scholarly journals Some new Ostrowski type fractional integral inequalities for generalized (s,m, φ)-preinvex functions via Caputo k -fractional derivatives

2018 ◽  
Vol 37 (1) ◽  
pp. 133-151
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Derivatives ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Preinvex Functions

scholarly journals On applications of Caputo k-fractional derivatives

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ghulam Farid ◽  
Naveed Latif ◽  
Matloob Anwar ◽  
Ali Imran ◽  
Muhammad Ozair ◽  
...  
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Chebyshev Inequality ◽  
Absolutely Continuous ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Type Variable

Abstract This research explores Caputo k-fractional integral inequalities for functions whose nth order derivatives are absolutely continuous and possess Grüss type variable bounds. Using Chebyshev inequality (Waheed et al. in IEEE Access 7:32137–32145, 2019) for Caputo k-fractional derivatives, several integral inequalities are derived. Further, Laplace transform of Caputo k-fractional derivative is presented and Caputo k-fractional derivative and Riemann–Liouville k-fractional integral of an extended generalized Mittag-Leffler function are calculated. Moreover, using the extended generalized Mittag-Leffler function, Caputo k-fractional differential equations are presented and their solutions are proposed by applying the Laplace transform technique.

scholarly journals Fractional trapezium-type inequalities for strongly exponentially generalized preinvex functions with applications

2020 ◽  
pp. 38-38
Author(s):  
Artion Kashuri ◽  
Themistocles Rassias
Keyword(s):  
Applied Sciences ◽  
Error Estimates ◽  
Quadrature Formula ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Trapezium Type ◽  
Type Integral ◽  
Preinvex Functions

The aim of this paper is to introduce a new extension of preinvexity called strongly exponentially generalized (m; !1; !2; h1; h2)-preinvexity. Some new integral inequalities of trapezium-type for strongly exponentially generalized (m; !1; !2; 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; !1; !2; h1; h2)-preinvex functions with modulus c via general fractional integrals are obtained. We show that the class of strongly exponentially generalized (m; !1; !2; 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.

scholarly journals Fractional integral identity, estimation of its bounds and some applications to trapezoidal quadrature rule

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2629-2641
Author(s):  
Artion Kashuri ◽  
Muhammad Awan ◽  
Muhammad Noor
Keyword(s):  
Applied Sciences ◽  
Integral Identity ◽  
Fractional Integral ◽  
Quadrature Rule ◽  
The Other ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Trapezium Type ◽  
Type Integral ◽  
Preinvex Functions

The aim of this paper is to introduce a new extension of preinvexity called exponentially (m,?1,?2, h1,h2)-preinvexity. Some new integral inequalities of Hermite-Hadamard type for exponentially (m,?1,?2,h1,h2)-preinvex functions via Riemann-Liouville fractional integral are established. Also, some new estimates with respect to trapezium-type integral inequalities for exponentially (m,?1,?2,h1,h2)-preinvex functions via general fractional integrals are obtained. We show that the class of exponentially (m,?1,?2, h1,h2)-preinvex functions includes several other classes of preinvex functions. We shown by two basic examples the efficiency of the obtained inequalities on the base of comparing those with the other corresponding existing ones. 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.

scholarly journals ON SOME FRACTIONAL INTEGRAL INEQUALITIES OF HERMITE-HADAMARD TYPE FOR r-PREINVEX FUNCTIONS

2016 ◽  
Author(s):  
Abdullah AKKURT ◽  
Hüseyin YILDIRIM
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Preinvex Functions

In this paper, we have established Hermite-Hadamard inequalities for r-preinvex functions via fractional integrals.

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