Open Access

Some Reverses of the Jensen Inequality for Functions of Selfadjoint Operators in Hilbert Spaces

Journal of Inequalities and Applications20102010:496821

DOI: 10.1155/2010/496821

Received: 22 September 2009

Accepted: 23 April 2010

Published: 26 May 2010

Abstract

Some reverses of the Jensen inequality for functions of self-adjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq1_HTML.gif be a selfadjoint linear operator on a complex Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq2_HTML.gif . The Gelfand map establishes a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq3_HTML.gif -isometrically isomorphism https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq4_HTML.gif between the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq5_HTML.gif of all continuous functions defined on the spectrum of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq6_HTML.gif denoted https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq7_HTML.gif and the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq8_HTML.gif -algebra https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq9_HTML.gif generated by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq10_HTML.gif and the identity operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq11_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq12_HTML.gif as follows (see e.g., [1, page 3]):

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq13_HTML.gif and any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq14_HTML.gif we have

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq15_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq17_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq18_HTML.gif

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq20_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq22_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq23_HTML.gif

With this notation we define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ1_HTML.gif
(1.1)

and we call it the continuous functional calculus for a selfadjoint operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq24_HTML.gif

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq25_HTML.gif is a selfadjoint operator and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq26_HTML.gif is a real valued continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq27_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq28_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq29_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq30_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq31_HTML.gif is a positive operator on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq32_HTML.gif Moreover, if both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq34_HTML.gif are real valued functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq35_HTML.gif then the following important property holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ2_HTML.gif
(P)

in the operator order of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq36_HTML.gif

For a recent monograph devoted to various inequalities for functions of selfadjoint operators, see [1] and the references therein. For other results, see [24].

The following result that provides an operator version for the Jensen inequality is due to [5] (see also [1, page 5]).

Theorem 1.1 (Mond and Pečarić, 1993, [5]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq37_HTML.gif be a selfadjoint operator on the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq38_HTML.gif and assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq39_HTML.gif for some scalars https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq40_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq41_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq42_HTML.gif is a convex function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq43_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ3_HTML.gif
(MP)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq44_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq45_HTML.gif

As a special case of Theorem 1.1 we have the following Hölder-McCarthy inequality.

Theorem 1.2 (Hölder-McCarthy, 1967, [6]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq46_HTML.gif be a selfadjoint positive operator on a Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq47_HTML.gif . Then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq48_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq50_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq51_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq52_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq54_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq55_HTML.gif

(iii)if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq56_HTML.gif is invertible, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq57_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq58_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq59_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq60_HTML.gif

The following theorem is a multiple operator version of Theorem 1.1 (see e.g., [1, page 5]).

Theorem 1.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq61_HTML.gif be selfadjoint operators with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq63_HTML.gif for some scalars https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq65_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq66_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq67_HTML.gif is a convex function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq68_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ4_HTML.gif
(1.2)

The following particular case is of interest. Apparently it has not been stated before either in the monograph [1] or in the research papers cited therein.

Corollary 1.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq69_HTML.gif be selfadjoint operators with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq70_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq71_HTML.gif for some scalars https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq72_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq73_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq74_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq75_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ5_HTML.gif
(1.3)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq76_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq77_HTML.gif

Proof.

It follows from Theorem 1.3 by choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq78_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq79_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq80_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq81_HTML.gif

Remark 1.5.

The above inequality can be used to produce some norm inequalities for the sum of positive operators in the case when the convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq82_HTML.gif is nonnegative and monotonic nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq83_HTML.gif Namely, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ6_HTML.gif
(1.4)

The inequality (1.4) reverses if the function is concave on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq84_HTML.gif .

As particular cases we can state the following inequalities:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ7_HTML.gif
(1.5)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq85_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ8_HTML.gif
(1.6)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq86_HTML.gif

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq87_HTML.gif are positive definite for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq88_HTML.gif , then (1.5) also holds for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq89_HTML.gif

If one uses the inequality (1.4) for the exponential function, then one obtains the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ9_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq90_HTML.gif are positive operators for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq91_HTML.gif

In Section https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq92_HTML.gif of the monograph [1] there are numerous and interesting converses of the Jensen type inequality from which we would like to mention one of the simplest (see [4] and [1, page 61]).

Theorem 1.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq93_HTML.gif be selfadjoint operators with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq94_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq95_HTML.gif , for some scalars https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq97_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq98_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq99_HTML.gif is a strictly convex function twice differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq100_HTML.gif , then for any positive real number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq101_HTML.gif one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ10_HTML.gif
(1.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ11_HTML.gif
(1.9)

The case of equality was also analyzed but will be not stated in here.

The main aim of the present paper is to provide different reverses of the Jensen inequality where some upper bounds for the nonnegative difference
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ12_HTML.gif
(1.10)

will be provided. Applications for some particular convex functions of interest are also given.

2. Reverses of the Jensen Inequality

The following result holds.

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq102_HTML.gif be an interval and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq103_HTML.gif a convex and differentiable function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq104_HTML.gif (the interior of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq105_HTML.gif whose derivative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq106_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq107_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq108_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq109_HTML.gif is a selfadjoint operator on the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq110_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq111_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ13_HTML.gif
(2.1)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq112_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq113_HTML.gif

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq114_HTML.gif is convex and differentiable, we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ14_HTML.gif
(2.2)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq115_HTML.gif

Now, if we chose in this inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq116_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq117_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq118_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq119_HTML.gif then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ15_HTML.gif
(2.3)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq120_HTML.gif any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq121_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq122_HTML.gif

If we fix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq123_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq124_HTML.gif in (2.3) and apply property (P), then we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ16_HTML.gif
(2.4)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq125_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq126_HTML.gif which is clearly equivalent to the desired inequality (2.1).

Corollary 2.2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq127_HTML.gif is as in Theorem 2.1. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq128_HTML.gif are selfadjoint operators with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq131_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq132_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ17_HTML.gif
(2.5)

Proof.

As in [1, page 6], if we put
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ18_HTML.gif
(2.6)
then we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq133_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq134_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ19_HTML.gif
(2.7)

and so on. The details are omitted.

Applying Theorem 2.1 for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq136_HTML.gif , we deduce the desired result (2.5).

Corollary 2.3.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq137_HTML.gif is as in Theorem 2.1. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq138_HTML.gif are selfadjoint operators with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq139_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq140_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq141_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq142_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq143_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ20_HTML.gif
(2.8)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq144_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq145_HTML.gif

Remark 2.4.

Inequality (2.8), in the scalar case, namely
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ21_HTML.gif
(2.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq146_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq147_HTML.gif has been obtained for the first time in 1994 by Dragomir and Ionescu, see [7].

The following particular cases are of interest.

Example 2.5.
  1. (a)
    Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq148_HTML.gif be a positive definite operator on the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq149_HTML.gif Then we have the following inequality:
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ22_HTML.gif
    (2.10)
     
for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq150_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq151_HTML.gif
  1. (b)
    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq152_HTML.gif is a selfadjoint operator on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq153_HTML.gif , then we have the inequality
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ23_HTML.gif
    (2.11)
     
for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq154_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq155_HTML.gif
  1. (c)
    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq156_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq157_HTML.gif is a positive operator on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq158_HTML.gif , then
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ24_HTML.gif
    (2.12)
     

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq159_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq160_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq161_HTML.gif is positive definite, then inequality (2.12) also holds for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq162_HTML.gif

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq164_HTML.gif is a positive definite operator then the reverse inequality also holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ25_HTML.gif
(2.13)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq165_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq166_HTML.gif

Similar results can be stated for sequences of operators; however the details are omitted.

3. Further Reverses

In applications would be perhaps more useful to find upper bounds for the quantity
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ26_HTML.gif
(3.1)

that are in terms of the spectrum margins https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq167_HTML.gif and of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq168_HTML.gif .

The following result may be stated.

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq169_HTML.gif be an interval and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq170_HTML.gif a convex and differentiable function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq171_HTML.gif (the interior of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq172_HTML.gif whose derivative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq173_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq174_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq175_HTML.gif is a selfadjoint operator on the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq176_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq177_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ27_HTML.gif
(3.2)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq178_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq179_HTML.gif

One also has the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ28_HTML.gif
(3.3)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq180_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq181_HTML.gif

Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq183_HTML.gif then one also has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ29_HTML.gif
(3.4)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq184_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq185_HTML.gif

Proof.

We use the following Grüss type result we obtained in [8].

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq186_HTML.gif be a selfadjoint operator on the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq187_HTML.gif and assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq188_HTML.gif for some scalars https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq189_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq190_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq191_HTML.gif are continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq193_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq194_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ30_HTML.gif
(3.5)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq195_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq196_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq197_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq198_HTML.gif

Therefore, we can state that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ31_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ32_HTML.gif
(3.7)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq199_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq200_HTML.gif which together with (2.1) provide the desired result (3.2).

On making use of the inequality obtained in [9]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ33_HTML.gif
(3.8)
for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq201_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq202_HTML.gif we can state that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ34_HTML.gif
(3.9)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq203_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq204_HTML.gif which together with (2.1) provides the desired result (3.3).

Further, in order to prove the third inequality, we make use of the following result of Grüss' type we obtained in [9].

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq205_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq206_HTML.gif are positive, then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ35_HTML.gif
(3.10)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq207_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq208_HTML.gif

Now, on making use of (3.10) we can state that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ36_HTML.gif
(3.11)

for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq209_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq210_HTML.gif which together with (2.1) provides the desired result (3.4).

Corollary 3.2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq211_HTML.gif is as in Theorem 3.1. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq212_HTML.gif are selfadjoint operators with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq213_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq214_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ37_HTML.gif
(3.12)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq215_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq216_HTML.gif

One also has the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ38_HTML.gif
(3.13)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq217_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq218_HTML.gif

Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq219_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq220_HTML.gif then one also has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ39_HTML.gif
(3.14)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq221_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq222_HTML.gif

The following corollary also holds.

Corollary 3.3.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq223_HTML.gif is as in Theorem 2.1. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq224_HTML.gif are selfadjoint operators with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq225_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq226_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq227_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq228_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq229_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ40_HTML.gif
(3.15)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq230_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq231_HTML.gif

One also has the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ41_HTML.gif
(3.16)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq232_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq233_HTML.gif

Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq234_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq235_HTML.gif then one also has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ42_HTML.gif
(3.17)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq236_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq237_HTML.gif

Remark 3.4.

Some of the inequalities in Corollary 3.3 can be used to produce reverse norm inequalities for the sum of positive operators in the case when the convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq238_HTML.gif is nonnegative and monotonic nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq239_HTML.gif

For instance, if we use inequality (3.15), then one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ43_HTML.gif
(3.18)
Moreover, if we use inequality (3.17), then we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ44_HTML.gif
(3.19)

4. Some Particular Inequalities of Interest

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq240_HTML.gif ) Consider the convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq241_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq242_HTML.gif On utilising inequality (3.2), then for any positive definite operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq243_HTML.gif on the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq244_HTML.gif we have the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ45_HTML.gif
(4.1)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq245_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq246_HTML.gif

However, if we use inequality (3.3), then we have the following result as well:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ46_HTML.gif
(4.2)
for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq247_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq248_HTML.gif
  1. (2)
    Finally, if we consider the convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq249_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq250_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq251_HTML.gif then on applying inequalities (3.2) and (3.3) for the positive operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq252_HTML.gif , we have the inequalities
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ47_HTML.gif
    (4.3)
     

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq253_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq254_HTML.gif respectively.

If the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq255_HTML.gif is positive definite https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq256_HTML.gif then, by utilising inequality (3.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ48_HTML.gif
(4.4)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq257_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq258_HTML.gif

Now, if we consider the convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq259_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq260_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq261_HTML.gif then from the inequalities (3.2) and (3.3) and for the positive definite operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq262_HTML.gif we have the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ49_HTML.gif
(4.5)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq263_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq264_HTML.gif respectively.

Similar results may be stated for the convex function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq265_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq266_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq267_HTML.gif However the details are left to the interested reader.

Declarations

Acknowledgment

The author would like to thank anonymous referee for valuable suggestions that have been implemented in the final version of this paper.

Authors’ Affiliations

(1)
Mathematics, School of Engineering & Science, Victoria University
(2)
School of Computational and Applied Mathematics, University of the Witwatersrand

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Copyright

© S. S. Dragomir. 2010

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