Abstract

For a given finite group G acting on an inner product space V, we introduce G-symmetrized convex functions and G-symmetrized increasing functions. We use them to establish some inequalities of Fejér and Hermite-Hadamard types. In particular, we apply Wright convexity. We interpret the obtained results for the group G = Cn\mathbb{C}_n of sign changes on V = Rn\mathbb{R}^n. The case n = 1 leads to some recent results by S. S. Dragomir [Symmetrized convexity and Hermite-Hadamard type inequalities, J. Math. Inequalities 10/4 (2016) 901–918] and S. Abramovich and L.-E. Persson [Some new Hermite-Hadamard and Fejer type inequalities without convexity/concavity, Math. Inequalities Appl. 23/2 (2020) 447–458].

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Fejer inequality
  • Hermite-Hadamard inequality
  • G-majorization
  • G-increaasing function
  • convex function
  • Wright convex function
  • G-symmetrized convex function
  • G-symmetrized increasing monotone function

Mathematics Subject Classification

52A41, 26D15; 26B25, 52A40

References

  1. S. Abramovich, J. Barić, J. Pečarić: Fejér and Hermite-Hadamard type inequalities for superquadratic functions, J. Math. Analysis Appl. 344 (2008) 1048–1056.https://doi.org/10.1016/j.jmaa.2008.03.051Google Scholar
  2. S. Abramovich, L.-E. Persson: Some new Hermite-Hadamard and Fejer type inequalities without convexity/concavity, Math. Inequalities Appl. 23/2 (2020) 447–458.https://doi.org/10.7153/mia-2020-23-36Google Scholar
  3. W. W. Breckner, T. Trif: Convex Functions and Related Functional Equations: Selected Topics, Cluj University Press, Cluj (2008).Google Scholar
  4. S. S. Dragomir: Symmetrized convexity and Hermite-Hadamard type inequalities, J. Math. Inequalities 10/4 (2016) 901–918.https://doi.org/10.7153/jmi-10-74Google Scholar
  5. M. L. Eaton, M. D. Perlman: Reflection groups, generalized Schur functions, and the geometry of majorization, Ann. Probability 5 (1977) 829–860.https://doi.org/10.1214/aop/1176995655Google Scholar
  6. A. Giovagnoli, H. P. Wynn: G-majorization with applications to matrix orderings, Linear Algebra Appl. 67 (1985) 111–135.https://doi.org/10.1016/0024-3795(85)90190-9Google Scholar
  7. G. M. Hardy, J. E. Littlewood, G. Pólya: Inequalities, 2nd ed., Cambridge University Press, Cambridge (1952).Google Scholar
  8. J. E. Klaričić-Bakula, J. E. Pečarić, J. Perić: Extensions of the Hermite-Hadamard inequality with applications, Math. Inequalities Appl. 12 (2012) 899–921.https://doi.org/10.7153/mia-15-77Google Scholar
  9. A. W. Marshall, I. Olkin, B. C. Arnold: Inequalities: Theory of Majorization and Its Applications, 2nd ed., Springer, New York (2011).https://doi.org/10.1007/978-0-387-68276-1Google Scholar
  10. M. Minculete, F.-C. Mitroi: Fejér-type inequalities, Austral. J. Math. Analysis Appl. 9/1 (2012) 1–8.Google Scholar
  11. E. Neuman: Inequalities involving multivariate convex functions II, Proc. Amer. Math. Soc. 109 (1990) 965–974.https://doi.org/10.1090/S0002-9939-1990-1009996-4Google Scholar
  12. C. T. Ng: Functions generating Schur-convex sums, Int. Ser. Num. Math. 80 (1985) 433–438.Google Scholar
  13. C. P. Niculescu, L.-E. Persson: Old and new on the Hermite-Hadamard inequality, Real Analysis Exchange 294/2 (2003/2004) 663–685.https://doi.org/10.14321/realanalexch.29.2.0663Google Scholar
  14. C. P. Niculescu, L.-E. Persson: Convex Functions and their Applications. A Contemporary Approach, Springer, New York (2006).https://doi.org/10.1007/0-387-31077-0_2Google Scholar
  15. M. Niezgoda: Group majorization and Schur type inequalities, Linear Algebra Appl. 268 (1998) 9–30.https://doi.org/10.1016/S0024-3795(97)89322-6Google Scholar
  16. M. Niezgoda: Fejér and Hermite-Hadamard like results for H-invex functions with applications, Positivity 23 (2019) 531–543.https://doi.org/10.1007/s11117-018-0623-0Google Scholar
  17. M. Niezgoda: Majorization and refinements of Hermite-Hadamard inequality, Math. Inequalities Appl. 23 (2020) 137–148.https://doi.org/10.7153/mia-2020-23-11Google Scholar
  18. A. Olbryś: On some inequalities equivalent to the Wright convexity, J. Math. Inequalities 9 (2015) 449–461.https://doi.org/10.7153/jmi-09-38Google Scholar
  19. A. G. M. Steerneman: G-majorization, group-induced cone orderings and reflection groups, Linear Algebra Appl. 127 (1990) 107–119.https://doi.org/10.1016/0024-3795(90)90338-DGoogle Scholar

Suggested citation

M. Niezgoda. “G-Majorization and Fejér and Hermite-Hadamard Like Inequalities for G-Symmetrized Convex Functions.” Journal of Convex Analysis 29 (2022), No. 1, 231–242. https://doi.org/10.68381/jca29014

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.