scholarly journals Some fractional proportional integral inequalities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Gauhar Rahman ◽  
Thabet Abdeljawad ◽  
Aftab Khan ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Exponential Functions ◽  
Proportional Integral ◽  
Conformable Derivatives ◽  
Fractional Integrals And Derivatives ◽  
Decreasing Functions

Abstract In the last few years, various researchers studied the so-called conformable integrals and derivatives. Based on that notion some authors used modified conformable derivatives (proportional derivatives) to generate nonlocal fractional integrals and derivatives, called fractional proportional integrals and derivatives, which contain exponential functions in their kernels. Our aim in this paper is to establish some new integral inequalities by utilizing the fractional proportional-integral operators. In fact, certain new classes of integral inequalities for a class of n ($n\in \mathbb{N}$ n ∈ N ) positive continuous and decreasing functions on $[a,b]$ [ a , b ] are presented. The inequalities presented in this paper are more general than the existing classical inequalities.

A distributional theory of fractional calculus

1985 ◽  
Vol 99 (3-4) ◽  
pp. 347-357 ◽  
Author(s):  
W. Lamb
Keyword(s):  
Fractional Calculus ◽  
Spectral Properties ◽  
Fractional Integration ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Fractional Powers ◽  
Fractional Powers Of Operators ◽  
Fractional Integrals And Derivatives

SynopsisIn this paper, a theory of fractional calculus is developed for certain spacesD′p,μof generalised functions. The theory is based on the construction of fractionalpowers of certain simple differential and integral operators. With the parameter μ suitably restricted, these fractional powers are shown to coincide with the Riemann-Liouville and Weyl operators of fractional integration and differentiation. Standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators; see [6], [7]. Some spectral properties are also obtained.

scholarly journals Different type parameterized inequalities via generalized integral operators with applications

2021 ◽  
Vol 66 (3) ◽  
pp. 423-440
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Integral Operator ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Recent Past ◽  
Special Means ◽  
Special Cases ◽  
Type Integral ◽  
Erentiable Function ◽  
Almost All

"The authors have proved an identity for a generalized integral operator via di erentiable function with parameters. By applying the established identity, the generalized trapezium, midpoint and Simpson type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identi ed. Some applications of presented results to special means and new error estimates for the trapezium and midpoint quadrature formula have been analyzed. The ideas and techniques of this paper may stimulate further research in the eld of integral inequalities."

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 the Hermite-Hadamard-Mercer Type Inequalities for generalized Proportional Fractional Integrals

2020 ◽  
Author(s):  
Seda KILINÇ YILDIRIM ◽  
Hüseyin Yıldırım
Keyword(s):  
Integral Inequalities ◽  
Fractional Integrals ◽  
Absolute Value ◽  
Proportional Integral ◽  
Type Integral

Our aim in this paper is to establish some new Hermite-Hadamard- Mercer type integral inequalities by utilizing the fractional proportional-integral operators.For this purpose, Hermite-Hadamard-Mercer inequalities for di¤er- antiable mappings whose derivatives in absolute value are convex.

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 Certain Fractional Proportional Integral Inequalities via Convex Functions

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 222 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad ◽  
Samee Ullah
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integral Operators ◽  
Proportional Integral ◽  
Special Cases ◽  
Classical Integral

The goal of this article is to establish some fractional proportional integral inequalities for convex functions by employing proportional fractional integral operators. In addition, we establish some classical integral inequalities as the special cases of our main findings.

scholarly journals Integral inequalities via Raina’s fractional integrals operator with respect to a monotone function

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shu-Bo Chen ◽  
Saima Rashid ◽  
Zakia Hammouch ◽  
Muhammad Aslam Noor ◽  
Rehana Ashraf ◽  
...  
Keyword(s):  
Fractional Integral ◽  
Monotone Function ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Liouville Type ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Generalized Fractional Integral

AbstractWe establish certain new fractional integral inequalities involving the Raina function for monotonicity of functions that are used with some traditional and forthright inequalities. Taking into consideration the generalized fractional integral with respect to a monotone function, we derive the Grüss and certain other associated variants by using well-known integral inequalities such as Young, Lah–Ribarič, and Jensen integral inequalities. In the concluding section, we present several special cases of fractional integral inequalities involving generalized Riemann–Liouville, k-fractional, Hadamard fractional, Katugampola fractional, $(k,s)$ ( k , s ) -fractional, and Riemann–Liouville-type fractional integral operators. Moreover, we also propose their pertinence with other related known outcomes.

scholarly journals Some new inequalities for generalized fractional conformable integral operators

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Kottakkaran Sooppy Nisar ◽  
Gauhar Rahman ◽  
Aftab Khan
Keyword(s):  
Integral Operators ◽  
Integral Inequalities ◽  
Decreasing Functions ◽  
New Inequalities

Abstract The present paper aims to establish certain new classes of integral inequalities for a class of n ($n\in \mathbb{N}$ n ∈ N ) positive continuous and decreasing functions by utilizing the generalized fractional conformable integral operators (FCIO) recently defined by Khan and Khan. From these results, we also derive several particular cases.

New integral inequalities for geometrically convex functions via conformable fractional integral operators

2021 ◽  
Vol 29 (2) ◽  
pp. 205-219 ◽  
Author(s):  
SAIMA RASHID ◽  
AHMET OCAK AKDEMIR ◽  
MUHAMMAD ASLAM NOOR ◽  
KHALIDA INAYAT NOOR
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Conformable Fractional Integral

We establish several basic inequalities versions of the Hermite-Hadamard type inequalities for GA− and GG−convexity for conformable fractional integrals. Several special cases are also discussed, which can be deduced from our main result.

scholarly journals Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications

2021 ◽  
Vol 5 (3) ◽  
pp. 80
Author(s):  
Hari Mohan Srivastava ◽  
Artion Kashuri ◽  
Pshtiwan Othman Mohammed ◽  
Dumitru Baleanu ◽  
Y. S. Hamed
Keyword(s):  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Wright Function ◽  
Nonconvex Functions ◽  
Special Cases ◽  
Type Integral ◽  
Generic Class ◽  
Two Parameters ◽  
Class Of Functions

In this paper, the authors define a new generic class of functions involving a certain modified Fox–Wright function. A useful identity using fractional integrals and this modified Fox–Wright function with two parameters is also found. Applying this as an auxiliary result, we establish some Hermite–Hadamard-type integral inequalities by using the above-mentioned class of functions. Some special cases are derived with relevant details. Moreover, in order to show the efficiency of our main results, an application for error estimation is obtained as well.

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