scholarly journals Integral Inequalities for s-Convexity via Generalized Fractional Integrals on Fractal Sets

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 53 ◽  
Author(s):  
Ohud Almutairi ◽  
Adem Kılıçman
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Fractal Sets ◽  
Special Means ◽  
Single Form ◽  
Hadamard Fractional Integrals ◽  
Katugampola Fractional Integral

In this study, we establish new integral inequalities of the Hermite–Hadamard type for s-convexity via the Katugampola fractional integral. This generalizes the Hadamard fractional integrals and Riemann–Liouville into a single form. We show that the new integral inequalities of Hermite–Hadamard type can be obtained via the Riemann–Liouville fractional integral. Finally, we give some applications to special means.

Some new fractional integral inequalities for generalized relative semi-m-(r; h1, h2)-preinvex mappings via generalized Mittag-Leffler function

2019 ◽  
Vol 26 (1/2) ◽  
pp. 41-55 ◽  
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Invex Set ◽  
Special Means ◽  
Special Cases ◽  
Differentiable Mappings

The authors discover a new identity concerning differentiable mappings defined on m-invex set via fractional integrals. By using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized relative semi- m-(r;h1,h2)-preinvex mappings by involving generalized Mittag-Leffler function are presented. It is pointed out that some new special cases can be deduced from main results of the paper. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

scholarly journals On the weighted fractional integral inequalities for Chebyshev functionals

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Sami Ullah Khan ◽  
Dumitru Baleanu ◽  
V. Vijayakumar
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Differentiable Functions ◽  
Hadamard Fractional Integrals ◽  
New Inequalities

AbstractThe goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integral involving another function $\mathcal{G}$ G in the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev’s functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev’s functionals with certain choices of $\omega (\theta )$ ω ( θ ) and $\mathcal{G}(\theta )$ G ( θ ) as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev’s functionals with certain choices of $\omega (\theta )$ ω ( θ ) and $\mathcal{G}(\theta )$ G ( θ ) .

scholarly journals New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
İmdat İşcan
Keyword(s):  
Integral Inequalities ◽  
Fractional Integrals ◽  
Real Numbers ◽  
Positive Real ◽  
General Integral ◽  
Lipschitzian Functions ◽  
Special Means ◽  
Hadamard Fractional Integrals

The author obtains new estimates on generalization of Hadamard, Ostrowski, and Simpson type inequalities for Lipschitzian functions via Hadamard fractional integrals. Some applications to special means of positive real numbers are also given.

scholarly journals Some New Generalized Integral Inequalities for GA-s-Convex Functions via Hadamard Fractional Integrals

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
İmdat İşcan ◽  
Mustafa Aydin
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Hadamard Fractional Integrals ◽  
Hadamard Fractional Integral ◽  
Generalized Integral Inequalities

We prove new generalization of Hadamard, Ostrowski, and Simpson inequalities in the framework of GA-s-convex functions and Hadamard fractional integral.

scholarly journals New generalized Pólya–Szegö and Čebyšev type inequalities with general kernel and measure

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
S. Iqbal ◽  
M. Samraiz ◽  
Thabet Abdeljawad ◽  
Kottakkaran Sooppy Nisar ◽  
G. Rahman ◽  
...  
Keyword(s):  
Measure Space ◽  
Fractional Integral ◽  
Integral Operators ◽  
Finite Measure ◽  
Integral Transforms ◽  
Fractional Integrals ◽  
Fractional Integral Operators ◽  
New Class ◽  
Hadamard Fractional Integrals ◽  
Katugampola Fractional Integral

AbstractIt is always attractive and motivating to acquire the generalizations of known results. In this article, we introduce a new class $\mathfrak{C(h)}$ C ( h ) of functions which can be represented in a form of integral transforms involving general kernel with σ-finite measure. We obtain some new Pólya–Szegö and Čebyšev type inequalities as generalizations to the previously proved ones for different fractional integrals including fractional integral of a function with respect to another function capturing Riemann–Liouville integrals, Hadamard fractional integrals, Katugampola fractional integral operators, and conformable fractional integrals. This new idea shall motivate the researchers to prove the results over a measure space with general kernels instead of special kernels.

scholarly journals Applying GG-Convex Function to Hermite-Hadamard Inequalities Involving Hadamard Fractional Integrals

2014 ◽  
Vol 2014 ◽  
pp. 1-20 ◽  
Author(s):  
Zhi Zhang ◽  
JinRong Wang ◽  
JianHua Deng
Keyword(s):  
Convex Function ◽  
Fractional Integral ◽  
Beta Function ◽  
Fractional Integrals ◽  
Incomplete Beta Function ◽  
Real Numbers ◽  
Special Means ◽  
New Type ◽  
Hadamard Fractional Integrals ◽  
Integral Identities

By virtue of fractional integral identities, incomplete beta function, useful series, and inequalities, we apply the concept of GG-convex function to derive new type Hermite-Hadamard inequalities involving Hadamard fractional integrals. Finally, some applications to special means of real numbers are demonstrated.

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 Fractional integral inequalities for generalized-$$\mathbf{m }$$-$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$-convex mappings via an extended generalized Mittag–Leffler function

2019 ◽  
Vol 9 (2) ◽  
pp. 231-243
Author(s):  
George Anastassiou ◽  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Convex Mappings ◽  
Invex Set ◽  
Special Means ◽  
Special Cases

AbstractThe authors discover a new identity concerning differentiable mappings defined on $$\mathbf{m }$$ m -invex set via general fractional integrals. Using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized-$$\mathbf{m }$$ m -$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$ ( ( h 1 p , h 2 q ) ; ( η 1 , η 2 ) ) -convex mappings by involving an extended generalized Mittag–Leffler function are presented. It is pointed out that some new special cases can be deduced from main results. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

scholarly journals Some new Hermite-Hadamard type inequalities for generalized harmonically convex functions involving local fractional integrals

2021 ◽  
Vol 6 (10) ◽  
pp. 10679-10695
Author(s):  
Wenbing Sun ◽  
◽  
Rui Xu
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Fractal Sets ◽  
Special Means ◽  
Local Fractional Integral ◽  
Harmonically Convex Functions

<abstract><p>In this paper, we establish a new integral identity involving local fractional integral on Yang's fractal sets. Using this integral identity, some new generalized Hermite-Hadamard type inequalities whose function is monotonically increasing and generalized harmonically convex are obtained. Finally, we construct some generalized special means to explain the applications of these inequalities.</p></abstract>

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.

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