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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq2_HTML.gif . The Gelfand map establishes a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq3_HTML.gif -isometrically isomorphism http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq4_HTML.gif between the set http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq6_HTML.gif denoted http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq7_HTML.gif and the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq8_HTML.gif -algebra http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq9_HTML.gif generated by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq10_HTML.gif and the identity operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq11_HTML.gif on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq13_HTML.gif and any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq14_HTML.gif we have

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

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

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

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

With this notation we define
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq24_HTML.gif

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

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

for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq44_HTML.gif with http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq47_HTML.gif . Then

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

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

(iii)if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq56_HTML.gif is invertible, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq57_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq59_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq61_HTML.gif be selfadjoint operators with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq63_HTML.gif for some scalars http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq65_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq66_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq67_HTML.gif is a convex function on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq68_HTML.gif , then
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq69_HTML.gif be selfadjoint operators with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq70_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq71_HTML.gif for some scalars http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq72_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq73_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq74_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq75_HTML.gif then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ5_HTML.gif
(1.3)

for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq76_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq78_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq79_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq80_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq82_HTML.gif is nonnegative and monotonic nondecreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq83_HTML.gif Namely, we have
http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ7_HTML.gif
(1.5)
for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq85_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ8_HTML.gif
(1.6)

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

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq87_HTML.gif are positive definite for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq88_HTML.gif , then (1.5) also holds for http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ9_HTML.gif
(1.7)

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

In Section http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq93_HTML.gif be selfadjoint operators with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq94_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq95_HTML.gif , for some scalars http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq97_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq98_HTML.gif . If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq100_HTML.gif , then for any positive real number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq101_HTML.gif one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ10_HTML.gif
(1.8)
where
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq102_HTML.gif be an interval and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq103_HTML.gif a convex and differentiable function on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq104_HTML.gif (the interior of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq105_HTML.gif whose derivative http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq106_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq107_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq108_HTML.gif If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq110_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq111_HTML.gif then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ13_HTML.gif
(2.1)

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

Proof.

Since http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ14_HTML.gif
(2.2)

for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq116_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq117_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq118_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq119_HTML.gif then we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ15_HTML.gif
(2.3)

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

If we fix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq123_HTML.gif with http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ16_HTML.gif
(2.4)

for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq125_HTML.gif with http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq128_HTML.gif are selfadjoint operators with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq131_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq132_HTML.gif , then
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ18_HTML.gif
(2.6)
then we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq133_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq134_HTML.gif
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq135_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq138_HTML.gif are selfadjoint operators with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq139_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq140_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq141_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq142_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq143_HTML.gif then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ20_HTML.gif
(2.8)

for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq144_HTML.gif with http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ21_HTML.gif
(2.9)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq146_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq149_HTML.gif Then we have the following inequality:
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ22_HTML.gif
    (2.10)
     
for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq150_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq151_HTML.gif
  1. (b)
    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq152_HTML.gif is a selfadjoint operator on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq153_HTML.gif , then we have the inequality
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ23_HTML.gif
    (2.11)
     
for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq154_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq155_HTML.gif
  1. (c)
    If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq156_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq157_HTML.gif is a positive operator on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq158_HTML.gif , then
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ24_HTML.gif
    (2.12)
     

for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq159_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq160_HTML.gif If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq162_HTML.gif

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq163_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ25_HTML.gif
(2.13)

for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq165_HTML.gif with http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq167_HTML.gif and of the function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq169_HTML.gif be an interval and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq170_HTML.gif a convex and differentiable function on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq171_HTML.gif (the interior of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq172_HTML.gif whose derivative http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq173_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq174_HTML.gif . If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq176_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq177_HTML.gif then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ27_HTML.gif
(3.2)

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

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

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

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

for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq184_HTML.gif with http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq187_HTML.gif and assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq188_HTML.gif for some scalars http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq189_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq191_HTML.gif are continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq192_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq193_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq194_HTML.gif then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ30_HTML.gif
(3.5)

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

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

for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq199_HTML.gif with http://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]:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ33_HTML.gif
(3.8)
for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq201_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq202_HTML.gif we can state that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ34_HTML.gif
(3.9)

for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq203_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq205_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq206_HTML.gif are positive, then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ35_HTML.gif
(3.10)

for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq207_HTML.gif with http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ36_HTML.gif
(3.11)

for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq209_HTML.gif with http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq212_HTML.gif are selfadjoint operators with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq213_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq214_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ37_HTML.gif
(3.12)

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

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

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

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

for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq221_HTML.gif with http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq224_HTML.gif are selfadjoint operators with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq225_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq226_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq227_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq228_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq229_HTML.gif then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ40_HTML.gif
(3.15)

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

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

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

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

for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq236_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq238_HTML.gif is nonnegative and monotonic nondecreasing on http://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
http://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
http://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

( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq240_HTML.gif ) Consider the convex function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq241_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq243_HTML.gif on the Hilbert space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq244_HTML.gif we have the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ45_HTML.gif
(4.1)

for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq245_HTML.gif with http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ46_HTML.gif
(4.2)
for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq247_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq249_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq250_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq252_HTML.gif , we have the inequalities
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ47_HTML.gif
    (4.3)
     

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

If the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq255_HTML.gif is positive definite http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ48_HTML.gif
(4.4)

for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq257_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq259_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq260_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq262_HTML.gif we have the inequalities
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_Equ49_HTML.gif
(4.5)

for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq263_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq265_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F496821/MediaObjects/13660_2009_Article_2172_IEq266_HTML.gif with http://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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