New Inequalities for Strongly r-Convex Functions

Copyright © 2019 Huriye Kadakal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, firstly we introduce a new concept called “strongly r-convex function.” After that we establishHermite-Hadamard-like inequalities for this class of functions. Moreover, by using an integral identity together with some well known integral inequalities, we establish several new inequalities for n-times differentiable strongly r-convex functions. In special cases, the results obtained coincide with the well-known results in the literature.


Introduction
A function  :  ⊆ R → R is said to be convex if the inequality

𝑓 (𝑡𝑥 + (1 − 𝑡) 𝑦) ≤ 𝑡𝑓 (𝑥) + (1 − 𝑡) 𝑓 (𝑦)
is valid for all ,  ∈  and  ∈ [0, 1].If this inequality reverses, then the function  is said to be concave on interval  ̸ = 0.This definition is well known in the literature.Convexity theory has appeared as a powerful technique to study a wide class of unrelated problems in pure and applied sciences.Many articles have been written by a number of mathematicians on convex functions and inequalities for their different classes, using, for example, the last articles [1][2][3][4][5][6] and the references in these papers.
Let  : [, ] → R be a convex function; then the inequality is known as the Hermite-Hadamard inequality (see [7] 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., [1,4,8]).In [9], the first author obtained a new refinement of the Hermite-Hadamard inequality for convex functions.The Hermite-Hadamard inequality was generalized in [10] to an -convex positive function which is defined on an interval [, ].( If the equality is reversed, then the function  is said to be -concave. It is obvious that 0-convex functions are simply logconvex functions, 1-convex functions are ordinary convex functions, and −1-convex functions are arithmetically harmonically convex.One should note that if the function  is -convex on [, ], then the function   is a convex function for  > 0 and   is a concave function for  < 0. We note that if the functions  and  are convex and  is increasing, then  is convex; moreover, since  = exp(log ), it follows that a log-convex function is convex.
The definition of -convexity naturally complements the concept of -concavity, in which the inequality is reversed [11] and plays an important role in statistics.
It is easily seen that if  is -convex on [, ], Some refinements of the Hadamard inequality for convex functions could be found in [12][13][14][15][16].In [14], the authors showed that if the function  is -convex in [, ] and 0 <  ≤ 1, then Theorem 2 (see [17]).Suppose that  is a positive -convex function on [, ].Then If the function  is a positive -concave function, then the inequality is reversed, where In this definition, if we take  = 0, we get the definition of convexity in the classical sense.Strongly convex functions have been introduced by Polyak [18], and they play an important role in optimization theory and mathematical economics.Since strong convexity is a strengthening of the notion of convexity, some properties of strongly convex functions are just "stronger versions" of known properties of convex functions.For more information on strongly convex functions, see [19][20][21] and references therein.Lemma 4. Let  ≥ 0,  ≥ 0. Then ( + )  ≤   +   , 0 <  ≤ 1.

Lemma 5 (Minkowski's integral inequality
Let 0 <  < , throughout this paper we will use for the arithmetic, geometric, and generalized logarithmic mean, respectively.Also for shortness we will use the following notation: where an empty sum is understood to be nil.

Main Results
In this section we introduce a new concept, which is called strongly -convex function, as follows.
In this definition, if we take  = 0, we get the definition of -convexity in the classical sense.Theorem 7. Let  : (0, ∞) → R be strongly -convex function with modulus  on [, ] with  < .Then the following inequality holds for 0 <  ≤ 1: Proof.Since the function  is strongly -convex function and  > 0, we have for all  ∈ [0, 1].It is easy to observe that Using Minkowski's integral inequality, we obtain Thus This proof is complete.
We will use the following lemma for obtaining our main results.
We note that the authors obtained several new integeral inequalities for -times differentiable log-convex, -convex functions in the first sense, strongly convex, -Convex and -Concave, and convex and concave functions using the above lemma (see [5,6,[22][23][24]).In this paper, we consider times differentiable strongly -convex function and establish several new inequalities for this class of functions.Obtained results in this paper coincide with the results of papers ([6, 23, 24]).
Remark 11.The following results are remarkable for Theorem 10.
(i) The results obtained in this paper reduce to the results of [24] in case of  = 0.
(ii) The results obtained in this paper reduce to the results of [23] in case of  = 1.
(iii) The results obtained in this paper reduce to the results of [6] in case of  = 1 and  = 0.
This completes the proof of theorem.
Remark 14.The following results are remarkable for Theorem 13.
(i) The results obtained in this paper reduce to the results of [24] in case of  = 0.
(ii) The results obtained in this paper reduce to the results of [23] in case of  = 1.
(iii) The results obtained in this paper reduce to the results of [6] in case of  = 1 and  = 0.

Corollary 15. Under the conditions Theorem 13 for 𝑛 = 1 we have the following inequalities:
(, , )     ≤  (−1)/ (, ) where Proof.Since | () |  for  > 1 is strongly -convex function on [, ], using Lemma 9 and the Hölder integral inequality, we have the following inequality: Here, using Lemma 4 we obtain For  ≥ 1, (i) The results obtained in this paper reduce to the results of [24] in case of  = 0.
(ii) The results obtained in this paper reduce to the results of [23] in case of  = 1.
(iii) The results obtained in this paper reduce to the results of [6]  (i) The results obtained in this paper reduce to the results of [24] in case of  = 0.
(ii) The results obtained in this paper reduce to the results of [23] in case of  = 1.
(iii) The results obtained in this paper reduce to the results of [6] in case of  = 1 and  = 0.