
-
Previous Article
The degenerate Monge-Ampère equations with the Neumann condition
- CPAA Home
- This Issue
-
Next Article
On the quotient quantum graph with respect to the regular representation
Inequalities of Hermite-Hadamard type for higher order convex functions, revisited
Intitute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland |
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.
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. 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. |
[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. 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.
|
[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. 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. |
[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. 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 |

[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
Tools
Metrics
Other articles
by authors
[Back to Top]