Exponential type convexity and some related inequalities

In this manuscript, we give and study the concept of exponential type convex functions and some of their algebraic properties. We prove two Hermite–Hadamard (H-H) type integral inequalities for the newly introduced class of functions. We also obtain some refinements of the H-H inequality for functions whose first derivative in absolute value at certain power is exponential type convex.


Introduction
A function f : I → R is said to be convex function if the following inequality holds: f km + (1k)n ≤ kf (m) + (1k)f (n) ( 1 ) for all m, n ∈ I and k ∈ [0, 1]. If (1) reverses, then f is said to be concave on I = ∅. Convexity theory provides powerful principles and techniques for studying a class of problems in mathematics. See articles [4,5,7,[9][10][11][12][13] and the references therein. Let f : I → R be a convex function. Then the following inequalities hold: for all m, n ∈ I with m < n. Inequality (2) is well known as the Hermite-Hadamard (H-H) integral inequality [6]. Some refinements of the H-H inequality for convex functions have been obtained [3,15]. The aim of this study is to submit the concept of exponential type convex functions and find some results connected with the right-hand side of new inequalities similar to the H-H inequality for this type of functions. Definition 1.1 ([14]) Let h : J → R be a nonnegative function and h = 0. We say that f : I → R is an h-convex function, or that f belongs to the class SX(h, I), if f is nonnegative and for all m, n ∈ I, k ∈ [0, 1] we have If (3) is reversed, then f is said to be h-concave, i.e., f ∈ SV (h, I). It is clear that, if h(u) = u, then the h-convexity reduces to convexity.
Readers can look at [1,8] for studies on h-convexity.

Main results
In this section, we give a new definition, which is called exponential type convexity, and we give it by setting some algebraic properties for the exponential type convex functions, as follows.

Definition 2.1
A nonnegative function f : I → R is called exponential type convex function if, for every m, n ∈ I and k ∈ [0, 1], The class of all exponential type convex functions on interval I is indicated by EXPC(I).
Remark 2.1 The range of the exponential type convex functions is [0, ∞).
Proof Let m ∈ I be arbitrary. Using the definition of the exponential type convex function We discuss some connections between the class of exponential type convex functions and other classes of generalized convex functions.
Proof The proof is obvious.

Proposition 2.1 Every nonnegative convex function is exponential type convex function.
Proof According to Lemma 2.1, since k ≤ e k -1 and 1k ≤ e 1-k -1 for all k ∈ [0, 1], we obtain

Proposition 2.2 Every exponential type convex function is an h-convex function with h(k) = e k -1.
Proof If we substitute e k -1 = h(k) and e 1-k -1 = h(1k) in inequality (3), an h-convex function is easily obtained. (i) f + g is exponential type convex, (ii) for c ≥ 0, cf is exponential type convex.
Proof (i) Let f , g be exponential type convex, then (ii) Let f be exponential type convex and c ∈ R (c ≥ 0), then Remark 2.2 Theorem 2.1 follows from the known fact that the space of an h-convex function is a convex cone for each h (see [14], Proposition 9).

Theorem 2.2
If f : I → J is convex and g : J → R is an exponential type convex function and nondecreasing, then g • f : I → R is an exponential type convex function.
Proof For m, n ∈ I and k ∈ [0, 1], we get Remark 2.3 The above theorem can also be derived from Theorem 15 in [14].
is an interval and f is an exponential type convex function on J.
Proof Let k ∈ [0, 1] and m, n ∈ J be arbitrary. Then This shows simultaneously that J is an interval, since it contains every point between any two of its points, and that f is an exponential type convex function on J.
Also, for every x ∈ [m, n], there exists λ ∈ [0, n-m 2 ] such that x = m+n 2 + λ or x = m+n 2λ. Without loss of generality, we suppose x = m+n 2 + λ. So, we have By using M as the upper bound, we get

Hermite-Hadamard inequality for exponential type convex functions
The aim of this section is to find some inequalities of H-H type for exponential type convex functions. In the next sections, we denote by L[m, n] the space of (Lebesgue) integrable functions on the interval [m, n].
Proof Firstly, from the property of the exponential type convex function of f , we get

Some new inequalities for exponential type convex functions
The aim of this section is to find new estimates that refine H-H inequality for functions whose first derivative in absolute value at certain power is exponential type convex. Dragomir and Agarwal [2] used the following lemma.

is the arithmetic mean of u and v.
Proof From Lemma 4.1 and the inequality we get holds for k ∈ [0, 1], where 1 p + 1 q = 1 and A is the arithmetic mean.
Proof From Lemma 4.1, Hölder's integral inequality, and the following inequality: which is the exponential type convex function of |f | q , we get holds for k ∈ [0, 1].
Proof Assume first that q > 1. By using Lemma 4.1, Hölder's inequality, and the property of the exponential type convex function of |f | q , we obtain For q = 1, we consider the estimates from the proof of Theorem 4.1, which also follows step by step the above estimates.

Applications for special means
Throughout this section, for the sake of simplicity, the following notations are used for special means of two nonnegative numbers m, n (n > m): 1. The arithmetic mean It is well known that L p is monotonically increasing over p ∈ R. Moreover, L 0 = I, L -1 = L. hold.

Conclusion
In this paper, we studied the concept of exponential type convex functions, which is a new concept. We proved some new Hermite-Hadamard type integral inequalities for the newly introduced class of functions using an identity together with Hölder's integral inequality. Especially, we would like to emphasize that different types of integral inequalities can be obtained using this new definition.