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February  2021, 20(2): 903-914. doi: 10.3934/cpaa.2020296

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

Tomasz Szostok

Intitute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland

Received  July 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

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.

Keywords: Convex functions, Hermite-Hadamard inequalities, Ohlin lemma, Stieltjes integral.
Mathematics Subject Classification: Primary: 26A51, 26D10; Secondary: 39B62.
Citation: Tomasz Szostok. Inequalities of Hermite-Hadamard type for higher order convex functions, revisited. Communications on Pure and Applied Analysis, 2021, 20 (2) : 903-914. doi: 10.3934/cpaa.2020296
References:
[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. Denuit, C. 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.

show all references

References:
[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. Denuit, C. 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.

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