Hermite-Hadamard Type Fractional Integral Inequalities for Generalized (r; g; s; m; ϕ)-Preinvex Functions

2017 ◽  
Vol 59 (1) ◽  
pp. 43-55
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Quadrature Formula ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Left Hand ◽  
New Class ◽  
Preinvex Functions

AbstractIn the present paper, a new class of generalized (r; g; s; m; ϕ)-preinvex functions is introduced and some new integral inequalities for the left hand side of Gauss-Jacobi type quadrature formula involving generalized (r; g; s; m; ϕ)-preinvex functions are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for generalized (r; g; s; m; ϕ)-preinvex functions via Riemann-Liouville fractional integrals are established. These results not only extend the results appeared in the literature (see [1],[2]), but also provide new estimates on these types.

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

2019 ◽  
Vol 25 (1) ◽  
pp. 59-72
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Quadrature Formula ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Left Hand ◽  
New Class ◽  
Special Means ◽  
Preinvex Functions

Abstract 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,\varphi) -preinvex functions, Aust. J. Math. Anal. Appl. 13 2016, 1, Article ID 16]. At the end, some applications to special means are given.

scholarly journals Hermite-Hadamard Type Inequalities for Mtm-Preinvex Functions

2017 ◽  
Vol 58 (1) ◽  
pp. 77-96
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Quadrature Formula ◽  
Beta Function ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Left Hand ◽  
Special Means ◽  
Preinvex Functions

AbstractIn the present paper, the notion of MTm-preinvex function is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving MTm-preinvex functions along with beta function are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for MTm-preinvex functions via classical integrals and Riemann-Liouville fractional integrals are established. At the end, some applications to special means are given. These results not only extend the results appeared in the literature (see [13]), but also provide new estimates on these types.

scholarly journals Hermite-Hadamard type integral inequalities for products of two generalized (s; m; ξ)-preinvex functions

2017 ◽  
Vol 3 (1) ◽  
pp. 102-115
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Quadrature Formula ◽  
Beta Function ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Left Hand ◽  
Type Integral ◽  
Preinvex Functions

Abstract In this paper, the notion of generalized (s; m; ξ)-preinvex function is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving generalized (s; m; ξ)-preinvex functions along with beta function are given. Moreover, we establish some new Hermite-Hadamard type integral inequalities for products of two generalized (s; m; ξ)-preinvex functions via classical and Riemann-Liouville fractional integrals. These results not only extend the results appeared in the literature (see [10],[11]), but also provide new estimates on these types. At the end, some conclusions are given.

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.

SOME LOCAL FRACTIONAL INTEGRAL INEQUALITIES FOR GENERALIZED PREINVEX FUNCTIONS AND APPLICATIONS TO NUMERICAL QUADRATURE

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950071 ◽  
Author(s):  
WENBING SUN
Keyword(s):  
Numerical Integration ◽  
Error Estimates ◽  
Fractional Integral ◽  
Numerical Quadrature ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Parallel Development ◽  
Special Values ◽  
Local Fractional Integral ◽  
Preinvex Functions

In this paper, a new identity with parameters involving local fractional integrals is derived. Using this identity, some general local fractional integral inequalities for generalized preinvex functions are established. A parallel development is deduced for generalized preconcave functions. Taking special values for the parameters, some generalized midpoint inequalities, trapezoidal inequalities and Simpson inequalities are obtained. Finally, as some applications, error estimates of numerical integration for local fractional integrals are given.

scholarly journals New conformable fractional integrals of Ostrowski type using new generalized (s, m, ϕ)-preinvex mappings

2017 ◽  
Vol 3 (2) ◽  
pp. 173-185
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Quadrature Formula ◽  
Beta Function ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Left Hand ◽  
Special Means

AbstractIn the present paper, the notion of new generalized (s, m, ϕ)-preinvex mapping is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving new generalized (s, m, ϕ)-preinvex mappings along with beta function are given. Moreover, some generalizations of Ostrowski type inequalities for new generalized (s, m, ϕ)-preinvex mappings via conformable fractional integrals are established. At the end, some applications to special means are given.

scholarly journals Fractional trapezium type inequalities for twice differentiable preinvex functions and their applications

2020 ◽  
Vol 10 (2) ◽  
pp. 226-236
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Integral Operator ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Recent Past ◽  
Trapezium Type ◽  
Special Means ◽  
Special Cases ◽  
Type Integral ◽  
Preinvex Functions ◽  
Almost All

Trapezoidal inequalities for functions of divers natures are useful in numerical computations. The authors have proved an identity for a generalized integral operator via twice differentiable preinvex 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 recent past decades. Various special cases have been identified. Some applications of presented results to special means have been analyzed. The ideas and techniques of this paper may stimulate further research.

scholarly journals Fractional Hermite–Hadamard–Fejer Inequalities for a Convex Function with Respect to an Increasing Function Involving a Positive Weighted Symmetric Function

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1503 ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Artion Kashuri
Keyword(s):  
Fractional Calculus ◽  
Convex Function ◽  
Symmetric Function ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Important Direction ◽  
Increasing Function

There have been many different definitions of fractional calculus presented in the literature, especially in recent years. These definitions can be classified into groups with similar properties. An important direction of research has involved proving inequalities for fractional integrals of particular types of functions, such as Hermite–Hadamard–Fejer (HHF) inequalities and related results. Here we consider some HHF fractional integral inequalities and related results for a class of fractional operators (namely, the weighted fractional operators), which apply to function of convex type with respect to an increasing function involving a positive weighted symmetric function. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities.

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