Advanced Search
Article Contents
Article Contents

Inequalities of Hermite-Hadamard type for higher order convex functions, revisited

Abstract Full Text(HTML) Figure(2) Related Papers Cited by
  • In this paper we present a very short proof of inequalities of Hermite-Hadamard type obtained by M. Bessenyei and Zs. Páles. This proof is based on the recently developed method connected with use of stochastic orderings of random variables. In the second part of the paper we present a way to extend these known inequalities. Namely, we describe completely the possible inequalities of Hermite-Hadamard type for longer expression than it was the case in the results of Bessenyei and Páles.

    Mathematics Subject Classification: Primary: 26A51, 26D10; Secondary: 39B62.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The graphs of functions $ F^{[1]} $ and $ G^{[1]} $ in the case $ 2\alpha_1\geq x_2 $

    Figure 2.  The graphs of functions $ F^{[1]} $ and $ G^{[1]} $ (with two crossing points in the interval $ (x_2,x_3) $) in the case $ 2\alpha_1<x_2 $

  • [1] M. Bessenyei and Zs. Páles, Higher-order generalizations of Hadamard's inequality, Publicationes Math. (Debrecen), 61 (2002), 623-643. 
    [2] M. Bessenyei and Zs. Páles, Characterization of higher order monotonicity via integral inequalities, P. Roy. Soc. Edinb. A, 140 (2010), 723-736.  doi: 10.1017/S0308210509001188.
    [3] M. DenuitC. Lefevre and M. Shaked, The s-convex orders among real random variables, with applications, Math. Inequal. Appl., 1 (1998), 585-613.  doi: 10.7153/mia-01-56.
    [4] C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer, New York, 2006. doi: 10.1007/0-387-31077-0.
    [5] J. Ohlin, On a class of measures of dispersion with application to optimal reinsurance, Astin Bull., 5 (1969), 249-266. 
    [6] A. Olbryś and T. Szostok, Inequalities of the Hermite-Hadamard type involving numerical differentiation formulas, Results Math., 67 (2015), 403-416.  doi: 10.1007/s00025-015-0451-5.
    [7] T. Rajba, On The Ohlin lemma for Hermite-Hadamard-Fejer type inequalities, Math. Inequal. Appl. 17, (2014), 557–571. doi: 10.7153/mia-17-42.
    [8] T. Rajba, On a generalization of a theorem of Levin and Stečkin and inequalities of the Hermite-Hadamard type, Math. Inequal. Appl., 20, (2017), 363–375. doi: 10.7153/mia-20-25.
    [9] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, NY, 2007. doi: 10.1007/978-0-387-34675-5.
    [10] T. Szostok, Ohlin's lemma and some inequalities of the Hermite-Hadamard type, Aequationes Math., 89 (2015), 915-926.  doi: 10.1007/s00010-014-0286-2.
    [11] T. Szostok, Levin-Stechkin theorem and inequalities of the Hermite-Hadamard type, arXiv: 1411.7708.
    [12] E. W. Weisstein, Legendre-Gauss quadrature, MathWorld, 2015.
    [13] E. W. Weisstein, Lobatto quadrature, MathWorld.
    [14] E. W. Weisstein, Radau quadrature, MathWorld.
  • 加载中



Article Metrics

HTML views(433) PDF downloads(241) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint