Certain inequalities for fractional $(p,q)$-calculus

<abstract><p>In this article, we introduce the notion of $ _{a}{\bar{T}}_{p,q} $-integrals. Using the definition of $ _{a}{\bar{T}}_{p,q} $-integrals, we derive some new post quantum analogues of some classical results of Young's inequality, Hölder's inequality, Minkowski's inequality, Ostrowski's inequality and Hermite-Hadamard's inequality.</p></abstract>

In this paper, we use the the Riemann-Liouville fractional integral to develop some new results related to the Hermite-Hadamard inequality. Other integral inequalities related to the Minkowsky inequality are also established. Our results have some relationships with [E. Set, M. E. Ozdemir and S.S. Dragomir, J. Inequal. Appl. 2010, Art. ID 148102, 9 pp.] and [L. BougoffaJ. qualPure and ApplMath 7 (2006), no. 2, Article 60, 3 pp.]. Some interested inequalities of these references can be deduced as some special cases.

We establish some new Hermite-Hadamard-type inequalities involving product of two functions. Other integral inequalities for two functions are obtained as well. The analysis used in the proofs is fairly elementary and based on the use of the Minkowski, Holder, and Young inequalities.