Abstract
A new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of sigma >0. We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.
Highlights
Introduction and preliminariesThe following inequality is known as Simpson’s inequality.Theorem 1.1 Let F : 1⁄2a1; a2 ! R be a four times continuously differentiable function on ða1; a2Þ and kF 4k1 1⁄4 sup jF 4ðxÞj\1, x2ða1 ;a2 Þ F ða1Þ þ 2 ða2Þ þ a1 þ 2 a2! À a2 1 À a1 Za2 ðxÞdx kF 4k1ða2
A new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)convex functions of higher order of r [ 0
In last we present applications of the obtained results in obtaining Simpson-like quadrature formula
Summary
The following inequality is known as Simpson’s inequality. Theorem 1.1 Let F : 1⁄2a1; a2 ! R be a four times continuously differentiable function on ða; a2Þ and kF 4k1 1⁄4 sup jF 4ðxÞj\1, x2ða1 ;a2 Þ. A1Þ4: Simpson’s inequality plays a significant role in modern analysis [4] In past it has been extended and generalized in different directions using novel and innovative approaches. Sarikaya et al [13] obtained Simpson’s inequality using differentiable convex functions. Fractional calculus is very old but in recent decades has received special attention. The classical Riemann-Liouville fractional integrals are defined as: Definition 1.1 ([7]) Let F 2 L11⁄2a1; a2. The main objective of this paper is to derive some new generalized variants of Simpson’s like inequalities for the functions belonging to Lebesgue L1, Lq and L1 spaces. In order to show the significance of the obtained results, we discuss some interesting applications of the main results. We hope that the ideas and the techniques of this paper will inspire interested readers working in this filed
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