Hermite–Hadamard type inequalities for m-convex and (α,m)$( \alpha , m)$-convex functions

holds for all u, v ∈ I and t ∈ [0, 1]. Convexity in connection with integral inequalities is an interesting research area since much attention has been given to studying the concept of convexity and its variant forms in recent years. Some of the most useful inequalities related to the integral mean of a convex function are Hermite–Hadamard’s inequality, Jensen’s inequality, and Hardy’s inequality (see [8, 23–25, 31]). Hermite–Hadamard’s inequality provides a necessary and sufficient condition for a function to be convex. This well-known result of Hermite and Hadamard is stated as follows: If f is a convex function on some nonempty interval I of real numbers and [u, v] ∈ I with u < v, then


Introduction
Let a real function f be defined on some nonempty interval I of real numbers. The function f : I → R is said to be convex if the inequality holds for all u, v ∈ I and t ∈ [0, 1].
Convexity in connection with integral inequalities is an interesting research area since much attention has been given to studying the concept of convexity and its variant forms in recent years. Some of the most useful inequalities related to the integral mean of a convex function are Hermite-Hadamard's inequality, Jensen's inequality, and Hardy's inequality (see [8,[23][24][25]31]). Hermite-Hadamard's inequality provides a necessary and sufficient condition for a function to be convex. This well-known result of Hermite and Hadamard is stated as follows: If f is a convex function on some nonempty interval I of real numbers and [u, v] ∈ I with u < v, then This double inequality may be regarded as a refinement of the concept of convexity, and it follows easily from Jensen's inequality. Recently, a remarkable variety of generalizations and extensions have been considered for the concept of convexity, and related Hermite-Hadamard type integral inequalities have been studied by many researchers (see, for example, [1,2,6,10,11,13,17,19,21,26,28,29,32,33] and the references cited therein).

Preliminaries
We recall the following well-known results and concepts. Toader [36] introduced the concept of m-convex functions as follows.
It can be easily seen that for m = 1, m-convexity reduces to the classical convexity of functions.
In [7], Dragomir and Agarwal proved the following result connected with the right part of (1).
Bakula et al. [4] established the following result by using Lemma 2.1 and Hölder's integral inequality.

Theorem 2.1 Suppose that I is an
open real interval such that [0, ∞) ⊂ I, and let 0 ≤ u < v < ∞. Consider the differentiable function f : I → R on I such that f ∈ L [u, v]. If |f | q is an m-convex function on [u, v] for some m ∈ (0, 1] and q ≥ 1, then İşcan [12] obtained the following integral inequality which gives better results than the classical Hölder integral inequality. [u, v]. If |f | p and |g| q are integrable functions on [u, v] for p > 1 and

Theorem 2.2 (Hölder-İşcan integral inequality) Let f and g be two real functions defined on
İşcan [12] proved the following Hermite-Hadamard type inequality by using Lemma 2.1 and the Hölder-İşcan integral inequality.

Theorem 2.3 Suppose that f
In [16], a different representation of the Hölder-İşcan integral inequality was given as follows.

Main results
Now we are in a position to establish some new Hermite-Hadamard type inequalities for the classes of m-convex and (α, m)-convex functions. where Proof From Lemma 2.1 and the Hölder-İşcan integral inequality, we have and analogously So we can write Similarly, we have Taking into account that we deduce from (4), (5), and (6) inequality (3). (3), we get inequality (2).

Theorem 3.2 Suppose that I is an open real interval such that [0, ∞) ⊂ I, and let
where Proof Using Lemma 2.1 and an improved power-mean integral inequality, we have and analogously So we obtain Similarly, we have By using inequalities (8), (9) and the fact that 1 0 t|1 -2t| dt = 1 4 , we get inequality (7).

Corollary 3.1
Let the assumptions of Theorem 3.2 be satisfied. If we take m = 1, then inequality (7) becomes the following inequality:

Theorem 3.3 Suppose that I is an open real interval such that [0, ∞) ⊂ I, and let
Consider the differentiable function f : I → R on I such that f ∈ L [u, v]. If |f | q is an (α, m)-convex function on [u, v] for some α, m ∈ (0, 1] and q > 1, q = p p-1 , then where Proof Using Lemma 2.1 and the Hölder-İşcan integral inequality, we have By (α, m)-convexity of |f | q on [u, v] for all t ∈ [0, 1], we get The proof of the first inequality in (11) is completed by the combination of inequalities (12) and (13). The proof of the second inequality in (11) is completed using the fact

Theorem 3.4 Suppose that I is an open real interval such that [0, ∞) ⊂ I, and let
Consider the differentiable function f : I → R on I such that f ∈ L [u, v]. If |f | q is an (α, m)-convex function on [u, v] for some α, m ∈ (0, 1] and q ≥ 1, then where Proof Similar to Theorem 3.2 and using (α, m)-convexity of |f | q , we get the desired result.

Applications to special means
We now consider the applications of our results to the following special means for positive real numbers u and v (u = v).
Proof The assertions follow from Theorem 3.3 and Corollary 3.2 applied respectively to the (α, m)-convex mapping f (x) = x n , x ∈ R, n ∈ Z.