scholarly journals Bounds for the Remainder in Simpson’s Inequality via n-Polynomial Convex Functions of Higher Order Using Katugampola Fractional Integrals

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yu-Ming Chu ◽  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javad ◽  
Awais Gul Khan
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Fractional Integral ◽  
Higher Order ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Simpson’S Inequality ◽  
Generalized Fractional Integral

The goal of this paper is to derive some new variants of Simpson’s inequality using the class of n-polynomial convex functions of higher order. To obtain the main results of the paper, we first derive a new generalized fractional integral identity utilizing the concepts of Katugampola fractional integrals. This new fractional integral identity will serve as an auxiliary result in the development of the main results of this paper.

scholarly journals A Study of Uniform Harmonic χ -Convex Functions with respect to Hermite-Hadamard’s Inequality and Its Caputo-Fabrizio Fractional Analogue and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Miguel Vivas-Cortez ◽  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Hadamard’S Inequality ◽  
Special Cases ◽  
Harmonic Convex

In this paper, we introduce the notion of uniform harmonic χ -convex functions. We show that this class relates several other unrelated classes of uniform harmonic convex functions. We derive a new version of Hermite-Hadamard’s inequality and its fractional analogue. We also derive a new fractional integral identity using Caputo-Fabrizio fractional integrals. Utilizing this integral identity as an auxiliary result, we obtain new fractional Dragomir-Agarwal type of inequalities involving differentiable uniform harmonic χ -convex functions. We discuss numerous new special cases which show that our results are quite unifying. Finally, in order to show the significance of the main results, we discuss some applications to means of positive real numbers.

scholarly journals Simpson type inequalities and applications

2021 ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Michael Th. Rassias ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Quadrature Formula ◽  
Convex Functions ◽  
Integral Identity ◽  
Higher Order ◽  
Auxiliary Result ◽  
First Order ◽  
New Class ◽  
Differentiable Functions

AbstractA new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of $$\sigma >0$$ σ > 0 . We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.

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 Hermite-Hadamard Type Inequalities for convex functions via generalized fractional integral operators

2016 ◽  
Vol 5 (1) ◽  
pp. 55-62
Author(s):  
Erhan Set ◽  
Abdurrahman Gözpinar
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integral Operators ◽  
Absolute Value ◽  
Type Integral ◽  
Generalized Fractional Integral ◽  
Fractional Type ◽  
Generalized Fractional Integral Operators

AbstractIn this present work, the authors establish a new integral identity involving generalized fractional integral operators and by using this fractional-type integral identity, obtain some new Hermite-Hadamard type inequalities for functions whose first derivatives in absolute value are convex. Relevant connections of the results presented here with those earlier ones are also pointed out.

scholarly journals On generalized Ostrowski, Simpson and Trapezoidal type inequalities for co-ordinated convex functions via generalized fractional integrals

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hüseyin Budak ◽  
Fatih Hezenci ◽  
Hasan Kara
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Special Choice ◽  
Ostrowski Inequality ◽  
Generalized Fractional Integral ◽  
Differentiable Mappings

AbstractIn this study, we prove an identity for twice partially differentiable mappings involving the double generalized fractional integral and some parameters. By using this established identity, we offer some generalized inequalities for differentiable co-ordinated convex functions with a rectangle in the plane $\mathbb{R} ^{2}$ R 2 . Furthermore, by special choice of parameters in our main results, we obtain several well-known inequalities such as the Ostrowski inequality, trapezoidal inequality, and the Simpson inequality for Riemann and Riemann–Liouville fractional integrals.

scholarly journals Hermite-Hadamard type inequalities for B-1-convex functions involving generalized fractional integral operators

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6457-6464 ◽  
Author(s):  
Ilknur Yesilce ◽  
Gabil Adilov
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Abstract Convexity ◽  
Hadamard Inequality ◽  
Fractional Integral Operators ◽  
General Integral ◽  
Generalized Fractional Integral ◽  
General Integral Operator

B??1-convexity is an abstract convexity type. We obtained Hermite-Hadamard inequality for B-1-convex functions. But now, there are new and more general integral operator types that are fractional integrals. Thus, we need to prove Hermite-Hadamard inequalities involving different fractional integral operator types with this article.

scholarly journals Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function

2020 ◽  
Vol 18 (1) ◽  
pp. 794-806 ◽  
Author(s):  
Jiangfeng Han ◽  
Pshtiwan Othman Mohammed ◽  
Huidan Zeng
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Primary Objective ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Generalized Fractional Integral

Abstract The primary objective of this research is to establish the generalized fractional integral inequalities of Hermite-Hadamard-type for MT-convex functions and to explore some new Hermite-Hadamard-type inequalities in a form of Riemann-Liouville fractional integrals as well as classical integrals. It is worth mentioning that our work generalizes and extends the results appeared in the literature.

scholarly journals NEW INEQUALITIES OF OSTROWSKI TYPE FOR CO-ORDINATED CONVEX FUNCTIONS VIA GENERALIZED FRACTIONAL INTEGRALS

2021 ◽  
pp. 899
Author(s):  
Muhammad Aamir Ali ◽  
Hüseyin Budak ◽  
Zhiyue Zhang
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Special Cases ◽  
Generalized Fractional Integral ◽  
New Inequalities

In this paper, we establish new inequalities of Ostrowski type for co-ordinated convex function by using generalized fractional integral. We also discuss some special cases of our established results.

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 Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka B. Malinowska ◽  
Delfim F. M. Torres
Keyword(s):  
Fractional Integral ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Fractional Integrals ◽  
Natural Boundary ◽  
Free Boundary Value Problems ◽  
Fractional Variational Problems ◽  
Generalized Fractional Integral ◽  
Fractional Calculus Of Variations ◽  
Natural Boundary Conditions

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed.

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