- Research Article
- Open Access

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

- SS Dragomir
^{1, 2}Email author

**2010**:496821

https://doi.org/10.1155/2010/496821

© S. S. Dragomir. 2010

**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.

## Keywords

- Hilbert Space
- Convex Function
- Differentiable Function
- Positive Operator
- Operator Version

## 1. Introduction

Let
be a selfadjoint linear operator on a complex Hilbert space
. The *Gelfand map* establishes a
-isometrically isomorphism
between the set
of all *continuous functions* defined on the *spectrum* of
denoted
and the
-algebra
generated by
and the identity operator
on
as follows (see e.g., [1, page 3]):

For any and any we have

(i)

(ii) and

(iii)

(iv) and where and for

and we call it the *continuous functional calculus* for a selfadjoint operator

in the operator order of

For a recent monograph devoted to various inequalities for functions of selfadjoint operators, see [1] and the references therein. For other results, see [2–4].

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]).

for each with

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

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

Let be a selfadjoint positive operator on a Hilbert space . Then

(i) for all and with

(ii) for all and with

(iii)if is invertible, then for all and with

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

Theorem 1.3.

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.

for any with

Proof.

It follows from Theorem 1.3 by choosing where with

Remark 1.5.

The inequality (1.4) reverses if the function is concave on .

for

If are positive definite for each , then (1.5) also holds for

where are positive operators for each

In Section 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.

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

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.

for any with

Proof.

for any

for any any with

for each with which is clearly equivalent to the desired inequality (2.1).

Corollary 2.2.

Proof.

and so on. The details are omitted.

Applying Theorem 2.1 for and , we deduce the desired result (2.5).

Corollary 2.3.

for each with

Remark 2.4.

where , has been obtained for the first time in 1994 by Dragomir and Ionescu, see [7].

The following particular cases are of interest.

- (a)

for each with If is positive definite, then inequality (2.12) also holds for

for each with

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

## 3. Further Reverses

that are in terms of the spectrum margins and of the function .

The following result may be stated.

Theorem 3.1.

for any with

for any with

for any with

Proof.

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

for each with where and

for each with which together with (2.1) provide the desired result (3.2).

for each with 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].

for each with

for each with which together with (2.1) provides the desired result (3.4).

Corollary 3.2.

for any with

for any with

for any with

The following corollary also holds.

Corollary 3.3.

for any with

for any with

for any with

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 is nonnegative and monotonic nondecreasing on

## 4. Some Particular Inequalities of Interest

for any with

- (2)

for any with respectively.

for any with

for any with respectively.

Similar results may be stated for the convex function with 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

## References

- Furuta T, Hot JM, Pečarić J, Seo Y:
*Mond-Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space*. Element, Zagreb, Croatia; 2005.MATHGoogle Scholar - Mond B, Pečarić JE: Classical inequalities for matrix functions.
*Utilitas Mathematica*1994, 46: 155–166.MathSciNetMATHGoogle Scholar - Matković A, Pečarić J, Perić I: A variant of Jensen's inequality of Mercer's type for operators with applications.
*Linear Algebra and Its Applications*2006, 418(2–3):551–564. 10.1016/j.laa.2006.02.030MathSciNetView ArticleMATHGoogle Scholar - Mićić J, Seo Y, Takahasi S-E, Tominaga M: Inequalities of Furuta and Mond-Pečarić.
*Mathematical Inequalities & Applications*1999, 2(1):83–111.MathSciNetMATHGoogle Scholar - Mond B, Pečarić JE: Convex inequalities in Hilbert space.
*Houston Journal of Mathematics*1993, 19(3):405–420.MathSciNetMATHGoogle Scholar - McCarthy CA: . Israel Journal of Mathematics 1967, 5: 249–271. 10.1007/BF02771613Google Scholar
- Dragomir SS, Ionescu NM: Some converse of Jensen's inequality and applications.
*Revue d'Analyse Numérique et de Théorie de l'Approximation*1994, 23(1):71–78.MathSciNetMATHGoogle Scholar - Dragomir SS: Grüss' type inequalities for functions of selfadjoint operators in Hilbert spaces.
*RGMIA Research Report Collection*2008., 11, article 11: PreprintGoogle Scholar - Dragomir SS: Some new Grüss' type inequalities for functions of selfadjoint operators in Hilbert spaces.
*RGMIA Research Report Collection*2008., 11, article 12: PreprintGoogle Scholar

## Copyright

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