Some reverses of the Jensen inequality for functions of selfadjoint 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 be a selfadjoint linear operator on a complex Hilbert space . The Gelfand mapestablishes 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
With this notation we define
and we call it the continuous functional calculus for a selfadjoint operator
If is a selfadjoint operator and is a real valued continuous function on , then for any implies that that is, is a positive operator on Moreover, if both and are real valued functions on then the following important property holds:
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]).
Let be a selfadjoint operator on the Hilbert space and assume that for some scalars with If is a convex function on then
for each with
As a special case of Theorem 1.1 we have the following HölderMcCarthy inequality.
Theorem 1.2 (HölderMcCarthy, 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.
Let be selfadjoint operators with , for some scalars and with . If is a convex function on , then
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 be selfadjoint operators with , for some scalars If with then
for any with
Proof.
It follows from Theorem 1.3 by choosing where with
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 is nonnegative and monotonic nondecreasing on Namely, we have
The inequality (1.4) reverses if the function is concave on .
As particular cases we can state the following inequalities:
for and
for
If are positive definite for each , then (1.5) also holds for
If one uses the inequality (1.4) for the exponential function, then one obtains the inequality
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.
Let be selfadjoint operators with , , for some scalars and with . If is a strictly convex function twice differentiable on , then for any positive real number one has
where
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
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 be an interval and a convex and differentiable function on (the interior of whose derivative is continuous on If is a selfadjoint operator on the Hilbert space with then
for any with
Proof.
Since is convex and differentiable, we have that
for any
Now, if we chose in this inequality for any with since then we have
for any any with
If we fix with in (2.3) and apply property (P), then we get
for each with which is clearly equivalent to the desired inequality (2.1).
Corollary 2.2.
Assume that is as in Theorem 2.1. If are selfadjoint operators with , and with , then
Proof.
As in [1, page 6], if we put
then we have
and so on. The details are omitted.
Applying Theorem 2.1 for and , we deduce the desired result (2.5).
Corollary 2.3.
Assume that is as in Theorem 2.1. If are selfadjoint operators with , and with then
for each with
Remark 2.4.
Inequality (2.8), in the scalar case, namely
where , 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.
(a) Let be a positive definite operator on the Hilbert space Then we have the following inequality:
for each with
(b) If is a selfadjoint operator on , then we have the inequality
for each with
(c) If and is a positive operator on , then
for each with If is positive definite, then inequality (2.12) also holds for
If and is a positive definite operator then the reverse inequality also holds
for each with
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
that are in terms of the spectrum margins and of the function .
The following result may be stated.
Theorem 3.1.
Let be an interval and a convex and differentiable function on (the interior of whose derivative is continuous on . If is a selfadjoint operator on the Hilbert space with then
for any with
One also has the inequality
for any with
Moreover, if and then one also has
for any with
Proof.
We use the following Grüss type result we obtained in [8].
Let be a selfadjoint operator on the Hilbert space and assume that for some scalars If and are continuous on and and then
for each with where and
Therefore, we can state that
for each with which together with (2.1) provide the desired result (3.2).
On making use of the inequality obtained in [9]:
for each with we can state that
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].
If and are positive, then
for each with
Now, on making use of (3.10) we can state that
for each with which together with (2.1) provides the desired result (3.4).
Corollary 3.2.
Assume that is as in Theorem 3.1. If are selfadjoint operators with , , then
for any with
One also has the inequality
for any with
Moreover, if and then one also has
for any with
The following corollary also holds.
Corollary 3.3.
Assume that is as in Theorem 2.1. If are selfadjoint operators with , and with then
for any with
One also has the inequality
for any with
Moreover, if and then one also has
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
For instance, if we use inequality (3.15), then one has
Moreover, if we use inequality (3.17), then we obtain
4. Some Particular Inequalities of Interest
() Consider the convex function , On utilising inequality (3.2), then for any positive definite operator on the Hilbert space we have the inequality
for any with
However, if we use inequality (3.3), then we have the following result as well:
for any with
(2) Finally, if we consider the convex function with then on applying inequalities (3.2) and (3.3) for the positive operator , we have the inequalities
for any with respectively.
If the operator is positive definite then, by utilising inequality (3.4), we have
for any with
Now, if we consider the convex function with then from the inequalities (3.2) and (3.3) and for the positive definite operator we have the inequalities
for any with respectively.
Similar results may be stated for the convex function with However the details are left to the interested reader.
Acknowledgment
The author would like to thank anonymous referee for valuable suggestions that have been implemented in the final version of this paper.
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