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

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

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

Received  July 2020 Revised  October 2020 Published  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.

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. 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.  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. 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.  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 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]

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

[2]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[3]

Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695

[4]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153

[5]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190

[6]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136

[7]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[8]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[9]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009

[10]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[11]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

[12]

Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021015

[13]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (51)
  • HTML views (58)
  • Cited by (0)

Other articles
by authors

[Back to Top]