Generalization and reverses of the left Fejer inequality for convex functions

Volume 10, Issue 6, pp 3231--3244 http://dx.doi.org/10.22436/jnsa.010.06.35

Publication Date: June 25, 2017 Submission Date: April 18, 2017

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Authors

S. S. Dragomir - Mathematics, School of Engineering & Science, Victoria University, P. O. Box 14428, Melbourne City, MC 8001, Australia. - chool of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa.

Abstract

In this paper we establish a generalization of the left Fej´er inequality for general Lebesgue integral on measurable spaces as well as various upper bounds for the difference \[\frac{1}{\int^b_a g(x)dx} \int^b_ah(x)g(x)dx-h\left(\frac{a+b}{2}\right),\] where \(h : [a, b] \rightarrow \mathbb{R}\) is a convex function and \(g : [a, b] \rightarrow [0,\infty)\) is an integrable weight. Applications for discrete means and Hermite-Hadamard type inequalities are also provided.

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Keywords

• Convex functions
• integral inequalities
• Jensen’s type inequalities
• Fejér type inequalities
• Lebesgue integral
• Hermite-Hadamard type inequalities
• special means.

MSC

•  26D15
•  26D20

References

• [1] A. G. Azpeitia, Convex functions and the Hadamard inequality, Rev. Colombiana Mat., 28 (1994), 7–12.
  • Google Scholar


• [2] S. S. Dragomir, C. E. M. Pearce, Selected topics on HermiteHadamard inequalities and applications, RGMIA Monographs, Victoria University, Melbourne City, Australia (2000)
  • Google Scholar


• [3] L. Fejér, Über die Fourierreihen, II, (Hungarian) Math. Natur. Ungar. Akad Wiss., 24 (1906), 369–390.
  • Google Scholar


• [4] A. Guessab, G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory, 115 (2002), 260–288.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] E. Kikianty, S. S. Dragomir, Hermite-Hadamard’s inequality and the p-HH-norm on the Cartesian product of two copies of a normed space, Math. Inequal. Appl., 13 (2010), 1–32.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] M. Merkle, Remarks on Ostrowski’s and Hadamard’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 10 (1999), 113–117.
  • MathSciNet
  • Google Scholar
  • MATH


• [7] C. E. M. Pearce, A. M. Rubinov, P-functions, quasi-convex functions, and Hadamard-type inequalities, J. Math. Anal. Appl., 240 (1999), 92–104.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] J. Pečarić, A. Vukelić , Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions, Functional equations, inequalities and applications, Kluwer Acad. Publ., Dordrecht, (2003), 105–137.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] G. Toader, Superadditivity and Hermite-Hadamard’s inequalities, Studia Univ. Babe-Bolyai Math., 39 (1994), 27–32.
  • MathSciNet
  • Google Scholar
  • MATH


• [10] G.-S. Yang, M.-C. Hong, A note on Hadamard’s inequality, Tamkang J. Math., 28 (1997), 33–37.
  • MathSciNet
  • Google Scholar


• [11] G.-S. Yang, K.-L. Tseng, On certain integral inequalities related to Hermite-Hadamard inequalities, J. Math. Anal. Appl., 239 (1999), 180–187.
  • View Article
  • MathSciNet
  • Google Scholar