Generalized integral inequalities for fractional calculus

In this paper, we present a variety of integral inequalities in and spaces for the integral operator involving generalized Mittag-Leffler function in its kernel, Hilfer fractional derivative, generalized Riemann-Liouville and Riemann-Liouville k-fractional integral operators.


Introduction
The importance of the fractional integral inequalities is enormous in establishing the uniqueness of solutions for certain fractional partial differential equations. This theory is also helpful in providing bounds for the solutions of fractional boundary value problems. In this era of progress and development, the theory of fractional integral inequalities catches the attention of many mathematicians and they provide plenty of applications of integral inequalities in fractional calculus. For more details see Anastassiou (2009), Ansari, Liu, and Mishra (2017), Mishra and Sen (2016), Iqbal, Pečarić, Samraiz, and Sultana (2015), Iqbal, Krulić, and Pečarić (2010), Niculescu and Persson (2006). Mitrinović and Pečarić (1991) introduced an integral inequality which later generalized by Farid, Iqbal, and Pečarić (2015). In the present work, we have paid attention to provide applications of the generalized integral inequality presented in (Farid et al., 2015) for fractional calculus.
We start with the definition of L p,r space given in Mubeen and Iqbal (2016).
ABOUT THE AUTHOR Sajid Iqbal is working as an assistant professor of Mathematics in University of Sargodha (Sub-Campus Bhakkar), Bhakkar, Pakistan. He is mainly known for works in Mathematical Inequalities involving convex functions and application in fractional calculus. He received his PhD degree from Abdus Salam School of Mathematical University, Government College University Lahore, Pakistan. He is a member of Pakistan Mathematical Society. He has published more than 30 research papers in high-quality international journals and works as a reviewer for in many international journals. He has supervised a PhD student and 13 MPhil students.

PUBLIC INTEREST STATEMENT
The importance of Mathematical inequalities is felt from the beginning and is now extensively known as one of the most important motivating forces behind the progress of current real analysis. This theory plays significant part in approximately all branches of Mathematics as well as in other areas of science. Here we have focused to present a variety of integral inequalities in generalized L p spaces involving fractional integral operators. The involvement of generalize fractional integral operator makes our results more general. where 1 ≤ p < ∞, and r ≥ 0.

Theorem 1.2
Let (Ω 1 , Σ 1 , 1 ), (Ω 2 , Σ 2 , 2 ) be measure spaces with -finite measures and f i :Ω 2 → ℝ, i = 1, 2, 3, 4, be non-negative measurable functions. Let g belongs to a particular class of functions U(f, k) which admits the representation where k:Ω 2 × Ω 1 → ℝ is a general non-negative kernel and f :Ω 1 → ℝ be a real valued function. If p, q are two real numbers such that 1 p + 1 q = 1, p > 1, then the inequality holds true, where , then we get the following inequality where The rest of the paper is organized as follows: In Section 2, we present the generalized integral inequality for six parameter fractional integral operator with Mittag-Leffler function in its kernel. Section 3 contains results for Hilfer fractional derivative. Section 4 consists of consequences for generalized Riemann-Liouville fractional integral operator. In the last section, we derive results for the Riemann-

Generalized integral inequality for fractional integral operator with six parameter Mittag-Leffler function in its kernel
First, we give the definition of the Mittag-Leffler function (see Mittag-Leffler, 1903) and fractional integral operator involving the generalized Mittag-Leffler function appearing in the kernel (see Salim & Faraj, 2012). • z n n! which was introduced by Shukla and Prajapati in (2007). In Srivastava and Tomovski (2009) investigated the properties of this function and its existence for a wider set of parameters.
• = p = q = 1, the operator (2.1) is defined by Prabhakar in (1971) and is denoted Wiman's function presented in Wiman (1905), and moreover, if = 1, then the Mittag-Leffler function E (z) will be the result.

Generalized integral inequality for Hilfer fractional derivative
In this section, we present the integral inequality (1.1) for the Hilfer fractional derivative. Let us recall the definition of Hilfer fractional derivative which is presented in Tomovski and Rudolf Hilfer Srivastava (2010).
The fractional derivative operator D , a+ of order 0 < < 1 and type 0 < ≤ 1 with respect to x ∈ [a, b] is defined by whenever the right hand side exists. The derivative (3.1) is usually called Hilfer fractional derivative.

Consequences for generalized Riemann-Liouville fractional integral operator
In this section, we find the applications of integral inequality (1.1) for generalized Riemann-Liouville fractional integral operator and extract the results of Farid et al. (2015) as special case. The generalized Riemann-Liouville fractional integral is defined as follows: b], then the left and right sided generalized Riemann-Liouville fractional integrals of order ≥ 0 and r ≥ 0 are given by where Γ is the Euler gamma function.
Theorem 4.2 Let f ∈ L 1,r [a, b] and the fractional integral operator I ,r a+ of order ≥ 0 and type r ≥ 0. Moreover p, q be two real numbers such that 1 p + 1 q = 1, p > 1, then the inequality