Some Integral Inequalities via Riemann Livouille Fractional Integrals

Abstract

Mathematical inequalities contribute highly to error estimation. In the past few decades, many authors have done their studies about the error analysis of some quadrature rules of Newton-Cotes type. Moreover, Convexity is highly recognized for its vital and primary task in the expansion of numerous section of mathematics, for example, mathematical finance, economics, management sciences and optimization theory. Hence, in this dissertation we are focusing on the detailed studies of the function of convexity in the theory of inequalities and generalizations of new integral inequalities for differentiable mapping along with the Simpsons type as well as Hermite-Hadamards type. Here, we derive some explicit bounds for the mid-point, Trapezoid and Simpsons quadrature rule with the aid of modern theory of inequalities and general Peano-Kernel approach in coherent with the convexity (s,m)−convex at most first derivative. In our monograph we shall generalize some integral inequalities for generalized convex function and, in parallel, developments shall be made on concavity. With the above mentioned technique, we are able to focus on the larger classes of functions allowing us to conduct different quadrature rule that have restrictions on the behavior of the integrand. By applying the Holder inequality and power mean equality, different generalizations and melioration for a formal inequalities and differentiable mapping of the functions where the function under specific condition is convex are considered. The results presented here extend various inequalities of the Simpson’s and Hermite-Hadamard.