Some Reverses of the Jensen Inequality for Functions of Selfadjoint Operators in Hilbert Spaces
© S. S. Dragomir. 2010
Received: 22 September 2009
Accepted: 23 April 2010
Published: 26 May 2010
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.
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]):
Theorem 1.1 (Mond and Pečarić, 1993, ).
As a special case of Theorem 1.1 we have the following Hölder-McCarthy inequality.
Theorem 1.2 (Hölder-McCarthy, 1967, ).
The following theorem is a multiple operator version of Theorem 1.1 (see e.g., [1, page 5]).
The following particular case is of interest. Apparently it has not been stated before either in the monograph  or in the research papers cited therein.
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.
and so on. The details are omitted.
where , has been obtained for the first time in 1994 by Dragomir and Ionescu, see .
The following particular cases are of interest.
Similar results can be stated for sequences of operators; however the details are omitted.
3. Further Reverses
The following result may be stated.
We use the following Grüss type result we obtained in .
Further, in order to prove the third inequality, we make use of the following result of Grüss' type we obtained in .
The following corollary also holds.
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
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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