• 호남수학학술지
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• 제40권1호
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• pp.125-138
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• 2018

In this present paper, we deal with the generalized (k, s)-fractional integral and differential operators recently defined by Nisar et al. and obtain some generalized (k, s)-fractional integral and differential formulas involving the class of a function as its kernels. Also, we investigate a certain number of their consequences containing the said function in their kernels.

• East Asian mathematical journal
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• 제30권3호
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• pp.283-291
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• 2014

A remarkably large number of inequalities involving the fractional integral operators have been investigated in the literature by many authors. Very recently, Baleanu et al. [2] gave certain interesting fractional integral inequalities involving the Gauss hypergeometric functions. Using the same fractional integral operator, in this paper, we present some (presumably) new fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Saigo, Erd$\acute{e}$lyi-Kober and Riemann-Liouville type fractional integral operators. Relevant connections of the results presented here with those earlier ones are also pointed out.

• 대한수학회보
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• 제53권1호
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• pp.181-193
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• 2016

During the past two decades or so, fractional integral inequalities have proved to be one of the most powerful and far-reaching tools for the development of many branches of pure and applied mathematics. Very recently, many authors have presented some generalized inequalities involving the fractional integral operators. Here, using the pathway fractional integral operator, we give some presumably new and potentially useful fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville type fractional integral operators. Relevant connections of the results presented here with those earlier ones are also pointed out.

• 호남수학학술지
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• 제43권4호
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• pp.587-596
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• 2021

In this article, using the concept proposed reciently by the author, of a Generalized k-Proportional Fractional Integral Operators with General Kernel, new integral inequalities are obtained for convex functions. It is shown that several known results are particular cases of the proposed inequalities and in the end new directions of work are provided.

• 호남수학학술지
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• 제42권2호
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• pp.359-376
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• 2020

The present research investigates the bounds of an integral operator for convex functions and a differentiable function f such that |f'| is convex. Further, these bounds of integral operators specifically produce estimations of various classical fractional and recently defined conformable integral operators. These results also contain bounds of Hadamard type for symmetric convex functions.

• Kyungpook Mathematical Journal
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• 제59권3호
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• pp.433-443
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• 2019

In this work, we evaluate the Mellin transform of the Marichev-Saigo-Maeda fractional integral operator with Appell's function $F_3$ type kernel. We then discuss six special cases of the result involving the Saigo fractional integral operator, the $Erd{\acute{e}}lyi$-Kober fractional integral operator, the Riemann-Liouville fractional integral operator and the Weyl fractional integral operator. We obtain new and known results as special cases of our main results. Finally, we obtain the images of S-generalized Gauss hypergeometric function under the operators of our study.

• 대한수학회보
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• 제59권4호
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• pp.905-915
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• 2022

This paper studies the boundedness of integral operators on the Cesàro function spaces. As applications of the main result, we obtain the Hilbert inequalities, the boundedness of the Erdélyi-Kober fractional integrals and the Mellin fractional integrals on the Cesàro function spaces.

• 대한수학회보
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• 제57권2호
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• pp.383-391
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• 2020

We study the mapping property of multilinear fractional maximal operators in Lipschitz spaces. It should be pointed out that some of the techniques employed in the study of fractional integral operators do not apply to fractional maximal operators.

• Korean Journal of Mathematics
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• 제29권2호
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• pp.227-237
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• 2021

Opial inequality and its consequences are useful in establishing existence and uniqueness of solutions of initial and boundary value problems for differential and difference equations. In this paper we analyze Opial-type inequalities for convex functions. We have studied different versions of these inequalities for a generalized integral operator. Further difference of Opial-type inequalities are utilized to obtain generalized mean value theorems, which further produce various interesting derivations for fractional and conformable integral operators.

• Kyungpook Mathematical Journal
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• 제60권1호
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• pp.73-116
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• 2020

The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind.