Fractional Minkowski-Type Integral Inequalities via the Unified Generalized Fractional Integral Operator

This paper is aimed at presenting the uni ﬁ ed integral operator in its generalized form utilizing the uni ﬁ ed Mittag-Le ﬄ er function in its kernel. We prove the boundedness of this newly de ﬁ ned operator. A fractional integral operator comprising a uni ﬁ ed Mittag-Le ﬄ er function is used to establish further Minkowski-type integral inequalities. Several related fractional integral inequalities that have recently been published in various articles can be inferred.


Introduction
Integral operators are useful in the study of differential equations and in the formation of real-world problems in integral equations. They also behave like integral transformations in particular cases. In the past few decades, fractional integral operators have been defined extensively (see [1][2][3][4]). Recently, in [5] a unified integral operator is studied which has interesting consequences in the theory of fractional integral operators. This paper is aimed at presenting a unified integral operator in the more generalized form via the unified Mittag-Leffler function introduced in [6]. The boundedness of the newly defined integral operator is studied. By taking the power function ξ β ; β > 1, a unified generalized extended fractional integral operator is deduced and analyzed to construct Minkowski-type integral inequalities. This is the extension of our previous work on Minkowski-type integral inequalities [7]. The connection of the results of this paper is established with many published results of references [7][8][9]. We begin by reviewing several key Minkowski-type inequalities as well as some definitions that will be useful in our subsequent work.
The well-known Minkowski inequality is given as follows: Theorem 1. Let ϕ, ψ∈L m ½u, v. Then for m ≥ 1, we have Some more Minkowski-type inequalities are stated in the next results.
In Section 2, we give the definition of further generalized integral operator containing the unified Mittag-Leffler function. The boundedness of this integral operator is proved under the conditions stated in the definition. In Section 3, by applying a particular fractional integral operator for the power function, Minkowski-type fractional integral inequalities are established. In Section 4, reverse Minkowski-type fractional integral inequalities are presented. The connection of these inequalities with previous work is stated in the form of remarks and corollaries.

Generalized Version of a Unified Integral Operator
In this section, we introduce a generalized version of a unified integral operator containing a unified Mittag-Leffler function in its kernel and also discuss its boundedness.

Journal of Function Spaces
Next, we discuss the boundedness of the newly defined generalized form of unified fractional integral operator.
Proof. According to the statement, ζ/ξ is an increasing function; therefore, the following inequality prevails: Since g is differentiable and increasing and ϕ is a positive function, so the above inequality remains preserved by multiplying it with g ′ ðtÞϕðtÞ. Therefore, we obtain the following inequality: Multiplying (23) by MðωðgðξÞ − gðtÞÞ α ; sÞ and integrating over ½u, ξ one can get Solving the above definite integral, we get Similarly, one can easily prove (21).

Unified Versions of Minkowski-Type Fractional Integral Inequalities
In this section, we give proof of unified versions of generalized Minkowski-type integral inequalities.

Reverse Minkowski-Type Fractional Integral Inequalities
In this section, we state and prove some reverse versions of Minkowski-type inequalities that are the generalizations of (2), (3), and (5).