continuous function, v W Œa; b ! C is of bounded variation, then the Riemann–Stieltjes integral Rb a p .t/ dv .t/ exists and the following inequality holds: ˇ ˇZ b ˇ ˇ b _ ˇ ˇ p .t/ dv .t/ˇ  max jp .t/j .v/ ; ˇ ˇ t 2Œa;b ˇ a

(3.123)

a

where

b _

.v/ denotes the total variation of v on Œa; b :

a

Now, on utilizing property (3.123) and identity (3.113) we have jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij ˇZ M ˇM ˇ ˇ_

˝ ˛ ˝ ˛ .f /  max ˇˇ hE x; xi x; F x  E x; F x d .g .//ˇˇ 2Œm;M 

m0

m

(3.124) for any x 2 Œm; M  : The same inequality (3.123) produces the bound ˇZ ˇ max ˇˇ 2Œm;M 

M m0

  max

2Œm;M 

ˇ ˇ ˝ ˛ ˝ ˛ hE x; xi x; F x  E x; F x d .g .//ˇˇ M ˇ ˝ ˛ ˝ ˛ˇ _ ˇ ˇ max hE x; xi x; F x  E x; F x .f /

2Œm;M 

m

3.8 Two Operators Gr¨uss’ Type Inequalities

D

max

.;/2Œm;M 2

109

M ˇ ˝ ˛ ˝ ˛ˇ _ ˇhE x; xi x; F x  E x; F x ˇ .f /

(3.125)

m

for any x 2 Œm; M  : By making use of (3.124) and (3.125) we deduce the first part of (3.120). Further, we notice that by the Schwarz inequality in H we have for any u; v; e 2 H with kek D 1 that

1=2

1=2 : (3.126) jhu; vi  hu; ei he; vij  kuk2  jhu; eij2 kvk2  jhv; eij2 Indeed, if we write Schwarz’s inequality for the vectors uhu; ei e and vhv; ei e we have jhu  hu; ei e; v  hv; ei eij  ku  hu; ei ek kv  hv; ei ek which, by performing the calculations, is equivalent with (3.126). Now, on utilizing (3.126), we can state that ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ

1=2   ˇ˝ ˛ˇ 1=2 F x 2  ˇ F x; x ˇ2  kE xk2  jhE x; xij2

(3.127)

for any ;  2 Œm; M  : Since E and F are projections and E ; F  0 then kE xk2  jhE x; xij2 D hE x; xi  hE x; xi2 1 4

(3.128)

 ˇ˝   ˛ ˛ˇ ˝ ˛ ˝ F x 2  ˇ F x; x ˇ2 D F x; x 1H  F x; x  1 4

(3.129)

D hE x; xi h.1H  E / x; xi  and

for any ;  2 Œm; M  and x 2 H with kxk D 1: Now, if we use (3.127)–(3.129) then we get the second part of (3.120). 2. Further, recall that if p W Œa; b ! C is a Riemann integrable function and v W Œa; b ! C is Lipschitzian with the constant L > 0, i.e. jf .s/  f .t/j  L js  tj for any t; s 2 Œa; b; Rb then the Riemann–Stieltjes integral a p .t/ dv .t/ exists and the following inequality holds: ˇZ ˇ Z b ˇ b ˇ ˇ ˇ p .t/ dv .t/ˇ  L (3.130) jp .t/j dt: ˇ ˇ a ˇ a

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

110

If we use inequality (3.130), then we have in the case when g is Lipschitzian with the constant K > 0 that ˇZ M ˇ ˇ ˇ

˝ ˛ ˝ ˛ ˇ max ˇ hE x; xi x; F x  E x; F x d .g .//ˇˇ 2Œm;M 

m0

Z

M

 K max

2Œm;M 

ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ d

(3.131)

m0

for any x 2 H with kxk D 1 and the first part of (3.121) is proved. Further, by employing (3.127)–(3.129) we also get that Z

M

max

2Œm;M  m0

ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ d

 max ŒhE x; xi h.1H  E / x; xi1=2 2Œm;M 

Z

M



˝

F x; x

˛ ˝

 ˛1=2 1H  F x; x d

(3.132)

m0

for any x 2 H with kxk D 1: If we use the Cauchy–Bunyakovsky–Schwarz integral inequality and the spectral representation theorem, then we have successively Z

M

˝

F x; x

 ˛1=2 ˛ ˝ 1H  F x; x d

m0

Z

M



˝

1=2 Z F x; x d ˛

m0

 D

˝



˝

1=2 ˝  ˛ 1H  F x; x d

m0

˛ ˇM F x; x ˇm0 



M

1H  F



Z

M m0

˛ ˇM x; x ˇ

D h.M1H  B/ x; xi

˝ ˛ d F x; x

 m0

1=2

Z

M

1=2

 ˛ ˝ d 1H  F x; x



m0

h.B  m1H / x; xi1=2

(3.133)

for any x 2 H with kxk D 1: On employing now (3.131)–(3.133) we deduce the second part of (3.121). The last part of (3.121) follows by the elementary inequality ˛ˇ 

1 .˛ C ˇ/2 ; ˛ˇ  0 4

(3.134)

3.8 Two Operators Gr¨uss’ Type Inequalities

111

for the choice ˛ D h.M1H  B/ x; xi and ˇ D h.B  m1H / x; xi and the details are omitted. 3. Further, from the theory of Riemann–Stieltjes integral it is also well known that if p W Œa; b ! C is of bounded variation and v W Œa; b ! R is continuous and Rb monotonic non-decreasing, then the Riemann–Stieltjes integrals a p .t/ dv .t/ Rb and a jp .t/j dv .t/ exist and ˇZ ˇ Z ˇ b ˇ b ˇ ˇ p .t/ dv .t/ˇ  jp .t/j dv .t/ : ˇ ˇ a ˇ a

(3.135)

Now, if we assume that g is monotonic non-decreasing on Œm; M  ; then by (3.135) we have that ˇZ M ˇ ˇ ˇ

˝ ˛ ˝ ˛ max ˇˇ hE x; xi x; F x  E x; F x d .g .//ˇˇ 2Œm;M 

m0

Z

M

 max

2Œm;M 

ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dg ./

(3.136)

m0

for any x 2 H with kxk D 1: Further, by employing (3.127)–(3.129) we also get that Z

M

max

2Œm;M  m0

ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dg ./

 max ŒhE x; xi h.1H  E / x; xi1=2 2Œm;M 

Z

M



˝

F x; x

 ˛1=2 ˛ ˝ 1H  F x; x dg ./

(3.137)

m0

for any x 2 H with kxk D 1: These prove the first part of (3.122). If we use the Cauchy–Bunyakovsky–Schwarz integral inequality for the Riemann–Stieltjes integral with monotonic non-decreasing integrators and the spectral representation theorem, then we have successively Z

M

˝

F x; x

˛ ˝

 ˛1=2 1H  F x; x dg ./

m0

Z

M



1=2 Z ˝ ˛ F x; x dg ./

m0



ˇM ˝ ˛ D F x; x g ./ˇm0 

M

˝

1=2  ˛ 1H  F x; x dg ./

m0

Z

M m0

˝ ˛ g ./ d F x; x

1=2

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

112



ˇM  ˛ ˝  1H  F x; x g ./ˇm0  D h.g .M / 1H  g .B// x; xi

1=2

Z

M

 ˛ ˝ g ./ d 1H  F x; x

1=2

m0

h.g .B/  g .m/ 1H / x; xi1=2

(3.138)

for any x 2 H with kxk D 1: Utilizing (3.138) we then deduce the last part of (3.122). The details are omitted.  Now, in order to provide other results that are similar to the Gr¨uss’ type inequalities stated in the introduction, we can state the following corollary: Corollary 3.34 (Dragomir, 2010, [35]). Let A be a selfadjoint operators in the Hilbert space H with the spectrum Sp .A/  Œm; M  for some real numbers m < M and let fE g be the spectral family of A: Also, assume that f W Œm; M  ! C is continuous and of bounded variation on Œm; M  : 1. If g W Œm; M  ! C is continuous and of bounded variation on Œm; M  ; then we have the inequality jhf .A/ x; g .A/ xi  hf .A/ x; xi hx; g .A/ xij 

max

.;/2Œm;M 2

M M _ ˇ ˝ ˛ ˝ ˛ˇ _ ˇhE x; xi x; E x  E x; E x ˇ .g/ .f / m

 max ŒhE x; xi h.1H  E / x; xi 2Œm;M 

.g/

m

M _

m

.f /

m

_ 1_ .g/ .f / 4 m m M



M _

M

(3.139)

for any x 2 H with kxk D 1: 2. If g W Œm; M  ! C is Lipschitzian with the constant K > 0 on Œm; M  ; then we have the inequality jhf .A/ x; g .A/ xi  hf .A/ x; xi hx; g .A/ xij Z M _ M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; E x  E x; E x ˇ d  K max .f / 2Œm;M 

K

M _

Z

m0

m

.f / max ŒhE x; xi h.1H  E / x; xi1=2 2Œm;M 

m M

 m0

˝

E x; x

˛ ˝

 ˛1=2 1H  E x; x d

3.8 Two Operators Gr¨uss’ Type Inequalities

113



1 _ K .f / h.M1H  A/ x; xi1=2 h.A  m1H / x; xi1=2 2 m



_ 1 K .M  m/ .f / 4 m

M

M

(3.140)

for any x 2 H with kxk D 1: 3. If g W Œm; M  ! R is continuous and monotonic non-decreasing on Œm; M  ; then we have the inequality jhf .A/ x; g .A/ xi  hf .A/ x; xi hx; g .A/ xij Z

M

 max

2Œm;M 



M _

_ M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; E x  E x; E x ˇ dg ./ .f /

m0

m

.f / max ŒhE x; xi h.1H  E / x; xi1=2 2Œm;M 

m

Z

M



˝

E x; x

˛ ˝

 ˛1=2 1H  E x; x dg ./

m0



1 2

M _

.f / h.g .M / 1H  g .A// x; xi1=2 h.g .A/  g .m/ 1H / x; xi1=2

m

_ 1 Œg .M /  g .m/ .f / 4 m M



(3.141)

for any x 2 H with kxk D 1: Remark 3.35. The following inequality for the variance of f .A/ under the assumptions that A is a selfadjoint operators in the Hilbert space H with the spectrum Sp .A/  Œm; M  for some real numbers m < M , fE g is the spectral family of A and f W Œm; M  ! C is continuous and of bounded variation on Œm; M  can be stated 0  kf .A/ xk2  hf .A/ x; xi2 

max

.;/2Œm;M 2

"M #2 ˇ ˝ ˛ ˝ ˛ˇ _ ˇhE x; xi x; E x  E x; E x ˇ .f /

 max ŒhE x; xi h.1H  E / x; xi 2Œm;M 

for any x 2 H with kxk D 1:

m

"M _ m

#2 .f /

"M #2 1 _  .f / (3.142) 4 m

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

114

3.8.3 Bounds for f Lipschitzian The case when the first function is Lipschitzian is as follows: Theorem 3.36 (Dragomir, 2010, [35]). Let A; B be two selfadjoint operators in the Hilbert space H with the spectra Sp .A/ ; Sp .B/  Œm;˚M  for some real numbers m < M and let fE g be the spectral family of A and F  the spectral family of B: Also, assume that f W Œm; M  ! C is Lipschitzian with the constant L > 0 on Œm; M  : 1. If g W Œm; M  ! C is Lipschitzian with the constant K > 0 on Œm; M  ; then we have the inequality jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij Z M Z M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dd  LK Z

m0 M

ŒhE x; xi h.1H  E / x; xi1=2 d

 LK Z

m0

m0 M



˝  ˛1=2 ˛ ˝ F x; x 1H  F x; x d

m0

 LK Œh.M1H  A/ x; xi h.A  m1H / x; xi1=2  Œh.M1H  B/ x; xi h.B  m1H / x; xi1=2 

1 LK .M  m/2 (3.143) 4

for any x 2 H with kxk D 1: 2. If g W Œm; M  ! R is continuous and monotonic non-decreasing on Œm; M  ; then we have the inequality jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij Z M Z M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dg ./ d L Z

m0 M

L Z 

m0

ŒhE x; xi h.1H  E / x; xi1=2 d

m0 M

˝

F x; x

 ˛1=2 ˛ ˝ 1H  F x; x dg ./

m0

 L Œh.M1H  A/ x; xi h.A  m1H / x; xi1=2  Œh.g .M / 1H  g .B// x; xi h.g .B/  g .m/ 1H / x; xi1=2 

1 L .M  m/ Œg .M /  g .m/ 4

(3.144)

3.8 Two Operators Gr¨uss’ Type Inequalities

115

for any x 2 H with kxk D 1: Proof. 1. We observe that, on utilizing property (3.130) and identity (3.113) we have jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij ˇ Z M ˇZ M ˇ ˇ

˝ ˛ ˝ ˛ ˇ L hE x; xi x; F x  E x; F x d .g .//ˇˇ d ˇ m0

(3.145)

m0

for any x 2 H; kxk D 1: By the same property (3.130) we also have ˇZ ˇ ˇ ˇ

ˇ ˇ

˝ ˛ ˝ ˛ hE x; xi x; F x  E x; F x d .g .//ˇˇ

M m0

Z

M

K

ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ d

(3.146)

m0

for any x 2 H; kxk D 1 and  2 Œm; M  : Therefore, by (3.145) and (3.146) we get jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij Z M Z M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dd  LK m0

(3.147)

m0

for any x 2 H; kxk D 1; which proves the first inequality in (3.143). From (3.127)–(3.129) we have ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ

˝  ˛1=2 ˛ ˝  ŒhE x; xi h.1H  E / x; xi1=2 F x; x 1H  F x; x

(3.148)

for any x 2 H; kxk D 1 and ;  2 Œm; M  : Integrating on Œm; M 2 inequality (3.148) and utilizing the Cauchy– Bunyakowsky–Schwarz integral inequality for the Riemann integral we have Z

M m0

Z Z

ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dd

M m0 M

ŒhE x; xi h.1H  E / x; xi1=2 d

 m0

Z



M m0

˝  ˛1=2 ˛ ˝ F x; x 1H  F x; x d

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

116

Z

1=2 Z hE x; xi d

M

 m0

Z

h.1H  E / x; xi d m0

˝

M



1=2

M

1=2 Z ˛ F x; x d

m0

M

1=2 ˝  ˛ : 1H  F x; x d

(3.149)

m0

Integrating by parts and utilizing the spectral representation theorem we have Z M Z M M d hE x; xi hE x; xi d D hE x; xi jm0  m0

m0

D M  hAx; xi D h.M1H  A/ x; xi; Z M h.1H  E / x; xi d D h.A  m1H / x; xi m0

and the similar equalities for B; providing the second part of (3.143). The last part follows from (3.134) and we omit the details. 2. Utilizing inequality (3.135) we have ˇZ M ˇ ˇ ˇ

˝ ˛ ˝ ˛ ˇ ˇ x; xi x; F x  E x; F x d .g .// hE     ˇ ˇ m0

Z

M



ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dg ./

(3.150)

m0

which, together with (3.145), produces the inequality jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij Z M Z M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dg ./ d L m0

(3.151)

m0

for any x 2 H; kxk D 1: Now, by utilizing (3.148) and a similar argument to the one outlined above, we deduce the desired result (3.144) and the details are omitted.  The case of one operator is incorporated in Corollary 3.37 (Dragomir, 2010, [35]). Let A be a selfadjoint operators in the Hilbert space H with the spectrum Sp .A/  Œm; M  for some real numbers m < M and let fE g be the spectral family of A: Also, assume that f W Œm; M  ! C is Lipschitzian with the constant L > 0 on Œm; M  : 1. If g W Œm; M  ! C is Lipschitzian with the constant K > 0 on Œm; M  ; then we have the inequality jhf .A/ x; g .A/ xi  hf .A/ x; xi hx; g .A/ xij Z M Z M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; E x  E x; E x ˇ dd  LK m0

m0

3.8 Two Operators Gr¨uss’ Type Inequalities

Z

M

 LK

117

2 ŒhE x; xi h.1H  E / x; xi1=2 d

m0

 LK Œh.M1H  A/ x; xi h.A  m1H / x; xi 

1 LK .M  m/2 (3.152) 4

for any x 2 H with kxk D 1: 2. If g W Œm; M  ! R is continuous and monotonic non-decreasing on Œm; M  ; then we have the inequality jhf .A/ x; g .A/ xi  hf .A/ x; xi hx; g .A/ xij Z M Z M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dg ./ d L Z

m0 M

ŒhE x; xi h.1H  E / x; xi1=2 d

L Z

m0

m0

˝

M



E x; x

˛ ˝

 ˛1=2 1H  E x; x dg ./

m0

 L Œh.M1H  A/ x; xi h.A  m1H / x; xi1=2  Œh.g .M / 1H  g .A// x; xi h.g .A/  g .m/ 1H / x; xi1=2 

1 L .M  m/ Œg .M /  g .m/ 4

(3.153)

for any x 2 H with kxk D 1: Remark 3.38. The following inequality for the variance of f .A/ under the assumptions that A is a selfadjoint operators in the Hilbert space H with the spectrum Sp .A/  Œm; M  for some real numbers m < M , fE g is the spectral family of A and f W Œm; M  ! C is Lipschitzian with the constant L > 0 on Œm; M  can be stated 0  kf .A/ xk2  hf .A/ x; xi2 Z M Z M ˇ ˝ ˛ ˝ ˛ˇ 2 ˇhE x; xi x; E x  E x; E x ˇ dd L m0

Z

M

 L2

m0

2 ŒhE x; xi h.1H  E / x; xi1=2 d

m0

 L Œh.M1H  A/ x; xi h.A  m1H / x; xi 2



1 2 L .M  m/2 4

for any x 2 H with kxk D 1:

(3.154)

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

118

3.8.4 Bounds for f Monotonic Non-decreasing Finally, for the case of two monotonic functions we have the following result as well: Theorem 3.39 (Dragomir, 2010, [35]). Let A; B be two selfadjoint operators in the Hilbert space H with the spectra Sp .A/ ; Sp .B/  Œm;˚M  for some real numbers m < M and let fE g be the spectral family of A and F  the spectral family of B: If f; g W Œm; M  ! C are continuous and monotonic non-decreasing on Œm; M  ; then jhf .A/ x; g .B/ xi  hf .A/ x; xi hx; g .B/ xij Z M Z M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; F x  E x; F x ˇ dg ./ df ./  Z

m0 M



m0

ŒhE x; xi h.1H  E / x; xi1=2 df ./

m0

Z

˝  ˛1=2 ˛ ˝ F x; x 1H  F x; x dg ./

M

 m0

 Œh.f .M / 1H  f .A// x; xi h.f .A/  f .m/ 1H / x; xi1=2  Œh.g .M / 1H  g .B// x; xi h.g .B/  g .m/ 1H / x; xi1=2 

1 Œf .M /  f .m/ Œg .M /  g .m/ 4

(3.155)

for any x 2 H; kxk D 1: The details of the proof are omitted. In particular we have: Corollary 3.40 (Dragomir, 2010, [35]). Let A be a selfadjoint operators in the Hilbert space H with the spectrum Sp .A/  Œm; M  for some real numbers m < M and let fE g be the spectral family of A: If f; g W Œm; M  ! C are continuous and monotonic non-decreasing on Œm; M  ; then jhf .A/ x; g .A/ xi  hf .A/ x; xi hx; g .A/ xij Z M Z M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; E x  E x; E x ˇ dg ./ df ./  Z

m0 M

m0

ŒhE x; xi h.1H  E / x; xi1=2 df ./

 m0

Z



M m0

˝

E x; x

 ˛1=2 ˛ ˝ 1H  E x; x dg ./

References

119

 Œh.f .M / 1H  f .A// x; xi h.f .A/  f .m/ 1H / x; xi1=2  Œh.g .M / 1H  g .A// x; xi h.g .A/  g .m/ 1H / x; xi1=2 

1 Œf .M /  f .m/ Œg .M /  g .m/ 4

(3.156)

for any x 2 H; kxk D 1: In particular, the following inequality for the variance of f .A/ in the case of monotonic non-decreasing functions f holds: 0  kf .A/ xk2  hf .A/ x; xi2 Z M Z M ˇ ˝ ˛ ˝ ˛ˇ ˇhE x; xi x; E x  E x; E x ˇ df ./ df ./  m0

Z

M



m0

2

ŒhE x; xi h.1H  E / x; xi1=2 df ./

m0

 Œh.f .M / 1H  f .A// x; xi h.f .A/  f .m/ 1H / x; xi 

1 Œf .M /  f .m/2 4

(3.157)

for any x 2 H; kxk D 1:

References 1. G.A. Anastassiou, Gr¨uss type inequalities for the Stieltjes integral. Nonlinear Funct. Anal. Appl. 12 (2007), no. 4, 583–593. 2. G. A. Anastassiou, Chebyshev-Gr¨uss type and comparison of integral means inequalities for the Stieltjes integral. Panamer. Math. J. 17 (2007), no. 3, 91–109. 3. G. A. Anastassiou, Chebyshev-Gr¨uss type inequalities via Euler type and Fink identities. Math. Comput. Modelling 45 (2007), no. 9-10, 1189–1200 4. G. A. Anastassiou, Chebyshev-Gr¨uss type inequalities via Euler type and Fink identities. Math. Comput. Modelling 45 (2007), no. 9-10, 1189–1200 5. M. Biernacki, Sur une in´egalit´e entre les int´egrales due a` Tchebyscheff. Ann. Univ. Mariae Curie-Sklodowska (Poland), A5(1951), 23–29. ˇ 6. K. Boukerrioua, and A. Guezane-Lakoud, On generalization of Cebyˇ sev type inequalities. J. Inequal. Pure Appl. Math. 8 (2007), no. 2, Article 55, 4 pp. ˇ 7. P.L. Cebyˇ sev, O pribliˇzennyh vyraˇzenijah odnih integralov cˇ erez drugie. Soobˇsc´ enija i protokoly zasedani˘ı Matemmatiˇceskogo obˇcestva pri Imperatorskom Har’kovskom Universitete ˇ seva. Moskva–Leningrad, 1948a, (1882), No. 2, 93–98; Polnoe sobranie soˇcineni˘ı P. L. Cebyˇ 128–131. ˇ 8. P.L. Cebyˇ sev, Ob odnom rjade, dostavljajuˇsc´ em predel’nye veliˇciny integralov pri razloˇzenii podintegral’no˘ı funkcii na mnoˇzeteli. Priloˇzeni k 57 tomu Zapisok Imp. Akad. Nauk, No. 4; ˇ seva. Moskva–Leningrad, 1948b, (1883),157–169. Polnoe sobranie soˇcineni˘ı P. L. Cebyˇ ˇ 9. P. Cerone, On some results involving the Cebyˇ sev functional and its generalisations. J. Inequal. Pure Appl. Math. 4 (2003), no. 3, Article 55, 17 pp.

120

ˇ 3 Inequalities of the Cebyˇ sev and Gr¨uss Type

10. P. Cerone, On Chebyshev functional bounds. Differential & difference equations and applications, 267–277, Hindawi Publ. Corp., New York, 2006. ˇ 11. P. Cerone, On a Cebyˇ sev-type functional and Gr¨uss-like bounds. Math. Inequal. Appl. 9 (2006), no. 1, 87–102. 12. P. Cerone and S. S. Dragomir, A refinement of the Gr¨uss inequality and applications. Tamkang J. Math. 38 (2007), no. 1, 37–49. 13. P. Cerone and S. S. Dragomir, Chebychev functional bounds using Ostrowski seminorms. Southeast Asian Bull. Math. 28 (2004), no. 2, 219–228. 14. S.S. Dragomir, Gr¨uss inequality in inner product spaces, The Australian Math Soc. Gazette, 26 (1999), No. 2, 66–70. 15. S.S. Dragomir, A generalization of Gr¨uss’ inequality in inner product spaces and applications, J. Math. Anal. Appl., 237 (1999), 74–82. 16. S.S. Dragomir, Some discrete inequalities of Gr¨uss type and applications in guessing theory, Honam Math. J., 21(1) (1999), 145–156. 17. S.S. Dragomir, Some integral inequalities of Gr¨uss type, Indian J. of Pure and Appl. Math., 31(4) (2000), 397–415. ˇ 18. S.S. Dragomir, On some improvements of Cebyˇ sev’s inequality for sequences and integrals, Studia Univ. Babes¸-Bolyai, Mathematica (Romania), XXXV (4)(1990), 35–40. ˇ 19. S.S. Dragomir, Some improvement of Cebyˇ sev’s inequality for isotonic functionals, Atti. Sem. Mat. Fis. Univ. Modena (Italy), 41 (1993), 473–481. 20. S.S. Dragomir, A Gr¨uss type integral inequality for mappings of r-H¨older’s type and applications for trapezoid formula, Tamkang J. of Math., 31(1) (2000), 43–47. Rb 21. S.S. Dragomir, On the Ostrowski inequality for the Riemann-Stieltjes integral a f .t / du .t /, where f is of H¨older type and u is of bounded variation and applications, J. KSIAM, 5(2001), No. 1, 35–45. 22. S.S. Dragomir, Some Gr¨uss type inequalities in inner product spaces, J. Inequal. Pure & Appl. Math., 4(2) (2003), Article 42. (Onlinehttp://jipam.vu.edu.au/article.php?sid$=$280). 23. S.S. Dragomir, Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces, J. Inequal. Pure & Appl. Math., 5(3) (2004), Article 76. (Online:http://jipam.vu.edu.au/article. php?sid$=$432). 24. S.S. Dragomir, New reverses of Schwarz, triangle and Bessel inequalities in inner product spaces, Austral. J. Math. Anal. & Applics., 1(1) (2004), Article 1. (Online:http://ajmaa.org/cgibin/paper.pl?string$=$nrstbiips.tex). ˇ 25. S. S. Dragomir, On the Cebyˇ sev’s inequality for weighted means. Acta Math. Hungar. 104 (2004), no. 4, 345–355. 26. S.S. Dragomir, Advances in Inequalities of the Schwarz, Gr¨uss and Bessel Type in Inner Product Spaces, Nova Science Publishers Inc, New York, 2005, xC249 p. 27. S.S. Dragomir, On Bessel and Gr¨uss inequalities for orthornormal families in inner product spaces, Bull. Austral. Math. Soc., 69(2) (2004), 327–340. 28. S.S. Dragomir, Reverses of the Schwarz inequality in inner product spaces generalising a Klamkin-McLenaghan result, Bull. Austral. Math. Soc. 73(1)(2006), 69–78. ˇ 29. S.S. Dragomir, Cebyˇ sev’s type inequalities for functions of selfadjoint operators in Hilbert spaces, Lin. and Multilin. Alg. 58(2010), No. 7, 805-814. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 9. 30. S.S. Dragomir, Gr¨uss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 11. 31. S.S. Dragomir, Some new Gr¨uss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, Sarajevo J. Math. 18(2010), 89-107. RGMIA Res. Rep. Coll., 11(e) (2008), Art. 12. ˇ 32. S.S. Dragomir, Inequalities for the Cebyˇ sev functional of two functions of selfadjoint operators in Hilbert spaces, Aust. J. Math. Anal. Appl. 6(2009), No. 1, Art 7. ˇ 33. S.S. Dragomir, Some inequalities for the Cebyˇ sev functional of two functions of selfadjoint operators in Hilbert spaces, RGMIA Res. Rep. Coll., 11(e) (2008), Art. 8.

References

121

34. S.S. Dragomir, Quasi Gr¨uss’ type inequalities for continuous functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., 13(2010), Sup. Art. 12. 35. S.S. Dragomir, Gr¨uss’ type inequalities for some classes of continuous functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., 13(2010), No. 2, Art. 15. 36. S.S. Dragomir and G.L. Booth, On a Gr¨uss-Lupas¸ type inequality and its applications for the estimation of p-moments of guessing mappings, Mathematical Communications, 5 (2000), 117–126. ˇ 37. S. S. Dragomir and B. Mond, Some mappings associated with Cebyˇ sev’s inequality for sequences of real numbers. Bull. Allahabad Math. Soc. 8/9 (1993/94), 37–55. 38. S.S. Dragomir and J.E. Peˇcari´c, Refinements of some inequalities for isotonic linear functionals, L’Anal. Num. Th´eor de L’Approx. (Romania) 18(1989) (1), 61–65. 39. S. S. Dragomir, J. Peˇcari´c and J. S´andor, The Chebyshev inequality in pre-Hilbertian spaces. II. Proceedings of the Third Symposium of Mathematics and its Applications (Timis¸oara, 1989), 75–78, Rom. Acad., Timis¸oara, 1990. MR1266442 (94m:46033) 40. S. S. Dragomir and J. S´andor, The Chebyshev inequality in pre-Hilbertian spaces. I. Proceedings of the Second Symposium of Mathematics and its Applications (Timis¸oara, 1987), 61–64, Res. Centre, Acad. SR Romania, Timis¸oara, 1988. MR1006000 (90k:46048). 41. A.M. Fink, A treatise on Gr¨uss’ inequality, Analytic and Geometric Inequalities, 93-113, Math. Appl. 478, Kluwer Academic Publ., 1999. 42. T. Furuta, J. Mi´ci´c Hot, J. Peˇcari´c and Y. Seo, Mond-Peˇcari´c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005. Rb 1 ¨ das Maximum des absoluten Betrages von ba 43. G. Gr¨uss, Uber a f .x/g.x/dx R R b b 1  .ba/ 2 a f .x/dx a g.x/dx , Math. Z. , 39(1935), 215–226. 44. G.H. Hardy, J.E. Littlewood and G. P´olya, Inequalities, 1st Ed. and 2nd Ed. Cambridge University Press, (1934, 1952) Cambridge, England. 45. Z. Liu, Refinement of an inequality of Gr¨uss type for Riemann-Stieltjes integral, Soochow J. Math., 30(4) (2004), 483–489. ˇ 46. D.S. Mitrinovi´c and J.E. Peˇcari´c, History, variations and generalizations of the Cebyˇ sev inequality and question of some priorities. II. Rad Jugoslav. Akad. Znan. Umjet. No. 450 (1990), 139–156. 47. D.S. Mitrinovi´c and J.E. Peˇcari´c, On an identity of D.Z. Djokovi´c, Prilozi Mak. Akad.Nauk. Umj. (Skopje), 12(1)(1991), 21–22. 48. D.S. Mitrinovi´c, J.E. Peˇcari´c and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993. ˇ 49. D.S. Mitrinovi´c and P.M. Vasi´c, History, variations and generalisations of the Cebyˇ sev inequality and the question of some priorities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 461–497 (1974), 1–30. 50. B. Mond and J. Peˇcari´c, Classical inequalities for matrix functions, Utilitas Math., 46(1994), 155–166. 51. B. Mond and J. Peˇcari´c, Convex inequalities in Hilbert spaces, Houston J. Math., 19(1993), 405–420. 52. B. G. Pachpatte, A note on Gr¨uss type inequalities via Cauchy’s mean value theorem. Math. Inequal. Appl. 11 (2008), no. 1, 75–80. ˇ sev type inequalities. An. S¸tiint¸. Univ. Al. I. Cuza Ias¸i. Mat. 53. B. G. Pachpatte, A note on Cebyˇ (N.S.)(Romania), 53 (2007), no. 1, 97–102. ˇ 54. B. G. Pachpatte, New Cebyˇ sev type inequalities involving functions of two and three variables. Soochow J. Math. 33 (2007), no. 4, 569–577. ˇ 55. J. Peˇcari´c, Remarks on Biernacki’s generalization of Cebyˇ sev’s inequality. Ann. Univ. Mariae Curie-Sklodowska Sect. A 47 (1993), 116–122. 56. C.-J. Zhao and W.-S. Cheung, On multivariate Gr¨uss inequalities. J. Inequal. Appl. 2008, Art. ID 249438, 8 pp.