Hermite-Hadamard and Jensen’s type inequalities for modiﬁed ( p , h ) -convex functions

: In this study, we will derive the conception of modiﬁed ( p , h )-convex functions which will unify p -convexity with modiﬁed h -convexity. We will investigate the fundamental properties of modiﬁed ( p , h )-convexity. Furthermore, we will derive the Hermite-Hadamard, Fej´er and Jensen’s type inequalities for this generalization.


Introduction
In non-linear programming and optimization theory, convexity plays an important role. Three important areas of non-linear analysis are monotone operator theory, convex analysis and theory of non-expansive mapping. In early 1960 these theories get emerged. Theses areas have got the attention of many researcher and many connections have been identified between them over the past few years. The notion of convexity has been expanded and generalization in numerous ways utilizing novel and modern methods in recent years. Convexity also plays vital part in fields outside mathematics such as chemistry, biology, physics and other sciences. It is always interesting to generalize the definition of convexity from different aspects because the wide range of applications. Recently the object of numerous studies have been the convexity of functions and sets. For further studies on generalization of convexity one can see [1][2][3][4][5][6][7][8][9] and references therein.
Let us see some basic definitions and generalizations of convex functions [21,22].
for all r, s ∈ I.
for all x, y ∈ J, where J ⊂ R. If inequality (1.1) is reversed, then h is said to be a submultiplicative function.
is the Gamma function. Definition 1.4. (Convex function) Let I ⊆ R be an interval, then a function f : I → R is called convex if the following inequality holds: for t ∈ [0, 1] and for all r, s ∈ I, whereas p = n m or p = 2k+1, m = 2t+1, n = 2u+1, and t, u, k∈ N.
for t ∈ (0, 1) and ∀ r, s ∈ I, where p > 0. Likewise, if the inequality sign in (1.2) is inverted, then f is known as a modified (p, h)-concave function.
Of course, if we put in (1.2) 1. p = 1 then we get modified h-convex function; 2. p = 1 and h(t) = t then we get classical convex function.
The paper is organized as follows: In next section, we will derive some basic properties of this generalization. However, the third, fourth and fifth sections are devoted to develop Hermite-Hadamard inequality, Jensen type inequality and Fejér type inequalities for modified (p, h)-convex functions.

Basic results
In this section, we will verify our basic properties.
Proposition 1. Assume f i : I ⊂ R → R be modified (p, h)-convex function, suppose µ i , · · · , µ n be positive scalers. Consider a function g from R to R so that ADD then g is modified (p, h)-convex function.
Proof. We know that f i : I ⊂ R → R be modified (p, h)-convex functions. Then t ∈ [0, 1] and ∀ , r, s ∈ I, we have The proof is completed.
Proof. Since g is modified (p, h)-convex function on I, we obtained Then by using the convexity of f , we obtain which implies that f • g is modified (p, h)-convex function. 1]. Further let f j : I → R, j ∈ N is non empty collection of modified (p, h)-convex functions such that for each x ∈ I, max j∈J f j (x) exists in R, then the function Proof. For any x, y ∈ I and t ∈ [0, 1], we have which is required.
Further let g and f are two modified (p, h)-convex functions.
Then the product of f and g will be a modified (p, h)-convex function if g and f are similarly ordered.
Proof. We know that g and f are modified (p, h)-convex function. Then That's the required result.
Remark 2. If we put p = 1 and h(t) = t in (3.1) then we attain classical Hermite-Hadamard type inequality.
We know that f is a modified (p, h)-convex function, then Multiplying above inequality by t α−1 and then integrating over t ∈ [0, 1], we get The result is completed.

Jensen type inequality
The following expression is useful to prove Jensen type inequality for modified (p, h)-convex functions.
Proof. By using (4.1) it follows that: using the fact that h is supermultiplicative function, we have using the fact that h is supermultiplicative function, The result is completed.
Remark 4. If we choose p = 1 and h(t) = t in (4.2) then we have classical Jensen type inequality.
By setting r = [tv p 1 + (1 − t)v p 2 ] 1 p , we get r p−1 f (r)g(r)dr ≤ f (v 2 )g(v 2 )   Remark 5. If we take p = 1 and h(t) = t in (5.1) then we will get the result for convex function.

Conclusions
Convexity play an important rule in applied sciences and mathematics. In this paper, we introduced modified (p, h)-convex functions which unify p-convexity with modified h-convexity. We investigated the fundamental properties of modified (p, h)-convexity and gave the Hermite-Hadamard, Fejér and Jensen's type inequalities.