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New generalized trapezoidal type integral inequalities with applications

Kashuri Artion, Farid Ghulam, Set Erhan

Journal of Applied Analysis

2021

Vol. 27, nr 1

35--46

Trapezoidal inequalities for functions of diverse nature are useful in numerical computations. The authors have proved an identity for a generalized integral operator via a differentiable function. By applying the established identity, the generalized trapezoidal type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in the recent decades. Various special cases have been identified. Some applications of presented results have been analyzed.

Fractional trapezium–type inequalities for strongly exponentially generalized preinvex functions with applications

Fundo Akli, Kashuri Artion

Fasciculi Mathematici

2020

nr 64

57--76

The aim of this paper is to introduce a new extension of preinvexity called strongly exponentially generalized (m, ω1, ω2, h1, h2)-preinvexity. Some new integral inequalities of trapezium-type for strongly exponentially generalized (m, ω1, ω2, h1, h2)-preinvex functions with modulus c via Riemann-Liouville fractional integral are established. Also, some new estimates with respect to trapezium-type integral inequalities for strongly exponentially generalized (m, ω1, ω2, h1, h2)-preinvex functions with modulus c via general fractional integrals are obtained. We show that the class of strongly exponentially generalized (m, ω1, ω2, h1, h2)-preinvex functions with modulus c includes several other classes of preinvex functions. At the end, some new error estimates for trapezoidal quadrature formula are provided as well. This results may stimulate further research in different areas of pure and applied sciences.