scholarly journals A basic study of a fractional integral operator with extended Mittag-Leffler kernel

2021 ◽  
Vol 6 (11) ◽  
pp. 12757-12770
Author(s):  
Gauhar Rahman ◽  
◽  
Iyad Suwan ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad ◽  
...  
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Basic Study ◽  
Basic Properties

<abstract><p>In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.</p></abstract>

scholarly journals Two-weight norm inequalities for the rough fractional integrals

2001 ◽  
Vol 25 (8) ◽  
pp. 517-524 ◽  
Author(s):  
Yong Ding ◽  
Chin-Cheng Lin
Keyword(s):  
Integral Operator ◽  
Maximal Operator ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Weight Functions ◽  
Norm Inequalities ◽  
Fractional Maximal Operator

The authors give the weighted(Lp,Lq)-boundedness of the rough fractional integral operatorTΩ,αand the fractional maximal operatorMΩ,αwith two different weight functions.

scholarly journals Generalized Fractional Inequalities of the Hermite–Hadamard Type for Convex Stochastic Processes

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
McSylvester Ejighikeme Omaba ◽  
Eze R. Nwaeze
Keyword(s):  
Stochastic Process ◽  
Integral Operator ◽  
Stochastic Processes ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Convex Stochastic Processes ◽  
Katugampola Fractional Integral ◽  
Convex Stochastic Process

AbstractA generalization of the Hermite–Hadamard (HH) inequality for a positive convex stochastic process, by means of a newly proposed fractional integral operator, is hereby established. Results involving the Riemann– Liouville, Hadamard, Erdélyi–Kober, Katugampola, Weyl and Liouville fractional integrals are deduced as particular cases of our main result. In addition, we also apply some known HH results to obtain some estimates for the expectations of integrals of convex and p-convex stochastic processes. As a side note, we also pointed out a mistake in the main result of the paper [Hermite–Hadamard type inequalities, convex stochastic processes and Katugampola fractional integral, Revista Integración, temas de matemáticas 36 (2018), no. 2, 133–149]. We anticipate that the idea employed herein will inspire further research in this direction.

scholarly journals Boundedness of homogeneous fractional integrals on Lp for N/α ≤ P ≤ ∞

2002 ◽  
Vol 167 ◽  
pp. 17-33 ◽  
Author(s):  
Yong Ding ◽  
Shanzhen Lu
Keyword(s):  
Integral Operator ◽  
Hardy Spaces ◽  
Fractional Integral ◽  
Riesz Potential ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Potential Operator ◽  
Riesz Potential Operator ◽  
Homogeneous Fractional Integral Operator

AbstractIn this paper we study the map properties of the homogeneous fractional integral operator TΩ, α on Lp(ℝn) for n/α ≤ p ≤ ∞.We prove that if Ω satisfies some smoothness conditions on Sn−1 then TΩ, α is bounded from Ln/α(ℝn) to BMO(ℝn), and from Lp(ℝn) (n/α < p ≤ ∞) to a class of the Campanato spaces l, λ (ℝn), respectively. As the corollary of the results above, we show that when Ω satisfies some smoothness conditions on Sn−1 the homogeneous fractional integral operator TΩ, α is also bounded from Hp(ℝn) (n/(n + α) ≤ p ≤ 1) to Lq(ℝn) for 1/q = 1/p-α/n. The results are the extensions of Stein-Weiss (for p = 1) and Taibleson-Weiss’s (for n/(n + α) ≤ p < 1) results on the boundedness of the Riesz potential operator Iα on the Hardy spaces Hp(ℝn).

On the range and inversion of fractional integrals in weighted spaces

1982 ◽  
Vol 92 (1-2) ◽  
pp. 51-64 ◽  
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Integral Operator ◽  
Lebesgue Space ◽  
Inversion Formula ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Weighted Spaces ◽  
Fractional Integrals ◽  
Weight Functions ◽  
And Inversion ◽  
Unweighted Case

SynopsisThe weight functions w(x) for which the Riemann fractional integral operator Iα is bounded from the Lebesgue space Lp(wp) into Lq(wq), l/q = l/p −, have been characterized by Muckenhoupt and Wheeden. In this paper, we prove an inversion formula for Iα in the context of these weighted spaces and we also characterize the range of Iα as a subset of Lq(wq) Similar results are proved for other fractional integrals. These results may be viewed as weighted analogues of certain results of Stein and Zygmund, Herson and Heywood, Heywood, and Kober who considered the unweighted case, w(x) = l.

A note on fractional integral operator associated with multiindex Mittag-Leffler functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1931-1939 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Fractional Integral ◽  
Fractional Integration ◽  
Fractional Integral Operator ◽  
Integration Formula ◽  
Special Cases

Recently Kiryakova and several other ones have investigated so-called multiindex Mittag-Leffler functions associated with fractional calculus. Here, in this paper, we aim at establishing a new fractional integration formula (of pathway type) involving the generalized multiindex Mittag-Leffler function E?,k[(?j,?j)m;z]. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.

scholarly journals New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1753
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Omar Bazighifan ◽  
Georgia Irina Oros
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Type Inequality ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Convex Mappings ◽  
Special Cases ◽  
Harmonically Convex Functions

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

scholarly journals FRACTIONAL INTEGRAL OPERATORS IN NONHOMOGENEOUS SPACES

2009 ◽  
Vol 80 (2) ◽  
pp. 324-334 ◽  
Author(s):  
H. GUNAWAN ◽  
Y. SAWANO ◽  
I. SIHWANINGRUM
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Type Inequality ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Fractional Integral Operator ◽  
Homogeneous Type ◽  
Fractional Integral Operators

AbstractWe discuss here the boundedness of the fractional integral operatorIαand its generalized version on generalized nonhomogeneous Morrey spaces. To prove the boundedness ofIα, we employ the boundedness of the so-called maximal fractional integral operatorIa,κ*. In addition, we prove an Olsen-type inequality, which is analogous to that in the case of homogeneous type.

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