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New generalized trapezoidal type integral inequalities with applications

Kashuri Artion, Farid Ghulam, Set Erhan

Journal of Applied Analysis

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2021

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Vol. 27, nr 1

35--46

EN

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.

2

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

Fundo Akli, Kashuri Artion

Fasciculi Mathematici

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2020

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nr 64

57--76

EN

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.

3

Fractional non conformable Hermite-Hadamard inequalities for generalized ϕ-convex functions

Ali Muhammad A., Valdés Juan E. Nápoles, Kashuri Artion, Zhang Zhiyue

Fasciculi Mathematici

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2020

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nr 64

5--16

EN

In this paper, we present new inequalities of the Hermite–Hadamard type for generalized ϕ-convex functions, within the framework of non-conformable fractional integrals.

4

On some k-fractional integral inequalities of Hermite-Hadamard type for twice differentiable generalized beta (r, g)-preinvex functions

Kashuri Artion, Liko Rozana

Journal of Applied Analysis

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2019

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Vol. 25, nr 1

59--72

EN

In the present paper, a new class of generalized beta (r, g)-preinvex functions is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving generalized beta (r, g)-preinvex functions are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for generalized beta (r, g)-preinvex functions that are twice differentiable via k-fractional integrals are established. These general inequalities give us some new estimates for Hermite-Hadamard type k-fractional integral inequalities and also extend some results appeared in the literature; see [A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized (s, m, φ)-preinvex functions, Aust. J. Math. Anal. Appl. 13 (2016), no. 1, Article ID 16]. At the end, some applications to special means are given.