New inequalities for operator convex functions
© Bacak and Türkmen; licensee Springer 2013
Received: 15 November 2012
Accepted: 15 January 2013
Published: 19 April 2013
The aim of this paper is to present some new inequalities of Hermite-Hadamard type inequalities for operator convex functions. In this paper, we use elementary operations and give some inequalities related to the Hermite-Hadamard type. We conclude that the results given in this work are the generalization of the recent results.
Note that f is convex on if and only if is convex on .
which can be derived from the classical Hermite-Hadamard inequality (1) for the convex function .
in the operator order for all and for every self-adjoint operator A and B on a Hilbert space H whose spectra are contained in I. Notice that a function f is operator concave if −f is operator convex.
In recent years, many authors have been interested in giving some refinements and extensions of the Hermite-Hadamard inequality in (1). For more about convex functions and the Hermite-Hadamard inequality, see [3–6].
The author in  shows some new integral inequalities analogous to the well-known Hermite-Hadamard inequality. We give a general form of the second of these inequalities and show that the inequalities therein are satisfied for operator convex functions.
The author in  shows some new Hermite-Hadamard inequalities similar to Pachpatte’s results.
Pachpatte (2003) gives some integral inequalities analogous to the well-known Hermite-Hadamard inequality by using a fairly elementary analysis in .
where , .
Tunç (2012) gives an inequality for convex functions in  as follows.
where , .
Tunç (2012) gives another inequality for convex functions in , too.
where , .
Ghazanfari (2012) gives an inequality for two operator convex functions in  as follows.
2 Main results
In this section, we give some new Hermite-Hadamard type inequalities for operator convex functions and mention the differences related to the results in recent papers. We emphasize the difference by giving an example.
The following theorem is a generalization for the product of two operator convex functions.
and k is the number of steps.
If we continue the same operations as above until the change of variable , we have some inequalities. And then, if we sum these obtained inequalities, we get the desired inequality. □
Remark 6 In inequality (8), if we take , we get the inequality in (7).
Now, we show the comparison between Theorems 4 and 5 utilizing self-adjoint operators (Hermitian matrices) as follows.
So, we can conclude that our result, Theorem 5, is more strict than Theorem 4 in this case.
The following theorem is a lower bound for the product of two operator convex functions.
When the above equalities are taken into account, the proof is complete. □
Remark 9 In inequality (18), if we take , and , we get the inequality in (5). Our result is more general than (5).
In Theorem 8, we give a lower bound. But now we give both lower and upper bounds for the product of two operator convex functions.
where and are defined in (9) and (10) and k is the number of steps.
If we continue the same operations as above until the change of variable , we have some inequalities. And then, if we integrate the multiplication inequalities, we get k inequalities. These inequalities are defined on , respectively. The sum of the integration parts of these k inequalities yields . Thus, the proof is complete. □
Remark 11 Inequality (23) is a general form of inequality (18). When in inequality (23), we get inequality (18).
where , , and are defined in (9) and (10) and k is the number of steps.
Proof The proof is obvious from the proofs of Theorem 3 and Theorem 5. □
Remark 13 In Theorem 12, if we take , we get (6). Theorem 12 is a generalization of Theorem 3. If we take k as the largest number we can take in Theorem 12, we near the exact solution.
This study was supported by the Coordinatorship of Selçuk University’s Scientific Research Project (BAP) and the Scientific and Technical Research Council of Turkey (TÜBİTAK). The authors would like to thank the referees for the very helpful comments and suggestions to improve this paper.
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