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Journal Article•DOI•

Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities

Mehmet Zeki Sarikaya1, Erhan Set1, Hatice Yaldiz1, Nagihan Basak1•
Düzce University1
01 May 2013-Mathematical and Computer Modelling (Pergamon)-Vol. 57, Iss: 9, pp 2403-2407
TL;DR: An integral identity and some Hermite–Hadamard type integral inequalities for the fractional integrals are obtained and these results have some relationships with S.S. Dragomir and R.P. Agarwal's inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula.
About: This article is published in Mathematical and Computer Modelling.The article was published on 2013-05-01 and is currently open access. It has received 633 citations till now. The article focuses on the topics: Hermite–Hadamard inequality & Fractional calculus.
Citations
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Journal Article•DOI•
New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals

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Erhan Set1•
Düzce University1
01 Apr 2012-Computers & Mathematics With Applications
TL;DR: A new identity similar to an identity proved in Alomari et al. (2010) for fractional integrals is established and some new Ostrowski type inequalities for Riemann-Liouville fractional integral are established by making use of the established identity.
Abstract: A new identity similar to an identity proved in Alomari et al. (2010) [15] for fractional integrals is established. Then by making use of the established identity, some new Ostrowski type inequalities for Riemann-Liouville fractional integral are established. Our results have some relationships with the results of Alomari et al. (2010), proved in [15] and the analysis used in the proofs is simple.

194 citations

Journal Article•DOI•
Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for generalized fractional integrals ☆

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Hua Chen1, Udita N. Katugampola1•
University of Delaware1
15 Feb 2017-Journal of Mathematical Analysis and Applications
TL;DR: In this article, the Hermite-Hadamard and Hermite Hadamard-Fejer type inequalities for fractional integrals were obtained, which generalize the Riemann-Liouville equivalence.

163 citations

Cites background or methods from "Hermite-Hadamard's inequalities for..."

  • ...Main Results First we generalize Sarikaya’s results [37] of the Hermite-Hadamard’s inequalities for the Katugampola fractional integrals....

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  • ...Further results involving the two inequalities in question with applications to fractional integrals can be found, for example, in [6, 14, 37, 38] and the references therein....

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  • ...This can be proved using a similar line of argument as in the proof of Lemma 2 in [37]....

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Journal Article•DOI•
On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals

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Mehmet Zeki Sarikaya, Hseyin Yildirim
01 Jan 2017-Miskolc Mathematical Notes
TL;DR: In this article, the Hermite-Hadamard-type inequalities for fractional integrals were given an identity and with the help of this fractional-type integral identity, they gave some integral inequalities connected with the left-side of Hermite Hadamard type inequalities for Riemann-Liouville fractional integral integrals.
Abstract: In this paper, we have established Hermite-Hadamard-type inequalities for fractional integrals and will be given an identity. With the help of this fractional-type integral identity, we give some integral inequalities connected with the left-side of Hermite–Hadamard-type inequalities for Riemann-Liouville fractional integrals. 2010 Mathematics Subject Classification: 26D07; 26D10; 26D15; 26A33

145 citations

Cites background or methods from "Hermite-Hadamard's inequalities for..."

  • ...Meanwhile, Sarikaya et al.[25] presented the following important integral identity including the first-order derivative of f to establish many interesting HermiteHadamard type inequalities for convexity functions via Riemann-Liouville fractional integrals of the order ̨ > 0: Lemma 2....

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  • ...It is remarkable that Sarikaya et al.[25] first give the following interesting integral inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals....

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  • ...˛C1/ 2.b a/˛ J ˛aCf .b/CJ ˛ b f .a/ D b a 2 Z 1 0 .1 t /˛ t˛ f 0 .taC .1 t /b/dt: It is remarkable that Sarikaya et al.[25] first give the following interesting integral inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals....

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  • ...If the mapping jf 0j p p 1 is convex on Œa;b , thenˇ̌̌̌ ˇ 1b a Z b a f .x/dx f aCb 2 ˇ̌̌̌ ˇ .b a/ 4 1 pC1 1 p ( ˇ̌ f 0.a/ ˇ̌ p p 1 C3 ˇ̌ f 0.b/ ˇ̌ p p 1 p 1 p C 3 ˇ̌ f 0.a/ ˇ̌ p p 1 C ˇ̌ f 0.b/ ˇ̌ p p 1 p 1 p ) b a 4 4 pC1 1 p ˇ̌ f 0.a/ ˇ̌ C ˇ̌ f 0.b/ ˇ̌ : (1.3) Meanwhile, Sarikaya et al.[25] presented the following important integral identity including the first-order derivative of f to establish many interesting HermiteHadamard type inequalities for convexity functions via Riemann-Liouville fractional integrals of the order ˛ > 0: Lemma 2....

    [...]

  • ...The aim of this paper is to establish Hermite-Hadamard’s inequalities for RiemannLiouville fractional integral similar to the method in [25] and we will investigate some integral inequalities connected with the left hand side of the Hermite-Hadamard type inequalities for fractional integrals....

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Journal Article•DOI•
Hermite–Hadamard-type inequalities for Riemann–Liouville fractional integrals via two kinds of convexity

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JinRong Wang1, Xuezhu Li1, Michal Fe kan2, Yong Zhou3•
Guizhou University1, Comenius University in Bratislava2, Xiangtan University3
15 Nov 2013-Applicable Analysis
TL;DR: In this paper, two fundamental integral identities including the second-order derivatives of a given function via Riemann-Liouville fractional integrals are established, and with the help of these two fractional-type integral identities, all kinds of Hermite-Hadamard-type inequalities involving left-sided and right-sided RCI for m-convex and (s, ǫ)-m convex functions, respectively, are investigated.
Abstract: In this article, two fundamental integral identities including the second-order derivatives of a given function via Riemann–Liouville fractional integrals are established. With the help of these two fractional-type integral identities, all kinds of Hermite–Hadamard-type inequalities involving left-sided and right-sided Riemann–Liouville fractional integrals for m-convex and (s, m)-convex functions, respectively. Our methods considered here may be a stimulant for further investigations concerning Hermite–Hadamard-type inequalities involving Hadamard fractional integrals.

126 citations

Journal Article•DOI•
On some new Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals

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Yuruo Zhang1, JinRong Wang1•
Guizhou University1
30 Apr 2013-Journal of Inequalities and Applications
TL;DR: In this paper, three fundamental and important Riemann-Liouville fractional integral identities including a twice differentiable mapping are established, and some interesting Hermite-Hadamard type inequalities are presented.
Abstract: In this paper, three fundamental and important Riemann-Liouville fractional integral identities including a twice differentiable mapping are established. Secondly, some interesting Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals for m-convexity and -convexity functions, respectively, by virtue of the established integral identities are presented. MSC:26A33, 26A51, 26D07.

120 citations

References
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Book•
Fractional differential equations

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Igor Podlubny
01 Jan 1999

15,898 citations

Additional excerpts

  • ...For more details, one can consult [14–16]....

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An Introduction to the Fractional Calculus and Fractional Differential Equations

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Kenneth S. Miller, Bertram Ross
19 May 1993
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Abstract: Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.

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Convex Functions

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Dale E. Varberg
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Posted Content•
Fractional Calculus: Integral and Differential Equations of Fractional Order

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Rudolf Gorenflo, Francesco Mainardi
25 May 2008-arXiv: Mathematical Physics
TL;DR: In this article, the authors introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus, and derive the analytical solutions of the most simple linear integral and differential equations in fractional order.
Abstract: We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

1,281 citations

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Convex Functions, Partial Orderings, and Statistical Applications

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Josip Pečarić, Frank Proschan, Y. L. Tong
01 Jan 1992
TL;DR: In this paper, the authors present a survey of the applicability of Jensen's inequality to means and H*adolder's Inequalities, as well as a discussion of Reversals, Refinements, and Converses of Jensen and Steffensen's inequalities.
Abstract: Convex Functions. Jensen's and Jensen-Steffensen's Inequalities. Reversals, Refinements, and Converses of Jensen's and Jensen-Steffensen's Inequalities. Applications of Jensen's Inequality to Means and H*adolder's Inequalities. Hermite-Hadamard's and Jensen-Petrovi*aac's Inequalities. Popoviciu's, Burkill's, and Steffensen's Inequalities. *ajCeby*ajsev-Gr*aduss', Favard's, Berwald's, Gauss-Winckler's, and Related Inequalities. Hardy's, Hilbert's, Opial's, Young's, Nanson's, and Related Inequalities. General Linear Inequalities for Convex Sequences and Functions. Orderings and Convexity-Preserving Transformations. Convex Functions and Geometric Inequalities. Convexity, Majorization, and Schur-Convexity. Convexity and Log-Concavity Related Moment and Probability Inequalities. Muirhead's Theorem and Related Inequalities. Arrangement Ordering. Applications of Arrangement Ordering. Multivariate Arrangement Increasing Functions. References. Author Index. Subject Index.

977 citations

Related Papers (5)
Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula

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01 Sep 1998-Applied Mathematics Letters
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Theory and Applications of Fractional Differential Equations

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02 Mar 2006
Anatoly A. Kilbas, Hari M. Srivastava, Juan J. Trujillo
Selected Topics on Hermite-Hadamard Inequalities and Applications

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01 Mar 2003-Social Science Research Network
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An Introduction to the Fractional Calculus and Fractional Differential Equations

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