Fractional versions of Minkowski-type integral inequalities via unified Mittag-Leffler function

*Correspondence: ayyaz3007@gmail.com 2Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan Full list of author information is available at the end of the article Abstract We present unified versions of Minkowski-type fractional integral inequalities with the help of fractional integral operator based on a unified Mittag-Leffler function. These inequalities provide new as well as already known fractional versions of Minkowski-type inequalities.


Introduction
A Swedish mathematician Gosta Mittag-Leffler introduced a function in the form of a power series [1] E α (t) = ∞ l=0 t l (αl + 1) , where t, α ∈ C and (α) > 0. This function is called the Mittag-Leffler function. It plays a vital role in the representation of solutions of fractional differential equations. Many researchers have given its various generalizations and extensions, which are used to formulate solutions of real-world problems in different fields of science and engineering [2,3]. The Mittag-Leffler function is also used to introduce new generalized fractional integral operators. These integral operators are frequently used for extensions and generalizations of well-known classical integral inequalities. For a detailed study of the Mittag-Leffler function, we refer the readers to [4][5][6][7][8].
In [9] a generalization of the Mittag-Leffler function is given in the form of Q-function. In [6] the extended generalized Mittag-Leffler function and its related fractional integral operator are described along with their applications to generalizing classical Opial-type inequalities. In [10] a unified form of the Mittag-Leffler function, which generates a generalized Q-function and the extended generalized Mittag-Leffler function, is studied; also, a fractional integral operator containing a unified Mittag-Leffler function is introduced, and its boundedness is proved. For some recent related work, we refer the readers to [11][12][13][14].
In this paper, we present Minkowski-type inequalities by using the fractional integral operator corresponding to the unified Mittag-Leffler function. The findings of this paper are implicitly related with several Minkowski-type inequalities already studied for different kinds of known fractional integral operators. Some particular cases of the main results are explicitly given in the form of corollaries. First, we give the definition of the unified Mittag-Leffler function and the associated fractional integral operator (see [10]).
A reversed Minkowski-type inequality is given as follows.
Another reversed Minkowski-type inequality is given as follows.

Theorem 3 ([17]) Under the assumptions of Theorem
In the next section, we give some generalized versions of Minkowski-type integral inequalities using fractional integral operators containing the unified Mittag-Leffler function (1.1) defined in Definition 2. Also, the reversed Minkowski-type integral inequalities for these fractional integral operators are proved.
Before moving toward the proof of our next result, we state a particular case of the GM-AM inequality for x, y ≥ 0 with r, s > 1 satisfying r -1 + s -1 = 1, xy ≤ r -1 x r + s -1 y s , and also the elementary inequality (x + y) r ≤ 2 r-1 x r + y r , x, y ≥ 0, r > 1. (2.4)

Conclusions
We have proved some generalized Minkowski-type integral inequalities using fractional integral operators associated with unified Mittag-Leffler function. A number of such inequalities already studied for various types of known fractional integral operators can be deduced from the results of this paper. The unified Mittag-Leffler function and associated integral operators can be applied to extend and generalize the classical concepts.