Estimations of fractional integral operators for convex functions and related results

This research investigates the bounds of fractional integral operators containing an extended generalized Mittag-Leffler function as a kernel via several kinds of convexity. In particular, the established bounds are studied for convex functions and further connected with known results. Furthermore, these results applied to the parabolic function and consequently recurrence relations for Mittag-Leffler functions are obtained. Moreover, some fractional differential equations containing Mittag-Leffler functions are constructed and their solutions are provided by Laplace transform technique.


Introduction
Fractional calculus is the generalization of classical calculus. Fractional integral/derivative operators play a key role in the development of fractional calculus. They have been used to formulate various physical and dynamic problems in fractional models. The complex behavior of physical systems can be represented in terms of fractional models. For applications of fractional calculus operators in sciences and engineering we refer to reader to [5-7, 19, 26, 51]. Physical properties of viscoelastic material can be interpreted by a model of fractional derivatives [4]. Furthermore, fractional calculus is applied to physics [21], bioengineering [29] optics [9,18,25], fluid flow [12], energy systems [11,28] and biology [22][23][24].
On the other hand fractional integral/derivative operators have been used to construct and formulate new results in the theory of inequalities. Many of the well-known inequalities and related results are generalized and extended via fractional integral/derivative operators; see [2, 14-17, 30-32, 44, 52] and the references therein. At the same time convexity plays a vital role in enhancing the theory of inequalities, and facilitates optimization theory, mathematical analysis, mathematical statistics, graph theory with many other subjects. Fractional integral inequalities being suitable constraints provide existence and uniqueness of solutions for several mathematical problems in the form of fractional models.
The goal of this paper is the study of fractional integral operators containing Mittag-Leffler functions in their kernels. These operators are comprised in a single definition (Eq. (2.4) and Remark 1). We will analyze them for a generalized notion of convexity called (hm)-convexity. The method of proving the results of this paper can be utilized to get the results for other kinds of fractional and conformable integrals/derivatives already exist in the literature which authors will may consider for their future work; for instance for convenience Caputo-Fabrizio derivatives [3,10] can be used.

Preliminary results
In this section we give definitions and notions which will be useful to establish the results of this paper.

Definition 1
Let I be an interval in R. A function f : I → R is said to be convex if, for all a, b ∈ I and 0 ≤ t ≤ 1, the following inequality holds: Convex functions are further generalized in different ways. One of the generalizations of convex functions is called (hm)-convexity that contains several kinds of convexity for example h-convexity, m-convexity, s-convexity defined on the right half of real line including zero (see [35,45]).

Definition 2
Let J ⊆ R be an interval containing (0, 1) and let h : In the solution of integral and differential equations, the exponential function arises while in the solutions of fractional integral and differential equations, Mittag-Leffler function appears naturally. The Mittag-Leffler function is defined as follows [33]: The Mittag-Leffler functions are used in many areas of science and engineering, especially in the theory of fractional differential equations, in solutions of generalized fractional kinetic equations (see [40]). The Mittag-Leffler function was generalized by many mathematicians: for example Wiman [46], Agarwal [1], Prabhakar [36], Shukla and Prajapati [41], Salim [38], Salim and Faraj [39], Rahman et al. [37]. For a detailed study of this function see [20,27,36,37,39,[41][42][43].
Then the generalized fractional integral operators γ ,δ,k,c μ,α,l,ω,a + f and The following remark provides the connection of integral operators with already known fractional integral operators.

Remark 1 The operator in (2.4) contains various fractional operators:
(i) Setting p = 0, it reduces to the fractional integral operator defined by Salim-Faraj in [39]. (ii) Setting l = δ = 1, it reduces to the fractional integral operator defined by Rahman et al. in [37]. (iii) Setting p = 0 and l = δ = 1, it reduces to the fractional integral operator defined by Srivastava and Tomovski in [42]. (iv) Setting p = 0 and l = δ = k = 1, it reduces to the fractional integral operator defined by Prabhakar in [36]. (v) Setting p = ω = 0, it reduces to the Riemann-Liouville fractional integral.
, then the following inequality holds: The rest of the paper is organized as follows: In Sect.

be a real valued function. If f is positive and
Then first we observe the function f on the interval [a, x]; for t ∈ [a, x] and α ≥ 1, one has the following inequality: Now on the other hand we address the function f on the interval [x, b]; for t ∈ [x, b] and β ≥ 1, one has the following inequality: Similarly multiplying (3.6) and (3.7), then integrating over [ Adding (3.5) and (3.8), inequality (3.1) is obtained.
If m = 1 and h(z) = z in (3.1), then the following result holds for a convex function.
Proof Let x ∈ [a, b] and t ∈ [a, x]. Then using (hm)-convexity of |f |, we have From (3.11), one has Multiplying (3.2) and (3.12), then integrating over [a, x], we get The left hand side is calculated thus: Put now by putting xz = t, in second term of the right hand side of the above equation and by using (2.4) and (2.6), we get Therefore, (3.13) takes the form (3.14) Also from (3.11), one has Following the same procedure as one did for (3.12), we also have From (3.14) and (3.16), we get Along the same lines as for (3.2), (3.12) and (3.15), one can get from (3.6) and (3.18) the following inequality: From (3.17) and (3.19) via the triangular inequality, inequality (3.10) is obtained.
If m = 1 and h(z) = z in (3.10), then the following result holds for a convex function. (3.20) From this we have On the other hand for x ∈ [a, b], we have Similarly multiplying (3.23) and (3.25), then integrating over [a, b], we get , we get by using (2.5) and (2.7), we get   in [44], as follows: Below, the results of Sect. 2 are applied to obtain recurrence inequalities for Mittag-Leffler functions.

Corollary 5
If m = 1 and h(z) = z in (4.6), then we have Theorem 6 Mittag-Leffler functions satisfy the following recurrence relation: Proof In (4.4), putting x = a and x = b, then adding for α = β, inequality (4.9) is obtained.

Corollary 6
If m = 1 and h(z) = z in (4.9), then we have Theorem 7 Mittag-Leffler functions satisfy the following recurrence relation: Proof In (4.7), putting x = a and x = b, then adding for α = β, inequality (4.11) is obtained. (4.14) Theorem 9 Mittag-Leffler functions satisfy the following recurrence relation: By applying Theorem 3 similar relations can be established; we leave these for the reader.

Fractional differential equations involving extended generalized Mittag-Leffler function
In this section, generalized fractional differential equations are solved. The Riemann-Liouville fractional derivative operator D ν a + is defined as follows: where (I ν a + f )(x) is the Riemann-Liouville fractional integral operator defined as follows: For a = 0 the operator (D ν a + f )(x) is represented by (D ν 0 + f )(x) and (I ν a + f )(x) is represented by The Laplace transform of a function f (x) is defined as follows: In [34], the Laplace transform of fractional derivative (D ν 0 + f )(x) is found to be For more information related to differential equations see Refs. [47][48][49][50].

10)
where C is an arbitrary constant.
Proof Using the generalized fractional integral operator ( γ ,δ,k,c μ,α,l,ω,a + 1)(x; p) given in (2.6) with a = 0 in (5.9), we have Applying the Laplace transform on both sides of (5.11), we have First we calculate the Laplace transform of Mittag-Leffler function as follows: Since L[ t n-1 Γ (n) ] = 1 s n (n > 0), using it in the above, we have (5.14) By using (5.3), (5.4) (for n = 1) and (5.14) in (5.12), we have which implies that Now taking the inverse Laplace transformation on both sides of (5.15), we have After simplification one can get (5.10).
inequalities for recurrence relations of Mittag-Leffler functions are obtained via a particular convex function x 2 . At the end some generalized fractional differential equations are solved.