Abstract
In this paper, we establish the Hermite-Hadamard inequality for r-convex functions. We prove that r-convexity implies s-convexity (). As a result, we obtain a refinement of the Hermite-Hadamard inequality for an r-convex function (). We also investigate the Hermite-Hadamard inequality for the product of an r-convex function f and an s-convex function g.
MSC: 26D15, 26D10.
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1 Introduction
Let be a convex function, then the inequality
is known as the Hermite-Hadamard inequality (see [1] for more information). Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [2, 3] and [4]). In [5], the first author obtained a new refinement of the Hermite-Hadamard inequality for convex functions. The Hermite-Hadamard inequality was generalized in [6] to an r-convex positive function which is defined on an interval . A positive function f is called r-convex on , if for each and ,
It is obvious 0-convex functions are simply log-convex functions and 1-convex functions are ordinary convex functions. One should note that if f is r-convex in , then is a convex function ().
Some refinements of the Hadamard inequality for r-convex functions could be found in [7] and [8]. In [9], Bessenyei studied Hermite-Hadamard-type inequalities for generalized 3-convex functions. In [7], the authors showed that if f is r-convex in and , then
In this paper, first we show that if f is r-convex in and , then
In Theorem 2.3, we prove the following inequality for r-convex functions:
The inequality (4) is an extension and refinement of (2) and (3). In Theorem 2.4, we show that r-convexity implies s-convexity (). We employ this result in Theorem 2.6 and Corollary 2.7 to refine the Hermite-Hadamard inequality by r-convexity (). Finally, we generalize some results in [7] without using Minkowski’s inequality. Indeed, we obtain refinements for the product of an r-convex function f and an s-convex function g ().
2 Main results
Theorem 2.1 Let be r-convex and . Then the following inequality holds:
Proof Since , by Jensen’s inequality, we have
By convexity of and the right side of (1), we obtain
Thus,
□
Corollary 2.2 Let be a 1-convex function. Then
Theorem 2.3 Let be r-convex and . Then the following inequalities hold:
Proof First, let . Since f is r-convex, for all , we have
It is easy to observe that
By substitution , we obtain
For , we have
So,
The proof is completed. □
With the hypotheses of Theorem 2.3, if , its proving process shows that can be dominated by where .
Note that if we put in Theorem 2.3, we can obtain again the inequality (5).
Theorem 2.4 Let be r-convex on and . Then f is s-convex. In particular, if f is r-convex and , then f is convex.
In order to prove the above theorem, we need the following lemma.
Lemma 2.5 If and , then the following inequalities hold for every pair of non-negative real numbers x and y:
Proof The left side of the inequality is clear by Young’s inequality. The right side is obvious if either x or y equals zero. So, let and . Consider defined by
Then . So, is a critical point of f. By an easy calculation, we see that . It follows that f attains its maximum at . Thus, . This shows that
Now, if we put in the above inequality, we get
Therefore, we can deduce the right side of (6) by taking rs th root. □
Proof of Theorem 2.4 Since f is r-convex, by Lemma 2.5 for all and , we have
Hence, f is s-convex. □
Theorem 2.6 Let be r-convex on and . Then the following inequalities hold:
Proof The left side of the inequalities is clear by Theorem 2.3. For the right side, by the inequality in (6), we have
By integrating it on , we obtain
Thus,
Also, another inequality can be deduced by integrating the inequalities in (6) if we replace x and y by and , respectively. □
Corollary 2.7 Let be r-convex and . Then
In other words, when f is r-convex and , we can refine the Hermite-Hadamard inequalities through Theorem 2.6.
Theorem 2.8 Let be r-convex and s-convex functions respectively on and . Then the following inequality holds:
Proof Since f is r-convex and g is s-convex, for all , we have
Thus,
Applying Cauchy’s inequality, we get
Similar to the proof of Theorem 2.3 and by substitution , we obtain
Similarly,
Using (7), (8) and (9), we can obtain the desired result. □
Remark 2.9 If the conditions of Theorem 2.8 hold, and , by Theorem 2.4, f is s-convex. Thus, the result of Theorem 2.8 could be as follows:
If , we have
Now, if and in Theorem 2.8, we have
which is the same result as in [[7], Corollary 2.5]. This shows that Theorem 2.8 is a generalization of [[7], Theorem 2.3]. In fact, the condition is redundant.
Theorem 2.10 Let be 0-convex on . Then the following inequality holds:
Proof Since f and g are 0-convex, for all , we have
For all , and thus
□
Corollary 2.11 With the hypotheses of the above theorem and , we have
Theorem 2.12 Let be r-convex and 0-convex functions respectively on and . Then the following inequality holds:
Proof Since f is r-convex and g is 0-convex, for all , we have
Thus,
Again, Cauchy’s inequality shows that
We have
Similar to the proof of Theorem 2.10, we can show that
Using (10), (11) and (12), we can obtain the desired result. □
References
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Acknowledgements
The authors express their sincere thanks to the referee(s) for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially. The third author also acknowledges that this project was partially supported by University Putra Malaysia.
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Zabandan, G., Bodaghi, A. & Kılıçman, A. The Hermite-Hadamard inequality for r-convex functions. J Inequal Appl 2012, 215 (2012). https://doi.org/10.1186/1029-242X-2012-215
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DOI: https://doi.org/10.1186/1029-242X-2012-215