• Bulletin of the Korean Mathematical Society
• /
• v.56 no.6
• /
• pp.1423-1433
• /
• 2019

Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.

• Honam Mathematical Journal
• /
• v.44 no.2
• /
• pp.271-280
• /
• 2022

In this study, We have applied the right operator k-Riemann-Liouville is involving two orders α and β with a positive parameter p > 0, further, the left operator k-Riemann-Liouville is used with the negative parameter p < 0 to introduce a new version related to Hardy-type inequalities. These inequalities are given and reversed for the cases 0 < p < 1 and p < 0. We then improved and generalized various consequences in the framework of Hardy-type fractional integral inequalities.

• Korean Journal of Mathematics
• /
• v.29 no.4
• /
• pp.687-704
• /
• 2021

The refinement of an inequality provides better convergence of one quantity towards the other one. We have established the refinements of Hadamard inequalities for Riemann-Liouville fractional integrals via strongly (α, m)-convex functions. In particular, we obtain two refinements of the classical Hadamard inequality. By using some known integral identities we also give refinements of error bounds of some fractional Hadamard inequalities.

• Journal of applied mathematics & informatics
• /
• v.33 no.1_2
• /
• pp.119-125
• /
• 2015

In this note, two new mappings associated with convexity are propoesd, by which we obtain some new Hermite-Hadamard type inequalities for convex functions via Riemann-Liouville fractional integrals. We conclude that the results obtained in this work are the refinements of the earlier results.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.1
• /
• pp.9-23
• /
• 2018

In this paper, inspired by the fractional Brownian sheet of Riemann-Liouville type, we introduce the operator fractional Brownian sheet of Riemman-Liouville type, and study some properties of it. We also present an approximation in law to it based on the martingale differences.

• Kyungpook Mathematical Journal
• /
• v.56 no.4
• /
• pp.1161-1168
• /
• 2016

Belarbi and Dahmani [3], recently, using the Riemann-Liouville fractional integral, presented some interesting integral inequalities for the Chebyshev functional in the case of two synchronous functions. Subsequently, Dahmani et al. [5] and Sulaiman [17], provided some fractional integral inequalities. Here, motivated essentially by Belarbi and Dahmani's work [3], we aim at establishing certain (presumably) new inequalities associated with pathway fractional integral operators by using synchronous functions which are involved in the Chebychev functional. Relevant connections of the results presented here with those involving Riemann-Liouville fractional integrals are also pointed out.

• Communications of the Korean Mathematical Society
• /
• v.35 no.1
• /
• pp.161-184
• /
• 2020

In this article, we establish some new weighted Hardy-type inequalities involving some variants of extended Riemann-Liouville fractional derivative operators, using convex and increasing functions. As special cases of the main results, we obtain the results of [18,19]. We also prove the boundedness of the k-fractional integral operator on L_p[a, b].

• Journal of applied mathematics & informatics
• /
• v.24 no.1_2
• /
• pp.31-48
• /
• 2007

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on bypassing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable function $f(x+h)=E_{\alpha}(h^{\alpha}{D_x^{\alpha})f(x)$.

• Communications of the Korean Mathematical Society
• /
• v.32 no.3
• /
• pp.677-688
• /
• 2017

The composition of Jacobi type finite classes of the classical orthogonal polynomials with two generalized Riemann-Liouville fractional derivatives are considered. The outcomes are expressed in terms of generalized Wright function or generalized hypergeometric function. Similar composition formulas are also obtained by considering the generalized Riemann-Liouville and $Erd{\acute{e}}yi-Kober$ fractional integral operators.

• Kyungpook Mathematical Journal
• /
• v.58 no.1
• /
• pp.55-66
• /
• 2018

By making use of the Riemann-Liouville fractional integrals, we establish further results on Chebyshev inequality. Other Steffensen integral results of the weighted Chebyshev functional are also proved. Some classical results of the paper:[ Steffensen's generalization of Chebyshev inequality. J. Math. Inequal., 9(1), (2015).] can be deduced as some special cases.