Annals of Functional Analysis

On Minkowski and Hermite-Hadamard integral inequalities via fractional integration

Zoubir Dahmani

Full-text: Open access

Abstract

In this paper, we use the the Riemann-Liouville fractional integral to develop some new results related to the Hermite-Hadamard inequality. Other integral inequalities related to the Minkowsky inequality are also established. Our results have some relationships with [E. Set, M. E. Ozdemir and S.S. Dragomir, J. Inequal. Appl. 2010, Art. ID 148102, 9 pp.] and [L. BougoffaJ. qualPure and ApplMath 7 (2006), no. 2, Article 60, 3 pp.]. Some interested inequalities of these references can be deduced as some special cases.

Article information

Source
Ann. Funct. Anal. Volume 1, Number 1 (2010), 51- 58.

Dates
First available in Project Euclid: 12 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1399900993

Digital Object Identifier
doi:10.15352/afa/1399900993

Mathematical Reviews number (MathSciNet)
MR2755459

Zentralblatt MATH identifier
1205.26031

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 26A33: Fractional derivatives and integrals

Keywords
Concave function Hermite-Hadamard inequality Minkowski inequality Riemann-Liouville fractional integral

Citation

Dahmani, Zoubir. On Minkowski and Hermite-Hadamard integral inequalities via fractional integration. Ann. Funct. Anal. 1 (2010), no. 1, 51-- 58. doi:10.15352/afa/1399900993. https://projecteuclid.org/euclid.afa/1399900993


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References

  • A. El Farissi, Z. Latreuch and B. Belaidi, Hadamard-type inequalities for twice differentiable functions, RGMIA 12 (2009), no. 1, 7 pp.
  • S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math. 10 (2009), no. 3, Article 86, 5 pp.
  • L. Bougoffa, On Minkowski and Hardy integral inequality, J. Inequal. Pure and Appl. Math. 7 (2006), no. 2, Article 60, 3 pp.
  • S.S. Dragomir, C.E.M. Pearse, Selected Topic in Hermite–Hadamard Inequalities, Monographs:http://rgmia.vu.edu.au/monographs/hermite hadamard.html, Victoria University, 2000.
  • A. Florea and C.P. Niculescu, A Hermite–Hadamard inequality for convex-concave symmetric functions, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 50(98) (2007), no. 2, 149–156.
  • R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics (Udine, 1996), 223–276, CISM Courses and Lectures, 378, Springer, Vienna, 1997.
  • J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d'une fonction considree par Riemann, J. Math. Pures et Appl. 58 (1893), 171–215.
  • Ch. Hermite, Sur deux limites d'une integrale definie, Mathesis 3 (1883), 82.
  • A.W. Marshall and I. Olkin, Inequalities: Theory of Majoration and Applications, Academic Press, 1979.
  • S. Marinkovic, P. Rajkovic and M. Stankovic, The inequalities for some types q-integrals, Comput. Math. Appl. 56 (2008), 2490–2498.
  • C.P. Niculescu and L.E. Persson, Convex functions and their applications, A comtemporary approach, CMS Books in Mathematics, vol. 23, Springer Verlag, New York, 2006.
  • E. Set, M. E. Ozdemir and S.S. Dragomir, On the Hermite–Hadamard Inequality and Other Integral Inequalities Involving Two Functions, J. Inequal. Appl. 2010, Art. ID 148102, 9 pp.