American Institute of Mathematical Sciences

Advanced
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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer, New York, 2006. doi: 10.1007/0-387-31077-0.  Google Scholar

[5]

J. Ohlin, On a class of measures of dispersion with application to optimal reinsurance, Astin Bull., 5 (1969), 249-266.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, NY, 2007. doi: 10.1007/978-0-387-34675-5.  Google Scholar

[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.  Google Scholar

[11]

T. Szostok, Levin-Stechkin theorem and inequalities of the Hermite-Hadamard type, arXiv: 1411.7708. Google Scholar

[12]

E. W. Weisstein, Legendre-Gauss quadrature, MathWorld, 2015. Google Scholar

[13]

E. W. Weisstein, Lobatto quadrature, MathWorld. Google Scholar

[14]

E. W. Weisstein, Radau quadrature, MathWorld. Google Scholar

show all references

References:
[1]

M. Bessenyei and Zs. Páles, Higher-order generalizations of Hadamard's inequality, Publicationes Math. (Debrecen), 61 (2002), 623-643.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer, New York, 2006. doi: 10.1007/0-387-31077-0.  Google Scholar

[5]

J. Ohlin, On a class of measures of dispersion with application to optimal reinsurance, Astin Bull., 5 (1969), 249-266.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, NY, 2007. doi: 10.1007/978-0-387-34675-5.  Google Scholar

[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.  Google Scholar

[11]

T. Szostok, Levin-Stechkin theorem and inequalities of the Hermite-Hadamard type, arXiv: 1411.7708. Google Scholar

[12]

E. W. Weisstein, Legendre-Gauss quadrature, MathWorld, 2015. Google Scholar

[13]

E. W. Weisstein, Lobatto quadrature, MathWorld. Google Scholar

[14]

E. W. Weisstein, Radau quadrature, MathWorld. Google Scholar

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

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ruonan Liu, Run Xu. Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions. Mathematical Foundations of Computing, 2021, 4 (2) : 89-103. doi: 10.3934/mfc.2021005
[2]

S. S. Dragomir, I. Gomm. Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 271-278. doi: 10.3934/naco.2012.2.271
[3]

Shu-Lin Lyu. On the Hermite--Hadamard inequality for convex functions of two variables. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 1-8. doi: 10.3934/naco.2014.4.1
[4]

Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175
[5]

Muhammad Aslam Noor, Khalida Inayat Noor. General biconvex functions and bivariational inequalities. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021041
[6]

Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671
[7]

Leandro M. Del Pezzo, Nicolás Frevenza, Julio D. Rossi. Convex and quasiconvex functions in metric graphs. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021019
[8]

Jagannathan Gomatam, Isobel McFarlane. Generalisation of the Mandelbrot set to integral functions of quaternions. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 107-116. doi: 10.3934/dcds.1999.5.107
[9]

Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004
[10]

Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070
[11]

Khalida Inayat Noor, Muhammad Aslam Noor. Higher order uniformly close-to-convex functions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1277-1290. doi: 10.3934/dcdss.2015.8.1277
[12]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021019
[13]

Kunquan Lan, Wei Lin. Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1943-1960. doi: 10.3934/dcdsb.2018256
[14]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164
[15]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067
[16]

Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom. More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2119-2135. doi: 10.3934/dcdss.2021063
[17]

Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012
[18]

Pablo L. De Nápoli, Irene Drelichman, Ricardo G. Durán. Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1629-1642. doi: 10.3934/cpaa.2012.11.1629
[19]

Bernhard Kawohl. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683
[20]

Minghua Li, Chunrong Chen, Shengjie Li. Error bounds of regularized gap functions for nonmonotone Ky Fan inequalities. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1261-1272. doi: 10.3934/jimo.2019001

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (136)
  • HTML views (79)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences