General fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function

In this paper some new general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function are presented. These results produce inequalities for several kinds of fractional integral operators. Some interesting special cases of our main results are also pointed out.


Introduction, definitions, and preliminaries
Convex functions are very important in the field of integral inequalities. A lot of fractional integral inequalities and novel results have been established due to convex functions (for more details, see [1,8,13,14]).
Definition 1 A function f : I → R, where I is an interval in R, is said to be a convex function if f tx + (1t)y ≤ tf (x) + (1t)f (y) ( 1 ) holds for t ∈ [0, 1] and x, y ∈ I.
A convex function f : I → R is also equivalently defined by the Hadamard inequality where a, b ∈ I, a < b.
The concept of m-convexity was introduced in [17] and since then many properties, especially inequalities, have been obtained for this class of functions (see [3,6,7,18]). If we denote by K m (b) the set of m-convex functions defined on [0, b] for which f (0) < 0, then whenever m ∈ (0, 1). Note that in the class K 1 (b) there are only convex functions f : [4]), while k 0 (b) contains star-shaped functions.
is a 16 17 -convex function but it is not m-convex for any m ∈ ( 16 17 , 1].
For more results and inequalities related to m-convex functions, one can consult, for example, [3,6,7] along with the references therein.
In this paper we give general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function and deduce some results already published in [1,5,6,8,13]. Also we give a Hadamard type inequality for convex and m-convex functions by involving an extended Mittag-Leffler function.

Main results
Here we give some fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function and corresponding fractional integral operators given in (3) and (4). The following lemma is useful to establish the results.
Proof On integrating by parts one can have Subtracting (9) from (8), we get (7) which is the required identity.
If we take m = 1 in (7), then we get the following identity for a convex function.
We use identity (7) to establish the following fractional integral inequality.
Since |f | is an m-convex function, we have for t ∈ [a, mb].
Using (14) in (13) After simple calculation of the above inequality, we get (11) which is required.
If we take m = 1 in (11), then we get the following result for a convex function.  (16) for k < δ + (μ) and g ∞ = sup t∈ [a,b] |g(t)|. Next we give the following fractional integral inequality.
Proof From Lemma 2.1 and by using Hölder's inequality, we have Since |f (t)| q is an m-convex function, we have Using (20) After simple calculation of the above inequality, we get (17) which is required.
If we take m = 1 in (17), then we get the following result for a convex function.
for k < δ + (μ) and g ∞ = sup t∈ [a,b] |g(t)| and 1 p + 1 q = 1. In the next result we give Hadamard type inequalities for m-convex functions via an extended Mittag-Leffler function.