Study of inequalities for unified integral operators of generalized convex functions

The aim of this paper is to study unified integral operators for generalized convex functions namely (α, h−m)-convex functions. We obtained upper as well as lower bounds of these integral operators in diverse forms. The results simultaneously hold for many kinds of well known fractional integral operators and for various kinds of convex functions.


Introduction
T he notion named convexity has applications in almost all branches of mathematics for instance in mathematical analysis, optimization theory, mathematical statistics, graph theory etc. It has been used elegantly for the improvement of solutions of the proposed problems. The fascinating and elegant appearance of convex functions in different forms provides the motivation and encouragement for defining new concepts and definitions. For example the notions of superquadratic function, quasi-convex function, (α, m)-convex function, (h − m)-convex function, (s, m)-convex function, exponentially convex function and many more are due to convex functions, see [1][2][3][4][5][6][7] and references therein.
Convex functions are directly related with many known inequalities including Hadamard inequality, Jensen inequality, Arithmetic mean-Geometric mean inequality, Holder inequality etc. Also they have been proved very useful in the establishment of new inequalities. In recent era fractional integral inequalities are in focus of researchers. Many interesting extensions and generalizations for many well known inequalities have been produced by using different types of fractional integrals, see [8][9][10][11][12][13] and references therein.
The aim of this paper is to establish integral inequalities for (α, h − m)-convex functions which have direct consequences to several known fractional integral operators and for functions deducible from (α, h − m)-convex functions. We proceed by giving definitions of generalized fractional integral operators. Definition 1. [14] 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 Definition 2. [13] 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), k > 0 are defined by where Γ k (.) is defined as follows A fractional integral operator containing an extended generalized Mittag-Leffler function in its kernel is defined as follows: Then the generalized fractional integral operators γ,δ,k,c µ,α,l,ω,a + f and where is the extended generalized Mittag-Leffler function.
The following integral operator unifies all above definitions in a single form.
Next we give some definitions that generalize the notion of convexity. and t ∈ (0, 1), one has By selecting different values of function h and parameter m, the above definition produces many well known definitions.

Remark 1.
(i) If m = 1, then h-convex function can be obtained. (ii) If h(t) = t, then m-convex function can be obtained. (iii) If h(t) = t and m = 1, then convex function can be obtained. (iv) If h(t) = 1 and m = 1, p-function can be obtained.
(v) If h(t) = t s and m = 1, then s-convex function can be obtained.
(vii) If h(t) = 1 t s and m = 1, then s-Godunova-Levin function of second kind can be obtained.
The next definition unifies the above two definitions elegantly.

Definition 7.
[2] Let J ⊆ R be an interval containing (0, 1) and let h : One can note that the definitions of (h − m)-convex and (α, m)-convex functions can be obtained by setting α = 1 and h(t) = t respectively in (14). Motivated by this generalized convexity we are interested to investigate the bounds of the sum of left and right sided integral operators for these functions. By using (α, h − m)-convex functions, upper bounds of (9) and (10) are obtained. Furthermore by using condition of symmetry, two sided Hadamard type bounds are obtained and then by using (α, h − m)-convexity of function | f | and by defining an integral operator for convolution of two functions further bounds are studied.
In [27], we studied the properties of the kernel given in (11). Here we are interested in the following property.
(P) Let g and φ I be increasing functions. Then for m < t < n, m, n ∈ [a, b] the kernel K n m (E γ,δ,k,c µ,α ,l , g; φ) satisfies the following inequality: This can be obtained from following two straightforward inequalities: The reverse of inequality (15) holds when g and φ I are of opposite monotonicity.

Main results
The following theorem provides upper bound for unified integral operators of (α, h − m)-convex functions.
Theorem 1. Let f ∈ L 1 [a, b] be a positive (α, h − m)-convex function, 0 ≤ a < mb, m = 0. Let g be differentiable and strictly increasing function and φ x be an increasing function on [a, b]. If α , β, l, γ, c ∈ C, p, µ ≥ 0, δ ≥ 0 and where H a Proof. By (P), the following inequalities hold From (19) and (21), the following integral inequality holds By using (9) of Definition 4 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 follows On the other hand, multiplying (20) and (22), the following inequality holds By using (10) of Definition 4 on left hand side and by setting z = t − x b − x on right hand side, the following inequality is obtained Above inequality can be written as follows By adding (25) and (27), (18) can be obtained.
The following remark summarizes the connection of Theorem 1 with fractional integral inequalities.
Therefore from linearity and boundedness, the continuity of unified integral operators is followed.
The following lemma is important to proof of the upcoming theorem: Then the following inequality holds: Proof. Since f is (α, h − m)-convex, therefore the following inequality is valid in above inequality, we get (28).

Remark 4.
(i) If we put h(x) = x and α = m = 1 in (28) Proof. By (P), the following inequalities hold Multiplying (30) and (32) and integrating the resulting inequality over [a, b], we obtain b a K a x (E γ,δ,k,c µ,β,l , By using (9) of Definition 4 on left hand side and by supposing z = x − a b − 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 (30) and (32), the following inequality can be obtained from (32) and (31) By adding (34) and (35), following inequality can be achieved Multiplying both sides of (28) by K a x (E γ,δ,k,c µ,β,l , g; φ)d(g(x)) and integrating over [a, b], we have (g(x)).
The following remark summarizes the connection of Theorem 3 with already known fractional integral inequalities.
By using (9) of Definition 4 on left hand side and by supposing 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 (42) and adopting the same pattern as we did for the right hand side inequality, we have (vi) If we put m = 1 and h(t) = t in the result of (v), then [30,Corollary 2] can be obtained.

Results for (h − m)-convex functions
In this section we give results for (h − m)-convex functions, that are deduced from main results.
In the following theorem, we establish Hadamard type inequality for (h − m)-convex functions: