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.
Keywords:Hermite-Hadamard inequality; Barnes-Godunova-Levin inequality; Minkowski integral inequality; Hölder inequality
The following inequality is well known in the literature as Hermite-Hadamard’s inequality .
Theorem 1.1Let be a convex function on an interval of real numbers. Then the following Hermite-Hadamard inequality for convex functions holds:
If the functionfis concave, the inequality (1.1) can be written as follows:
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 .
Let be a duality function on the convex region . is called a duality convex function on the convex region D if
whenever and .
If the function is concave, the inequality (2.1) can be written as follows:
Let and be two positive nm-tuples, and let . Then, on putting , it easy follows that if , then
(also see, e.g., [, p.15]). Here, the rth power mean of x with weights p is the following: if ; if ; if and if .
Let , and . Now, we define the p-norm of the function on as follows:
and is the set of all functions such that .
Lemma 2.1 (see ) (Barnes-Godunova-Levin inequality)
Let , be nonnegative concave functions on , then for we have
Lemma 2.2 (see ) (Hermite-Hadamard inequality)
Let be a convex function. Then the following dual Hermite-Hadamard inequality for convex functions holds:
The inequality is reversed if the function is concave.
Lemma 2.3 (see ) (A reversed Minkowski integral inequality)
Let and be positive functions satisfying
3 Main results
Our main results are established in the following theorems.
Theorem 3.1Let and let be nonnegative functions such that and are concave on . Then
where is the Barnes-Godunova-Levin constant given by (2.4).
Proof Observe that whenever is concave on , the nonnegative function is also concave on . Namely,
and , using the power-mean inequality (2.3), we obtain
For , similarly, if is concave on , the nonnegative function is concave on .
In view that and are concave functions on , from Lemma 2.2, we get
By multiplying the above inequalities, we obtain
If , then it is easy to show that
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 3.1 By multiplying inequalities (3.2), (3.3), we obtain
By applying the Hölder inequality to the left-hand side of (3.7) with , we get
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.
Corollary 3.1Let and let , , be nonnegative functions such that and are concave on . Then
and if , then one has
This is just Theorem 2.1 established by Özdemir and Dragomir .
Theorem 3.2Let and let and , and let be positive functions with
Proof Since , are positive, as in the proof of Lemma 2.3 (see [, p.2]), we have
By multiplying the above inequalities and in view of the Minkowski inequality, we get
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 .
Theorem 3.3If and are as in Theorem 3.1, then the following inequality holds:
Proof If and are concave on , then from Lemma 2.2, we get
which imply that
On the other hand, if , from (2.3) we get
which imply that
Combining (3.12) and (3.13), we obtain the desired inequality as
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 .
Theorem 3.4Let be functions such that , and are in , and
and with .
Proof Since , , we have
In view of the Young-type inequality and using the elementary inequality
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 authors declare that they have no competing interests.
C-JZ, W-SC and X-YL jointly contributed to the main results Theorems 3.1-3.4. All authors read and approved the final manuscript.
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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