On generalization of an integral inequality and its applications

ABOUT THE AUTHOR I am working as Assistant Professor in the department of Mathematics COMSATS Institute of Information Technology, Attock Campus, Pakistan. I have completed my PhD in the subject of Mathematics from Abdus Salam School of Mathematical Sciences, Govt. College University, Lahore, Pakistan with specialization in Mathematical inequalities. My research interest area is Mathematical analysis, Functional analysis, Fractional Calculus, Mathematical Statistics. Recent paper is on generalization of an integral inequality and its applications in fractional calculus. Fractional integral inequalities are useful in establishing the uniqueness of solutions of certain partial differential equations also provides upper and lower bounds for the solutions of fractional boundary value problems. PUBLIC INTEREST STATEMENT Fractional calculus is as important as calculus and a lot of work has been published in the favor of fractional calculus. Fractional integral inequalities are useful in establishing the uniqueness of solutions of certain partial differential equations, and also provide upper and lower bounds for solutions of fractional boundary value problems. In this paper, an integral inequality is generalized and its applications in fractional calculus are found. Received: 31 December 2014 Accepted: 19 June 2015 Published: 12 August 2015


Introduction
In the ocean of inequalities, integral inequalities received great attention by many scientists, for example mathematicians, physicists, and statisticians. Here, we want to pay our attention to an integral inequality by Mitrinović and Pečarić (1991).

t)dh(t),
Additional information is available at the end of the article ABOUT THE AUTHOR I am working as Assistant Professor in the department of Mathematics COMSATS Institute of Information Technology, Attock Campus, Pakistan. I have completed my PhD in the subject of Mathematics from Abdus Salam School of Mathematical Sciences, Govt. College University, Lahore, Pakistan with specialization in Mathematical inequalities. My research interest area is Mathematical analysis, Functional analysis, Fractional Calculus, Mathematical Statistics. Recent paper is on generalization of an integral inequality and its applications in fractional calculus. Fractional integral inequalities are useful in establishing the uniqueness of solutions of certain partial differential equations also provides upper and lower bounds for the solutions of fractional boundary value problems.

PUBLIC INTEREST STATEMENT
Fractional calculus is as important as calculus and a lot of work has been published in the favor of fractional calculus. Fractional integral inequalities are useful in establishing the uniqueness of solutions of certain partial differential equations, and also provide upper and lower bounds for solutions of fractional boundary value problems. In this paper, an integral inequality is generalized and its applications in fractional calculus are found.
The organization of the paper is as follows: In Section 2, we give a generalization of integral inequality (Equation 1.2) and some remarks. In Section 3, we give applications for fractional integrals and fractional derivatives, and in Section 4, we give applications for Widder derivatives and linear differential operators. In the last Section 5, we give improvements of these results. In the whole paper, we suppose that all integrals exist.

Main results
Here, we give a generalization of integral inequality (Equation 1.2) and some remarks.

Applications for fractional integrals
, the left-sided and right-sided Riemann-Liouville fractional integrals of order > 0 are defined by    where Next, we observe the Caputo fractional derivatives (for details see Kilbas et al., 2006, Section 2.4, also Anastassiou, 2009, p. 449;Baleano et al., 2012, p. 16) Theorem 3.10 Let p, q be two real numbers such that 1 p + 1 q = 1, p > 1. Let > ≥ 0, m and n are given by Equation 3.3 for and , respectively. Let f ∈ AC m [a, b] be such that f (i) (a) = 0 for i = n, n + 1, … , m − 1. a, b]. Then, we have where Proof Applying Theorem 2.1 with Ω 1 = Ω 2 = (a, b), d 1 (x) = dx, d 2 (t) = dt and the kernel and replacing f by C D a+ f , g becomes C D a+ f and we get the inequality (Equation 3.8). □ , then we get where Remark 3.12 Using Theorem 2.1 and composition identities for the right-sided Caputo fractional derivatives given in Andrić et al. (2013b, Theorem 2.2), similar results can be stated and proved for the right-sided Caputo fractional derivatives (for details see Andrić et al., 2014a;Farid, & Pečarić, 2012b).
Results given for the Caputo fractional derivatives can be analogously done for two other types of fractional derivative that we observe: Canavati type and Riemann-Liouville type. Here, as an example inequality for each type of fractional derivatives, we give inequality analogous to the Equation 3.8 obtained with composition identity for the left-sided fractional derivatives. Proofs are omitted.

, the left-sided Riemann-Liouville fractional derivative of order is defined by
The following lemma summarizes conditions in the composition identity for the left-sided Riemann-Liouville fractional derivatives (for details see Andrić et al., 2013a).
Theorem 3.16 Let p, q be two real numbers such that 1 p + 1 q = 1, p > 1. Also, let > ≥ 0, m = [ ] + 1, and n = [ ] + 1. Suppose that one of the conditions in (i)-(vii) in Lemma 3.15 holds for { , , f } and let D a+ f ∈ L 1 [a, b], then we havē where Next, we give the results for a generalized fractional integral operator, the Saigo fractional integral operator (for details see, Saigo, 1978).
Let > 0, , ∈ ℝ. Then, the Saigo fractional integrals I , , a,t of order for a real-valued continuous function f are defined by: where, the function 2 F 1 (..) appearing as the kernal for operator (Equation 3.9) is the Gaussian hypergeometric function defined by and (a) n is the Pochhammer symbol: (a) n = a(a + 1) … (a + n − 1), (a) 0 = 1.