On an integral and consequent fractional integral operators via generalized convexity

Abstract: Fractional calculus operators are very useful in basic sciences and engineering. In this paper we study an integral operator which is directly related with many known fractional integral operators. A new generalized convexity namely exponentially (α, h−m)-convexity is defined which has been applied to obtain the bounds of unified integral operators. A generalized Hadamard inequality is established for the generalized convex functions. The established theorems reproduce several known results.


Introduction
Convex functions have wide applications in mathematical analysis, optimization theory, mathematical statistics, graph theory and many other subjects. Convex function is expressed and visualized in different ways, its analytic representation (1.1) provides motivation to define new concepts and notions. It is generalized in different forms, the (h − m)-convex function is defined as follows: Remark 3. i) If we take m = 1, then exponentially s-convex function defined by Mehreen et al. in [5], can be achieved. ii) If we take s = m = 1, then exponentially convex function defined by Awan et al. in [6], can be achieved. iii) If we take α = 0, then (s, m)-convex function defined by Efthekhari in [7], can be achieved. iv) If we take α = 0 and m = 1, then s-convex function defined by Hudzik in [8], can be achieved. v) If we take α = 0 and s = 1, then m-convex function defined by Toader in [9], can be achieved. vi) If we take α = 0 and s = m = 1, then convex function (9), can be achieved.
We will unify all above generalizations of convex functions in a single notion which will be called exponentially (α, h − m)-convex function (see Definition 12). Further we will use this generalized convexity for getting bounds of a unified integral operator. In the following, we give the definition of this unified integral operator and definitions of some of associated fractional integral operators. Definition 6.
[10] Let f, g : [a, b] −→ R, 0 < a < b, be the functions such that f be positive and f ∈ L 1 [a, b], and g be differentiable and strictly increasing. Also let φ x be an increasing function on [a, ∞) and α, l, γ, c ∈ C, p, µ, δ ≥ 0 and 0 < k ≤ δ + µ. Then for x ∈ [a, b] the left and right integral operators are defined by where the involved kernel is defined by and E γ,δ,k,c µ,α,l is the Mittag-Leffller function given in (1.18).
For suitable settings of functions φ, g and certain values of parameters included in Mittag-Leffler function, several recently defined known fractional integrals can be reproduced, see [11,Remarks 6 & 7].
An analogue k-fractional Riemann-Liouville integral operators are given in next definition.
[12] Let f : [a, b] → R be an integrable function. Also let g be an increasing and positive function on (a, b], having a continuous derivative g on (a, b). The left-sided and right-sided fractional integrals of a function f with respect to another function g on [a, b] of order µ where (µ) > 0 are defined by: (1.13) Definition 10.
Fractional integrals have a great importance in the field of mathematical inequalities. In recent decades many researchers introduced new fractional integral operators which have been used to obtain several types of fractional integral inequalities, see [12,13,[17][18][19][20][21][22][23][24][25]. The objective of this paper is to obtain the bounds of a unified integral operator utilizing exponentially (α, h − m)-convexity which are associated with many fractional integral inequalities.
In Section 2 we will list some properties of the kernel involved in the unified integral operators, which will be helpful in proving the main results of the paper. In Section 3 by using exponentially (α, h − m)-convex functions, upper bounds of unified integral operators (1.5) and (1.6) are obtained. Furthermore, by using condition of symmetry, two sided (upper an lower) bounds in the form of Hadamard inequality are obtained. Also we establish a related inequality by using exponentially (α, h − m)-convexity of function | f | and by defining integral operators for convolution of two functions.

Main results
First we define a generalized convexity namely exponentially (α, h − m)-convexity as follows: Definition 12. Let J ⊆ R be an interval containing (0, 1) and let h : Remark 4. All kinds of convex functions which are defined in the introduction section are deducible from above definition.
The following result provides upper bound for unified integral operators of (α, h − m)-convex functions.
] −→ R be differentiable and strictly increasing function, also let φ x be an increasing function. Then for unified integral operators the following inequality holds: The following result provides generalized Hadamard inequality for exponentially (α, h − m)-convex functions.
Theorem 3. The conditions on f, g and φ are same as in Theorem 1 and in addition if f is exponentially m-symmetric, then we have  . Then for unified integral operators the following inequality holds: (g(t)).

Proofs of main results
In this section we give the proofs of the results stated in aforementioned section.
By using (1.5) of Definition 6 on left hand side, and by setting z = x − t x − a on right hand side, the following inequality is obtained: Above inequality can be written as On the other hand, multiplying (4.2) and (4.4), by using (1.6) of Definition 6 on left hand side and integrating over (x, b] on right hand side, we obtain: Above inequality can be written as By adding (4.7) and (4.9), (3.2) can be obtained.
Proof of Theorem 2. From (4.7) we have Similarly, from (4.9) the following inequality holds: The boundedness with linearity provides the continuity.
By using (1.5) of Definition 6 on left hand side, and by setting z = x − t x − a on right hand side, the following inequality is obtained: Above inequality can be written as Adopting the same pattern of simplification as we did for (4.11) and (4.13), the following inequality can be observed for (4.13) and (4.12): By adding (4.15) and (4.16), following inequality can be achieved: Multiplying both sides of (4.10) by K a x (E γ,δ,k,c µ,α,l , g; φ)d(g(x)), and integrating over [a, b] we have f (x) e ηx d(g(x)).
By using (1.5) of Definition 6 on left hand side, and by setting z = x − t x − a on right hand side, the following inequality is obtained: Above inequality can be written as If we consider the left hand side from the inequality (4.22), and adopt the same pattern as we did for the right hand side inequality, we have g F φ,γ,δ,k,c µ,α,l,a + f (x, ω; p) ≥ −K a x (E γ,δ,k,c µ,α,l , g; φ)(x − a) (4.25) From (4.24) and (4.25), following inequality is observed: (4.27) On the same pattern as we did for (4.1) and (4.21), one can get following inequality from (4.2) and (4.27): Above inequality can be written as By adding (4.26) and (4.28), inequality (3.4) can be achieved.

Applications of main results
In this section by applying Theorem 1 we give some interesting consequences. The reader can obtain the applications of Theorems 2, 3 and 4. Some Hadamard Inequalities for exponentially (α, h − m)-convex functions: By applying Theorem 3 we give fractional Hadamard inequalities for exponentially (α, h − m)convex functions.

Conclusions
This paper provides estimates of an integral operator via generalized convexity namely exponentially (α, h − m)-convexity. The given results consist of bounds of this generalized integral operator fo exponentially (α, h − m)-convex functions. All the results hold for various associated fractional integral operators and notions of convexities; namely (α, h − m)-convexity, (h − m)-convexity, (α, m)-convexity, (s, m)-convexity and related convex functions. The reader can get the results for several kinds of fractional integral operators of convex and related functions given in Remarks 1-3.