scholarly journals Fractional Versions of Hadamard-Type Inequalities for Strongly Exponentially α , h − m -Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Shasha Li ◽  
Ghulam Farid ◽  
Atiq Ur Rehman ◽  
Hafsa Yasmeen
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Strongly Convex ◽  
Error Estimations ◽  
Integral Identities

In this article, we prove some fractional versions of Hadamard-type inequalities for strongly exponentially α , h − m -convex functions via generalized Riemann–Liouville fractional integrals. The outcomes of this paper provide inequalities of strongly convex, strongly m -convex, strongly s -convex, strongly α , m -convex, strongly s , m -convex, strongly h − m -convex, strongly α , h − m -convex, strongly exponentially convex, strongly exponentially m -convex, strongly exponentially s -convex, strongly exponentially s , m -convex, strongly exponentially h − m -convex, and exponentially α , h − m -convex functions. The error estimations are also studied by applying two fractional integral identities.

scholarly journals Inequalities for generalized Riemann–Liouville fractional integrals of generalized strongly convex functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ghulam Farid ◽  
Young Chel Kwun ◽  
Hafsa Yasmeen ◽  
Abdullah Akkurt ◽  
Shin Min Kang
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Constant Multiplier ◽  
Error Estimations ◽  
Integral Identities

AbstractSome new integral inequalities for strongly $(\alpha ,h-m)$ ( α , h − m ) -convex functions via generalized Riemann–Liouville fractional integrals are established. The outcomes of this paper provide refinements of some fractional integral inequalities for strongly convex, strongly m-convex, strongly $(\alpha ,m)$ ( α , m ) -convex, and strongly $(h-m)$ ( h − m ) -convex functions. Also, the refinements of error estimations of these inequalities are obtained by using two fractional integral identities. Moreover, using a parameter substitution and a constant multiplier, k-fractional versions of established inequalities are also given.

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 Refinements and Generalizations of Some Fractional Integral Inequalities via Strongly Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Ghulam Farid ◽  
Hafsa Yasmeen ◽  
Chahn Yong Jung ◽  
Soo Hak Shim ◽  
Gaofan Ha
Keyword(s):  
Error Bounds ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Integral Identities

In this article, we have established the Hadamard inequalities for strongly convex functions using generalized Riemann–Liouville fractional integrals. The findings of this paper provide refinements of some fractional integral inequalities. Furthermore, the error bounds of these inequalities are given by using two generalized integral identities.

scholarly journals Hermite–Hadamard-Type Inequalities for Convex Functions via the Fractional Integrals with Exponential Kernel

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 845 ◽  
Author(s):  
Xia Wu ◽  
JinRong Wang ◽  
and Jialu Zhang
Keyword(s):  
Real Number ◽  
Convex Functions ◽  
Fractional Integral ◽  
Second Order ◽  
Fractional Integrals ◽  
Exponential Kernel ◽  
Special Means ◽  
Integral Identities

In this paper, we establish three fundamental integral identities by the first- and second-order derivatives for a given function via the fractional integrals with exponential kernel. With the help of these new fractional integral identities, we introduce a few interesting Hermite–Hadamard-type inequalities involving left-sided and right-sided fractional integrals with exponential kernels for convex functions. Finally, some applications to special means of real number are presented.

scholarly journals Generalizations of Hermite–Hadamard like inequalities involving $\chi _{{\kappa }}$-Hilfer fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yu-Ming Chu ◽  
Muhammad Uzair Awan ◽  
Sadia Talib ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Hadamard’S Inequality ◽  
Integral Identities

Abstract The main objective of this paper is to obtain a new κ-fractional analogue of Hermite–Hadamard’s inequality using the class of s-convex functions and $\chi _{{\kappa }}$ χ κ -Hilfer fractional integrals. In order to obtain other main results of the paper we derive two new fractional integral identities using the definitions of $\chi _{{\kappa }}$ χ κ -Hilfer fractional integrals. For the validity of these identities we also take some particular examples. Using these identities we then obtain some more new variants of Hermite–Hadamard’s inequality using s-convex functions.

scholarly journals Ostrowski Type Inequalities Pertaining Strongly Convex Functions Via Conformable Fractional Integrals and Their Applications

2019 ◽  
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Fractional Integrals ◽  
Strongly Convex Functions ◽  
Strongly Convex Function ◽  
Strongly Convex ◽  
Error Estimations

In the article, by applied the concept of strongly convex function and one known identity, we establish several Ostrowski type inequalities involving conformable fractional integrals. As applications, some new error estimations for the midpoint formula are provided as well.

scholarly journals New fractional identities, associated novel fractional inequalities with applications to means and error estimations for quadrature formulas

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Artion Kashuri ◽  
Kottakkaran Sooppy Nisar ◽  
Muhammad Zakria Javed ◽  
Sabah Iftikhar ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Quadrature Formulas ◽  
Real Numbers ◽  
Positive Real ◽  
Special Means ◽  
Error Estimations ◽  
Harmonic Convex ◽  
Integral Identities

AbstractIn this paper, the authors derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given, and on using them as auxiliary results some interesting integral inequalities are found. Finally, in order to show the efficiency of our main results, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained.

scholarly journals On Strongly Convex Functions via Caputo–Fabrizio-Type Fractional Integral and Some Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Qi Li ◽  
Muhammad Shoaib Saleem ◽  
Peiyu Yan ◽  
Muhammad Sajid Zahoor ◽  
Muhammad Imran
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Convex Functions ◽  
Fractional Integral ◽  
Optimization Problems ◽  
Fractional Integral Operator ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Special Means ◽  
New Inequalities

The theory of convex functions plays an important role in the study of optimization problems. The fractional calculus has been found the best to model physical and engineering processes. The aim of this paper is to study some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator. In this paper, we present Hermite–Hadamard-type inequalities for strongly convex functions via the Caputo–Fabrizio fractional integral operator. Some new inequalities of strongly convex functions involving the Caputo–Fabrizio fractional integral operator are also presented. Moreover, we present some applications of the proposed inequalities to special means.

scholarly journals Inequalities for Riemann–Liouville Fractional Integrals of Strongly s , m -Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Fuzhen Zhang ◽  
Ghulam Farid ◽  
Saira Bano Akbar
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Error Estimations

The results of this paper provide two Hadamard-type inequalities for strongly s , m -convex functions via Riemann–Liouville fractional integrals and error estimations of well-known fractional Hadamard inequalities. Their special cases are given and connected with the results of some published papers.

scholarly journals Generalized Hadamard Fractional Integral Inequalities for Strongly s , m -Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-19
Author(s):  
Chao Miao ◽  
Ghulam Farid ◽  
Hafsa Yasmeen ◽  
Yanhua Bian
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Hadamard Inequality ◽  
Hadamard Fractional Integral

This article deals with Hadamard inequalities for strongly s , m -convex functions using generalized Riemann–Liouville fractional integrals. Several generalized fractional versions of the Hadamard inequality are presented; we also provide refinements of many known results which have been published in recent years.

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