On the Hermite-Hadamard type inequalities
© Zhao et al.; licensee Springer. 2013
Received: 20 October 2012
Accepted: 18 April 2013
Published: 7 May 2013
In the present paper, we establish some new Hermite-Hadamard type inequalities involving two functions. Our results in a special case yield recent results on Hermite-Hadamard type inequalities.
The following inequality is well known in the literature as Hermite-Hadamard’s inequality .
Recently, many generalizations, extensions and variants of this inequality have appeared in the literature (see, e.g., [2–10]) and the references given therein. In particular, in 2010, Özdemir and Dragomir  established some new Hermite-Hadamard inequalities and other integral inequalities involving two functions in ℝ. Following this work, the main purpose of the present paper is to establish some dual Hermite-Hadamard type inequalities involving two functions in . Our results provide some new estimates on such type of inequalities.
A region is called convex if it contains the close line segment joining any two of its points, or equivalently, if whenever and .
whenever and .
(also see, e.g., [, p.15]). Here, the r th power mean of x with weights p is the following: if ; if ; if and if .
and is the set of all functions such that .
Lemma 2.1 (see ) (Barnes-Godunova-Levin inequality)
Lemma 2.2 (see ) (Hermite-Hadamard inequality)
The inequality is reversed if the function is concave.
Lemma 2.3 (see ) (A reversed Minkowski integral inequality)
3 Main results
Our main results are established in the following theorems.
where is the Barnes-Godunova-Levin constant given by (2.4).
For , similarly, if is concave on , the nonnegative function is concave on .
Thus, by applying Barnes-Godunova-Levin inequality to the right-hand side of (3.4) with (3.5), (3.6), we get (3.1).
The proof is complete. □
Remark 3.2 Let and change to and , respectively, and with suitable changes in Theorem 3.1 and Remark 3.1, we have the following.
This is just Theorem 2.1 established by Özdemir and Dragomir .
This proof is complete. □
Remark 3.3 Let and change to and , respectively, and with suitable changes in (3.9), (3.9) reduces to an inequality established by Özdemir and Dragomir .
This proof is complete. □
Remark 3.4 Let and change to and , respectively, and with suitable changes in (3.11), (3.11) reduces to an inequality established by Özdemir and Dragomir .
and with .
This completes the proof. □
Remark 3.5 Let and change to and , respectively, and with suitable changes in (3.14), (3.14) reduces to an inequality established by Özdemir and Dragomir .
The first author’s research is supported by Natural Science Foundation of China (10971205). The second author’s research is partially supported by a HKU Seed Grant for Basic Research.
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