Refinements of Hermite-Hadamard type inequalities for operator convex functions
© Bacak and Türkmen; licensee Springer 2013
Received: 6 August 2012
Accepted: 9 May 2013
Published: 24 May 2013
The purpose of this paper is to present some new versions of Hermite-Hadamard type inequalities for operator convex functions. We give refinements of Hermite-Hadamard type inequalities for convex functions of self-adjoint operators in a Hilbert space analogous to well-known inequalities of the same type. The results presented in this paper are more general than known results given by several authors.
is known in the literature as the Hermite-Hadamard inequality for convex functions, see . Such inequality is very useful in many mathematical contexts and contributes as a tool for establishing some interesting estimations. Both inequalities in (1.1) hold in the reversed direction if f is concave.
Note that f is convex on if and only if is convex on .
which can be derived from the classical Hermite-Hadamard inequality (1.1) for the convex function .
On a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose.
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 (1.1). For more about convex functions and the Hermite-Hadamard inequality, see [2–6].
The author in  presents the Hermite-Hadamard type inequality for convex functions by sequences. But the inequality therein is established on . In this paper, a new refinement of the Hermite-Hadamard type inequality is presented. Our inequality is an improved version of the inequality given in . Namely, this inequality includes not only , but also all positive real numbers as the number of partition.
The author in  shows some new integral inequalities analogous to the well-known Hermite-Hadamard inequality. We give a general form of the first of these inequalities and show that the inequalities therein are satisfied for operator convex functions.
Dragomir proved the following theorem in .
Zabandan gave a refinement of the Hermite-Hadamard inequality for convex functions in .
Pachpatte gave some integral inequalities analogous to the well-known Hermite-Hadamard inequality by using a fairly elementary analysis in  as follows.
where , .
2 Main results
where k is the number of steps.
Proof The function f is continuous, exists for any self-adjoint operators A and B with spectra in I.
- 1.From the definition of operator convex functions, we have the inequalities(2.2)
- 2.Let , and let A and B be two self-adjoint operators with spectra in I. Define the real-valued function by . Since f is operator convex, then for any and with , we have
for any , and any two self-adjoint operators A and B with spectra in I, from (2.9) we get the desired result in (2.1). □
Remark 5 Our result for operator convex functions in Theorem 4 is more general than the inequality in Theorem 1. In the inequality (2.1) if we take , we get the inequality in (1.3).
Remark 6 Our result for operator convex functions in Theorem 4 is more general than the inequality in Theorem 2. In the inequality (2.1), if we take , we get the inequality in (1.4). In Theorem 2, there are no cases of . But our result involves these statements.
When above equalities are taken into account, the proof is complete. □
Remark 8 In the inequality (2.10), if we take , and , we get the inequality (1.5).
where k is the number of steps.
Proof The proof is obvious from the proof of Theorem 4 and Theorem 7. □
Remark 10 The inequality (2.14) is a general form of the inequality (2.10). When in the inequality (2.14), we get the inequality (2.10).
This study is a part of corresponding author’s MSc thesis.
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 wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.
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