Some Simpson’s Riemann–Liouville Fractional Integral Inequalities with Applications to Special Functions

Jamshed Nasir,Rostin Matendo Mabela,Khuram Ali Khan,Shahid Qaisar,Saad Ihsan Butt,Muhammad Arif

doi:10.1155/2022/2113742

Abstract

Based on the Riemann–Liouville fractional integral, a new form of generalized Simpson-type inequalities in terms of the first derivative is discussed. Here, some more inequalities for convexity as well as concavity are established. We expect that present outcomes are the generalization of already obtained results. Applications to beta, q-digamma, and Bessel functions are also provided.

Highlights

Integral inequalities have been widely used in various sciences, including mathematical sciences, applied sciences, differential equations, and functional analysis
In most mathematical analysis areas, many types of integral inequalities are used. ey are very important in approximation theory and numerical analysis, which estimate the error’s approximation [1,2,3]
Many scholars are interested in the Simpson-type inequality since it has been examined and studied for numerous classes of functions. Due to their efficacy and usefulness in pure and applied mathematics, Simpson-type and Newton-type results have been keenly interpolated for convex functions. e first striking result about the Simpson-type inequality along with its applications to the quadrature formula in numerical integration was given by Dragomir et al in [14]

Summary

Introduction

Integral inequalities have been widely used in various sciences, including mathematical sciences, applied sciences, differential equations, and functional analysis. Ere are several important inequalities due to their direct applications in applied sciences (see [4,5,6,7,8]) Several integral inequalities, such as Holder’s inequality, Simpson’s inequality, Newton’s inequality, the Hermite–Hadamard inequality, Ostrowski’s inequality, Cauchy–Schwarz, and Chebyshev, are well known in classical analysis and have been proven and applied in the setup of q-calculus using classical convexity [9,10,11,12]. Many scholars are interested in the Simpson-type inequality since it has been examined and studied for numerous classes of functions. Due to their efficacy and usefulness in pure and applied mathematics, Simpson-type and Newton-type results have been keenly interpolated for convex functions.

Inequalities for Simpson’s Type

Applications