scholarly journals Some new inequalities for convex functions via Riemann-Liouville fractional integrals

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mustafa Gürbüz ◽  
Çetin Yıldız
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Power Mean ◽  
Semigroup Property ◽  
Fractional Analysis ◽  
Power Mean Inequality ◽  
Fractional Orders ◽  
Scientific Fields ◽  
New Inequalities

AbstractFractional analysis has evolved considerably over the last decades and has become popular in many technical and scientific fields. Many integral operators which ables us to integrate from fractional orders has been generated. Each of them provides different properties such as semigroup property, singularity problems etc. In this paper, firstly, we obtained a new kernel, then some new integral inequalities which are valid for integrals of fractional orders by using Riemann-Liouville fractional integral. To do this, we used some well-known inequalities such as Hölder's inequality or power mean inequality. Our results generalize some inequalities exist in the literature.

scholarly journals New refinements of Chebyshev–Pólya–Szegö-type inequalities via generalized fractional integral operators

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Saad Ihsan Butt ◽  
Ahmet Ocak Akdemir ◽  
Muhammad Yousaf Bhatti ◽  
Muhammad Nadeem
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integral Operators ◽  
Inequality Theory ◽  
Synchronous Development ◽  
Fractional Analysis ◽  
Generalized Fractional Integral ◽  
Developing Area ◽  
Generalized Fractional Integral Operators ◽  
New Inequalities

Abstract Fractional analysis, as a rapidly developing area, is a tool to bring new derivatives and integrals into the literature with the effort put forward by many researchers in recent years. The theory of inequalities is a subject of many mathematicians’ work in the last century and has contributed to other areas with its applications. Especially in recent years, these two fields, fractional analysis and inequality theory, have shown a synchronous development. Inequality studies have been carried out by using new operators revealed in the fractional analysis. In this paper, by combining two important concepts of these two areas we obtain new inequalities of Chebyshev–Polya–Szegö type by means of generalized fractional integral operators. Our results are concerned with the integral of the product of two functions and the product of two integrals. They improve the results in the paper (J. Math. Inequal. 10(2):491–504, 2016).

scholarly journals Some novel inequalities involving a function’s fractional integrals in relation to another function through generalized quasiconvex mappings

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3349-3360
Author(s):  
Eze Nwaeze ◽  
Artion Kashuri
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Fractional Integral Operators ◽  
Absolute Value ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators ◽  
New Inequalities

In this paper, we establish new inequalities of the Hermite-Hadamard, midpoint and trapezoid types for functions whose first derivatives in absolute value are ?-quasiconvex by means of generalized fractional integral operators with respect to another function ? : [?,?] ? (0,?). Our theorems reduce to results involving the Riemann-Liouville fractional integral operators if ? is the identity map, and results involving the Hadamard operators if ?(x) = ln x. More inequalities can be deduced by choosing different bifunctions for ?. To the best of our knowledge, the results obtained herein are new and we hope that they will stimulate further interest in this direction.

scholarly journals New Chebyshev type inequalities via a general family of fractional integral operators with a modified Mittag-Leffler kernel

2021 ◽  
Vol 6 (10) ◽  
pp. 11167-11186
Author(s):  
Hari M. Srivastava ◽  
◽  
Artion Kashuri ◽  
Pshtiwan Othman Mohammed ◽  
Abdullah M. Alsharif ◽  
...  
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Chebyshev Inequality ◽  
Fractional Integral Operators ◽  
General Family ◽  
Chebyshev Functional ◽  
Bounded Below

<abstract><p>The main goal of this article is first to introduce a new generalization of the fractional integral operators with a certain modified Mittag-Leffler kernel and then investigate the Chebyshev inequality via this general family of fractional integral operators. We improve our results and we investigate the Chebyshev inequality for more than two functions. We also derive some inequalities of this type for functions whose derivatives are bounded above and bounded below. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. Finally, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view.</p></abstract>

scholarly journals On certain new CAUCHY–TYPE fracitioanl integral inequalities and OPIAL–TYPE fractional derivative inequalities

2015 ◽  
Vol 46 (1) ◽  
pp. 67-73 ◽  
Author(s):  
Amit Chouhan
Keyword(s):  
Fractional Derivative ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Future Research ◽  
Cauchy Type ◽  
Fractional Integral Operators ◽  
Integrable Functions ◽  
New Inequalities

The aim of this paper is to establish several new fractional integral and derivative inequalities for non-negative and integrable functions. These inequalities related to the extension of general Cauchy type inequalities and involving Saigo, Riemann-Louville type fractional integral operators together with multiple Erdelyi-Kober operator. Furthermore the Opial-type fractional derivative inequality involving H-function is also established. The generosity of H-function could leads to several new inequalities that are of great interest of future research.

scholarly journals Some New Tempered Fractional Pólya-Szegö and Chebyshev-Type Inequalities with Respect to Another Function

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad ◽  
Muhammad Samraiz
Keyword(s):  
Present Article ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Bounded Functions ◽  
New Inequalities

In this present article, we establish certain new Pólya–Szegö-type tempered fractional integral inequalities by considering the generalized tempered fractional integral concerning another function Ψ in the kernel. We then prove certain new Chebyshev-type tempered fractional integral inequalities for the said operator with the help of newly established Pólya–Szegö-type tempered fractional integral inequalities. Also, some new particular cases in the sense of classical tempered fractional integrals are discussed. Additionally, examples of constructing bounded functions are considered. Furthermore, one can easily form new inequalities for Katugampola fractional integrals, generalized Riemann–Liouville fractional integral concerning another function Ψ in the kernel, and generalized fractional conformable integral by applying different conditions.

scholarly journals Consensus of classical and fractional inequalities having congruity on time scale calculus

2019 ◽  
Vol 27 (1) ◽  
pp. 57-69
Author(s):  
Muhammad Jibril Shahab Sahir
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Hölder’S Inequality ◽  
Fractional Integrals ◽  
Power Mean ◽  
Extended Form ◽  
Hölder's Inequality ◽  
Time Scale Calculus ◽  
Power Mean Inequality ◽  
Radon’S Inequality

Abstract In this paper, we find accordance of some classical inequalities and fractional dynamic inequalities. We find inequalities such as Radon’s inequality, Bergström’s inequality, Rogers-Hölder’s inequality, Cauchy-Schwarz’s inequality, the weighted power mean inequality and Schlömilch’s inequality in generalized and extended form by using the Riemann-Liouville fractional integrals on time scales.

scholarly journals Some new inequalities for generalized convex functions pertaining generalized fractional integral operators and their applications

2021 ◽  
Vol 17 (1) ◽  
pp. 37-64
Author(s):  
A. Kashuri ◽  
M.A. Ali ◽  
M. Abbas ◽  
M. Toseef
Keyword(s):  
Differentiable Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Generalized Convex Functions ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators ◽  
New Inequalities

Abstract In this paper, authors establish a new identity for a differentiable function using generic integral operators. By applying it, some new integral inequalities of trapezium, Ostrowski and Simpson type are obtained. Moreover, several special cases have been studied in detail. Finally, many useful applications have been found.

scholarly journals Some Estimates for Generalized Riemann-Liouville Fractional Integrals of Exponentially Convex Functions and Their Applications

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 807 ◽  
Author(s):  
Saima Rashid ◽  
Thabet Abdeljawad ◽  
Fahd Jarad ◽  
Muhammad Aslam Noor
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Order Derivative ◽  
Fundamental Identity ◽  
First Order ◽  
Special Means ◽  
Exponentially Convex Functions ◽  
New Inequalities

In the present paper, we investigate some Hermite-Hadamard ( HH ) inequalities related to generalized Riemann-Liouville fractional integral ( GRLFI ) via exponentially convex functions. We also show the fundamental identity for GRLFI having the first order derivative of a given exponentially convex function. Monotonicity and exponentially convexity of functions are used with some traditional and forthright inequalities. In the application part, we give examples and new inequalities for the special means.

scholarly journals On New Inequalities via Riemann-Liouville Fractional Integration

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Mehmet Zeki Sarikaya ◽  
Hasan Ogunmez
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integration ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
New Inequalities

We extend the Montgomery identities for the Riemann-Liouville fractional integrals. We also use these Montgomery identities to establish some new integral inequalities. Finally, we develop some integral inequalities for the fractional integral using differentiable convex functions.

Fractional powers of operators and Riesz fractional integrals

1989 ◽  
Vol 112 (3-4) ◽  
pp. 237-247 ◽  
Author(s):  
S. E. Schiavone
Keyword(s):  
Banach Spaces ◽  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Fréchet Spaces ◽  
Test Functions ◽  
Fractional Powers ◽  
Fractional Integral Operators ◽  
Fractional Powers Of Operators

SynopsisIn this paper, a theory of fractional powers of operators due to Balakrishnan, which is valid for certain operators on Banach spaces, is extended to Fréchet spaces. The resultingtheory is shown to be more general than that developed in an earlier approach by Lamb, and is applied to obtain mapping properties of certain Riesz fractional integral operators on spaces of test functions.

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