Abstract
In the present paper, we establish some new HermiteHadamard type inequalities involving two functions. Our results in a special case yield recent results on HermiteHadamard type inequalities.
MSC: 26D15.
Keywords:
HermiteHadamard inequality; BarnesGodunovaLevin inequality; Minkowski integral inequality; Hölder inequality1 Introduction
The following inequality is well known in the literature as HermiteHadamard’s inequality [1].
Theorem 1.1Letbe a convex function on an interval of real numbers. Then the following HermiteHadamard 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., [210]) and the references given therein. In particular, in 2010, Özdemir and Dragomir [11] established some new HermiteHadamard 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 HermiteHadamard type inequalities involving two functions in . Our results provide some new estimates on such type of inequalities.
2 Preliminaries
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
If the function is concave, the inequality (2.1) can be written as follows:
Let and be two positive nmtuples, and let . Then, on putting , it easy follows that if , then
(also see, e.g., [[1], 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 pnorm of the function on as follows:
and
and is the set of all functions such that .
Lemma 2.1 (see [12]) (BarnesGodunovaLevin inequality)
Let, be nonnegative concave functions on, then forwe have
where
Lemma 2.2 (see [1]) (HermiteHadamard inequality)
Letbe a convex function. Then the following dual HermiteHadamard inequality for convex functions holds:
The inequality is reversed if the functionis concave.
Lemma 2.3 (see [13]) (A reversed Minkowski integral inequality)
Letandbe positive functions satisfying
Then
3 Main results
Our main results are established in the following theorems.
Theorem 3.1Letand letbe nonnegative functions such thatandare concave on. Then
whereis the BarnesGodunovaLevin constant given by (2.4).
Proof Observe that whenever is concave on , the nonnegative function is also concave on . Namely,
that is,
and , using the powermean 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
and
By multiplying the above inequalities, we obtain
If , then it is easy to show that
and
Thus, by applying BarnesGodunovaLevin inequality to the righthand 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 lefthand 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 3.1Letand let, , be nonnegative functions such thatandare concave on. Then
This is just Theorem 2.1 established by Özdemir and Dragomir [11].
Theorem 3.2Letand letand, and letbe positive functions with
Then
Proof Since , are positive, as in the proof of Lemma 2.3 (see [[13], p.2]), we have
and
By multiplying the above inequalities and in view of the Minkowski inequality, we get
Hence
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 [11].
Theorem 3.3Ifandare as in Theorem 3.1, then the following inequality holds:
Proof If and are concave on , then from Lemma 2.2, we get
and
which imply that
On the other hand, if , from (2.3) we get
and
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 [11].
Theorem 3.4Letbe functions such that, andare in, and
Then
where
and
In view of the Youngtype inequality and using the elementary inequality
we have
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 [11].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
CJZ, WSC and XYL jointly contributed to the main results Theorems 3.13.4. All authors read and approved the final manuscript.
Acknowledgements
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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