New fractal–fractional Simpson estimates for twice differentiable functions with applications

Abstract

In this article, we establish a new auxiliary identity on fractal sets for twice local differentiable function involving extended fractal integral operators. Testing this identity together with generalized fractal Hölder’s and Power-mean integral inequalities, we develop some new fractal–fractional Simpson’s type inequalities. Furthermore, we use modified Yang Hölder’s and Power-mean inequality to create new fractal estimates. We also give comparison analysis of bounds and show how the modified form of Yang Hölder’s and Power-mean integral inequalities can result in improved lower upper bounds. We also provide concrete examples to examine the validity of obtain results numerically and also justify them by 2D and 3D graphical analysis. As implementations, we operate our findings to get new applications in form of $\zeta$-type special means, moment of random variables and wave equations.