Fejér–Hadamard Type Inequalities for (α, h-m)-p-Convex Functions via Extended Generalized Fractional Integrals

2021 ◽  
Vol 5 (4) ◽  
pp. 253
Author(s):  
Ghulam Farid ◽  
Muhammad Yussouf ◽  
Kamsing Nonlaopon
Keyword(s):  
Fractional Order ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Weighted Version ◽  
Hadamard Inequality ◽  
Fractional Integral Operators ◽  
Different Types ◽  
Inequality Type

Integral operators of a fractional order containing the Mittag-Leffler function are important generalizations of classical Riemann–Liouville integrals. The inequalities that are extensively studied for fractional integral operators are the Hadamard type inequalities. The aim of this paper is to find new versions of the Fejér–Hadamard (weighted version of the Hadamard inequality) type inequalities for (α, h-m)-p-convex functions via extended generalized fractional integrals containing Mittag-Leffler functions. These inequalities hold simultaneously for different types of well-known convexities as well as for different kinds of fractional integrals. Hence, the presented results provide more generalized forms of the Hadamard type inequalities as compared to the inequalities that already exist in the literature.

scholarly journals Boundedness of Fractional Integral Operators Containing Mittag-Leffler Function via Exponentially s-Convex Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
Gang Hong ◽  
G. Farid ◽  
Waqas Nazeer ◽  
S. B. Akbar ◽  
J. Pečarić ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Symmetric Functions ◽  
Integral Operators ◽  
Arbitrary Point ◽  
Fractional Integrals ◽  
Hadamard Inequality ◽  
Operator Inequalities ◽  
Fractional Integral Operators ◽  
Exponentially Convex Functions

The main objective of this paper is to obtain the fractional integral operator inequalities which provide bounds of the sum of these operators at an arbitrary point. These inequalities are derived for s-exponentially convex functions. Furthermore, a Hadamard inequality is obtained for fractional integrals by using exponentially symmetric functions. The results of this paper contain several such consequences for known fractional integrals and functions which are convex, exponentially convex, and s-convex.

Integral inequalities in a generalized context

2020 ◽  
Vol 57 (3) ◽  
pp. 312-320
Author(s):  
Péter Kórus ◽  
Luciano M. Lugo ◽  
Juan E. Nápoles Valdés
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Hadamard Inequality ◽  
Fractional Integral Operators

AbstractIn this paper we present different variants of the well-known Hermite–Hadamard inequality, in a generalized context. We consider general fractional integral operators for h-convex and r-convex functions.

scholarly journals On Some Generalized Fractional Integral Inequalities for p-Convex Functions

2019 ◽  
Vol 3 (2) ◽  
pp. 29
Author(s):  
Seren Salaş ◽  
Yeter Erdaş ◽  
Tekin Toplu ◽  
Erhan Set
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Hadamard Inequality ◽  
Fractional Integral Operators ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators

In this paper, firstly we have established a new generalization of Hermite–Hadamard inequality via p-convex function and fractional integral operators which generalize the Riemann–Liouville fractional integral operators introduced by Raina, Lun and Agarwal. Secondly, we proved a new identity involving this generalized fractional integral operators. Then, by using this identity, a new generalization of Hermite–Hadamard type inequalities for fractional integral are obtained.

scholarly journals Certain inequalities via generalized proportional Hadamard fractional integral operators

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Gauhar Rahman ◽  
Thabet Abdeljawad ◽  
Fahd Jarad ◽  
Aftab Khan ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Fractional Integral Operators ◽  
The Given ◽  
Hadamard Fractional Integrals ◽  
Hadamard Fractional Integral

Abstract In the article, we introduce the generalized proportional Hadamard fractional integrals and establish several inequalities for convex functions in the framework of the defined class of fractional integrals. The given results are generalizations of some known results.

scholarly journals Integral inequalities involving generalized Erdélyi-Kober fractional integral operators

2016 ◽  
Vol 14 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Dumitru Baleanu ◽  
Sunil Dutt Purohit ◽  
Jyotindra C. Prajapati
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Weighted Version ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Chebyshev Functional

AbstractUsing the generalized Erdélyi-Kober fractional integrals, an attempt is made to establish certain new fractional integral inequalities, related to the weighted version of the Chebyshev functional. The results given earlier by Purohit and Raina (2013) and Dahmani et al. (2011) are special cases of results obtained in present paper.

scholarly journals On Hermite-Hadamard type inequalities via fractional integral operators

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 837-854
Author(s):  
Tuba Tunç ◽  
Mehmet Sarıkaya
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Hadamard Inequality ◽  
Fractional Integral Operators

In this paper, we give new definitions related to fractional integral operators for two variables functions using the class of integral operators. We are interested to give the Hermite-Hadamard inequality for a rectangle in plane via convex functions on co-ordinates involving fractional integral operators.

scholarly journals Certain Inequalities Pertaining to Some New Generalized Fractional Integral Operators

2021 ◽  
Vol 5 (4) ◽  
pp. 160
Author(s):  
Hari Mohan Srivastava ◽  
Artion Kashuri ◽  
Pshtiwan Othman Mohammed ◽  
Kamsing Nonlaopon
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Chebyshev Inequality ◽  
Fractional Integral Operators ◽  
Chebyshev Functional ◽  
Generalized Fractional Integral ◽  
Finite Products ◽  
Generalized Fractional Integral Operators

In this paper, we introduce the generalized left-side and right-side fractional integral operators with a certain modified ML kernel. We investigate the Chebyshev inequality via this general family of fractional integral operators. Moreover, we derive new results of this type of inequalities for finite products of functions. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. From our above-mentioned results, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view. Furthermore, two important results and some interesting consequences for convex functions in the framework of the defined class of generalized fractional integral operators are established. Finally, two basic examples demonstrated the significance of our results.

scholarly journals Riemann-Liouville Fractional integral operators with respect to increasing functions and strongly $ (\alpha, m) $-convex functions

2021 ◽  
Vol 6 (10) ◽  
pp. 11403-11424
Author(s):  
Ghulam Farid ◽  
◽  
Hafsa Yasmeen ◽  
Hijaz Ahmad ◽  
Chahn Yong Jung ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Fractional Integral Operators ◽  
Strongly Convex ◽  
Increasing Functions ◽  
Integral Identities

<abstract><p>In this paper Hadamard type inequalities for strongly $ (\alpha, m) $-convex functions via generalized Riemann-Liouville fractional integrals are studied. These inequalities provide generalizations as well as refinements of several well known inequalities. The established results are further connected with fractional integral inequalities for Riemann-Liouville fractional integrals of convex, strongly convex and strongly $ m $-convex functions. By using two fractional integral identities some more Hadamard type inequalities are proved.</p></abstract>

scholarly journals The multiple composition of the left and right fractional Riemann-Liouville integrals - analytical and numerical calculations

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6087-6099 ◽  
Author(s):  
Mariusz Ciesielski ◽  
Tomasz Blaszczyk
Keyword(s):  
Numerical Calculation ◽  
Fractional Order ◽  
Fractional Integral ◽  
Integral Operators ◽  
Numerical Calculations ◽  
Fractional Integrals ◽  
Numerical Accuracy ◽  
Fractional Integral Operators ◽  
Left And Right ◽  
The Right

New fractional integral operators of order ? ? R+ are introduced. These operators are defined as the composition of the left and right (or the right and left) Riemann-Liouville fractional order integrals. Some of their properties are studied. Analytical results of fractional integrals of several functions are presented. For a numerical calculation of fractional order integrals, two numerical procedures are given. In the final part of this paper, examples of numerical evaluations of these operators of three different functions are shown in plots and the comparison of the numerical accuracy was analyzed in tables.

scholarly journals Hadamard and Fejér-Hadamard inequalities for generalized $ k $-fractional integrals involving further extension of Mittag-Leffler function

2021 ◽  
Vol 7 (1) ◽  
pp. 681-703
Author(s):  
Ye Yue ◽  
◽  
Ghulam Farid ◽  
Ayșe Kübra Demirel ◽  
Waqas Nazeer ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Fractional Integral Operators

<abstract><p>In this paper, $ k $-fractional integral operators containing further extension of Mittag-Leffler function are defined firstly. Then, the first and second version of Hadamard and Fejér-Hadamard inequalities for generalized $ k $-fractional integrals are obtained. Finally, by using these generalized $ k $-fractional integrals containing Mittag-Leffler functions, results for $ p $-convex functions are obtained. The results for convex functions can be deduced by taking $ p = 1 $.</p></abstract>

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