Some Novel Fractional Integral Inequalities over a New Class of Generalized Convex Function

The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on the Hermite–Hadamard-type inequalities concerning a monotonic increasing function. The proposed methodology deals with a new class of convexity and related integral and fractional inequalities. There exists a solid connection between fractional operators and convexity because of its fascinating nature in the numerical sciences. Some special cases have also been discussed, and several already-known inequalities have been recaptured to behave well. Some applications related to special means, q-digamma, modified Bessel functions, and matrices are discussed as well. The aftereffects of the plan show that the methodology can be applied directly and is computationally easy to understand and exact. We believe our findings generalise some well-known results in the literature on s-convexity.

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Hermite-Hadamard like inequalities for fractional integral operator via convexity and quasi-convexity with their applications

AIMS Mathematics ◽

10.3934/math.2022190 ◽

2021 ◽

Vol 7 (3) ◽

pp. 3418-3439

Author(s):

Jamshed Nasir ◽

◽

Shahid Qaisar ◽

Saad Ihsan Butt ◽

Hassen Aydi ◽

...

Keyword(s):

Integral Operator ◽

Fractional Integral ◽

Bessel Functions ◽

Fractional Operator ◽

Real Numbers ◽

Hadamard Inequality ◽

Special Means ◽

Special Cases ◽

Derivatives Of ◽

Quasi Convexity

<abstract><p>Since the supposed Hermite-Hadamard inequality for a convex function was discussed, its expansions, refinements, and variations, which are called Hermite-Hadamard type inequalities, have been widely explored. The main objective of this article is to acquire new Hermite-Hadamard type inequalities employing the Riemann-Liouville fractional operator for functions whose third derivatives of absolute values are convex and quasi-convex in nature. Some special cases of the newly presented results are discussed as well. As applications, several estimates concerning Bessel functions and special means of real numbers are illustrated.</p></abstract>

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Some New Generalizations for Exponentially s-Convex Functions and Inequalities via Fractional Operators

Fractal and Fractional ◽

10.3390/fractalfract3020024 ◽

2019 ◽

Vol 3 (2) ◽

pp. 24 ◽

Cited By ~ 13

Author(s):

Saima Rashid ◽

Muhammad Aslam Noor ◽

Khalida Inayat Noor ◽

Ahmet Ocak Akdemir

Keyword(s):

Convex Functions ◽

Fractional Integral ◽

Fractional Operators ◽

Special Means ◽

Special Cases ◽

Katugampola Fractional Integral ◽

Hadamard Fractional Integral

The main objective of this paper is to obtain the Hermite–Hadamard-type inequalities for exponentially s-convex functions via the Katugampola fractional integral. The Katugampola fractional integral is a generalization of Riemann–Liouville fractional integral and Hadamard fractional integral. Some special cases and applications to special means are also discussed.

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Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1959.0023 ◽

1959 ◽

Vol 249 (1257) ◽

pp. 284-292 ◽

Cited By ~ 1

Keyword(s):

Asymptotic Expansions ◽

Bessel Functions ◽

Modified Bessel Functions ◽

Lommel Functions ◽

Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

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Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations

Journal of Systems Science and Information ◽

10.21078/jssi-2018-069-16 ◽

2018 ◽

Vol 6 (1) ◽

pp. 69-84

Author(s):

Jia Xu ◽

Liwei Liu ◽

Taozeng Zhu

Keyword(s):

Generating Functions ◽

Transient Analysis ◽

Bessel Functions ◽

Queueing System ◽

System Size ◽

Heterogeneous Servers ◽

Modified Bessel Functions ◽

Multiple Vacations ◽

Special Cases ◽

Mean And Variance

AbstractWe consider anM/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities, mean and variance of the system size at timetby employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.

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

Symmetry ◽

10.3390/sym12091503 ◽

2020 ◽

Vol 12 (9) ◽

pp. 1503 ◽

Cited By ~ 3

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

Arab Journal of Mathematical Sciences ◽

10.1016/j.ajmsc.2018.12.003 ◽

2019 ◽

Vol 26 (1/2) ◽

pp. 41-55 ◽

Cited By ~ 1

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.

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New Modified Conformable Fractional Integral Inequalities of Hermite–Hadamard Type with Applications

Journal of Function Spaces ◽

10.1155/2020/4352357 ◽

2020 ◽

Vol 2020 ◽

pp. 1-14 ◽

Cited By ~ 4

Author(s):

Thabet Abdeljawad ◽

Pshtiwan Othman Mohammed ◽

Artion Kashuri

Keyword(s):

Fractional Integral ◽

Integral Inequalities ◽

Fractional Operators ◽

Special Means ◽

Conformable Fractional Integral ◽

Fractional Parameter

In this study, a few inequalities of Hermite–Hadamard type are constructed via the conformable fractional operators so that the normal version is recovered in its limit for the conformable fractional parameter. Finally, we present some examples to demonstrate the usefulness of conformable fractional inequalities in the context of special means of the positive numbers.

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Certain Fractional Integral Formulas Involving the Product of Generalized Bessel Functions

The Scientific World JOURNAL ◽

10.1155/2013/567132 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

D. Baleanu ◽

P. Agarwal ◽

S. D. Purohit

Keyword(s):

Bessel Function ◽

Fractional Integral ◽

Bessel Functions ◽

Fractional Integration ◽

Fractional Integrals ◽

General Character ◽

Generalized Bessel Function ◽

Integral Formulas ◽

Generalized Bessel Functions ◽

Special Cases

We apply generalized operators of fractional integration involving Appell’s functionF3(·)due to Marichev-Saigo-Maeda, to the product of the generalized Bessel function of the first kind due to Baricz. The results are expressed in terms of the multivariable generalized Lauricella functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.

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Generalized proportional fractional integral Hermite–Hadamard’s inequalities

Advances in Difference Equations ◽

10.1186/s13662-021-03651-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Tariq A. Aljaaidi ◽

Deepak B. Pachpatte ◽

Thabet Abdeljawad ◽

Mohammed S. Abdo ◽

Mohammed A. Almalahi ◽

...

Keyword(s):

Continuous Function ◽

Differential Equations ◽

Fractional Calculus ◽

Approximation Theory ◽

Fractional Differential Equations ◽

Fractional Integral ◽

Fractional Operators ◽

Uniqueness Of Solutions ◽

Special Cases ◽

Fractional Differential

AbstractThe theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work.

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Hermite-Hadamard inequalities for relative semi-convex functions and applications

Filomat ◽

10.2298/fil1402221n ◽

2014 ◽

Vol 28 (2) ◽

pp. 221-230 ◽

Cited By ~ 8

Author(s):

Muhammad Noor ◽

Khalida Noor ◽

Muhammad Awan

Keyword(s):

Applied Sciences ◽

Convex Functions ◽

Special Means ◽

Special Cases

In this paper, we prove some Hermite-Hadamard inequalities for the class of relative semiconvex functions. Several special cases are also discussed. Thus it is worth mentioning that our results can be viewed as a generalization of previous results. Some applications to special means are also presented. Ideas and techniques of this paper may inspire further research in various branches of pure and applied sciences.

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