On boundedness of unified integral operators for quasiconvex functions

This work deals with the bounds of a unified integral operator with which several fractional and conformable integral operators are directly associated. By using quasiconvex and monotone functions we establish bounds of these integral operators. We prove their boundedness and continuity. The results of this paper generalize already published results and have direct consequences for fractional and conformable integrals

An k-fractional analogues of the Riemann-Liouville integral operators are given in the next definition. [a, b]. Then the k-fractional Riemann-Liouville integrals of order μ with (μ) > 0, k > 0, are defined by We go ahead by defining the following generalized fractional integral operators:

Definition 3 ([15]) Let f : [a, b]
→ R be an integrable function. Let g be an increasing 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 a function g on [a, b] of order μ with (μ) > 0 are defined by (1.8) A generalized fractional integral operator containing an extended Mittag-Leffler function is defined as follows.
Recently, Farid in [7] studied the unified integral operator stated as follows (see also, [17]): , and g is differentiable and strictly increasing. Also, let φ x be an increasing function on [a, ∞), and let 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 For suitable settings of functions φ, g and certain values of parameters included in the Mittage-Leffler function (1.11), many of the fractional integral operators defined in recent decades can be obtained simultaneously; see [17,Remarks 1 and 2].
The aim of this paper is the study of bounds of a unified integral operator by using quasiconvex functions. The results we intend to establish are directly related with fractional and conformable integral operators. All the fractional and conformable integral operators defined in [2, 3, 6, 10, 13-15, 18, 20, 21, 23-26] satisfy the results of this paper for quasiconvex functions in particular cases.

Definition 7 ([22]) A function f satisfying the inequality
for λ ∈ [0, 1] and x, y ∈ C, where C is a convex set, is called a convex function on C.
A geometric interpretation of a convex function f : [a, b] → R is visualized by the wellknown Hadamard inequality Finite convex functions defined on a finite closed interval are quasiconvex functions, whereas quasiconvex functions are defined as follows.

Definition 8 ([12]) A function f satisfying the inequality
for λ ∈ [0, 1] and x, y ∈ C, where C is a convex set, is called a quasiconvex function on C.
The following example distinguishes the above two definitions.
Thus the class of quasiconvex functions contains the class of finite convex functions defined on finite closed intervals. The investigation of Hadamard inequality for quasiconvex functions is an implicit topic, and related results have been obtained independently by various authors; see, for example, [5,11,12] and references therein.
To get results for unified integral operators of quasiconvex functions, we follow the method from [17]. The paper is organized as: First, we obtain upper bounds of unified integral operators defined in (1.12) and (1.13), which lead to the boundedness and continuity of these operators. Then we obtain bounds in the form of a Hadamard-type inequality by imposing the symmetric property on quasiconvex functions. Finally, by defining the convolution of two functions we obtain a modulus inequality. All these results hold for almost all kinds of associated fractional and conformable integral operators. Also, some very particular cases of the proved results are already published in [4,9,27], and connection with them is stated in remarks.

2)
and hence Proof Under the assumptions of the theorem, we can obtain the inequality we get the following inequality: Using the quasiconvexity of f , for t ∈ [a, x], we have f (t) ≤ M x a (f ). Therefore we get the inequality By using (1.12) of Definition 6 on the left-hand side and integrating the right-hand side we obtain the inequality Now, on the other hand, for t ∈ (x, b], we have the following inequality: Using the quasiconvexity of f , for t ∈ [x, b], we also have f (t) ≤ M b x (f ). From (2.8), using (1.13) of Definition 6, we obtain By adding (2.6) and (2.9) we can achieve (2.3).
The following remark establishes connections with already known results. Further consequences of Theorem 1 are studied in the following results.
Theorem 2 Under the assumption of Theorem 1, we have Similarly, by putting x = a in (2.9) we obtain By adding (2.11) and (2.12) we obtain (2.10).  (1.12) and (1.13) are continuous. Also, we have
To prove the next result, we need the following lemma.
The following result provides upper and lower bounds of operators (1.12) and (1.13) in the form of Hadamard inequality.

Theorem 4 Under the assumptions of Theorem 1, if in addition f
Proof Under the assumptions of the theorem, we can obtain the inequality By using E γ ,δ,k,c μ,α,l (ω(g(x)g(a)) μ ; p) ≤ E γ ,δ,k,c μ,α,l (ω(g(b)g(a)) μ ; p) we get the inequality Using the quasiconvexity of f , for By using Definition 6 on the left-hand side and integrating the right-hand side we obtain the inequality On the other hand, for x ∈ (a, b), we have the inequality Adopting the same pattern of simplification as we did for (2.20), we can observe the following inequality for (2.23): By adding (2.21) and (2.24) we arrive at the inequality Multiplying both sides of (2.17) by φ(g(x)-g(a)) g(x)-g(a) g (x)E γ ,δ,k,c μ,α,l (ω(g(x)g(a)) μ ; p) and integrating over [a, b], we have From Definition 6 we obtain the inequality , ω; p). (2.26) Similarly, multiplying both sides of (2.17) by ) μ ; p) and integrating over [a, b], we have By adding (2.26) and (2.27) we obtain the inequality (2.28) Using (2.25) and (2.28), we arrive at (2.18).