Abstract

This note presents a complete solution of the support problem for functions that are generalized monotone in the sense of Beckenbach. The key tool of the proof is Tornheim's uniform convergence theorem. As applications, we improve some known support results and give an abstract version of the Hermite-Hadamard inequality.

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Keywords

  • Beckenbach family
  • Chebyshev system
  • generalized monotonicity and convexity
  • quadrature formula
  • Hermite-Hadamard inequality

Mathematics Subject Classification

26A51; 26D15, 39B62, 41A05, 41A55

References

  1. E. F. Beckenbach: Generalized convex functions, Bull. Amer. Math. Soc. 43 (1937) 363–371.https://doi.org/10.1090/S0002-9904-1937-06549-9Google Scholar
  2. M. Bessenyei: Hermite-Hadamard-type inequalities for generalized convex functions, Ph. D. Dissertation (2005), J. Inequal. Pure Appl. Math. 9/3 (2008), art. no. 63.Google Scholar
  3. M. Bessenyei: The Hermite-Hadamard inequality in Beckenbach’s setting, J. Math. Analysis Appl. 364 (2010) 366–383.https://doi.org/10.1016/j.jmaa.2009.11.015Google Scholar
  4. J. Hadamard: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann, J. Math. Pures Appl. 58 (1893) 171–215.Google Scholar
  5. Ch. Hermite: Sur deux limites d’une integrale define, Mathesis 3 (1883) 82.Google Scholar
  6. E. Hopf: Über die Zusammenhänge zwischen gewissen höheren Differenzenquotienten reeller Funktionen einer reellen Variablen und deren Differenzierbarkeitseigenschaften, Dissertation, Friedrich-Wilhelms-Universität, Berlin (1926).Google Scholar
  7. S. Karlin, W. J. Studden: Tchebycheff systems: With Applications in Analysis and Statistics, Pure and Applied Mathematics Vol. XV, John Wiley & Sons, New York (1966).Google Scholar
  8. M. Kuczma: An Introduction to the Theory of Functional Equations and Inequalities, Państwowe Wydawnictwo Naukowe, Uniwersytet Śląski, Warszawa-Kraków-Katowice (1985).Google Scholar
  9. D. S. Mitrinović, I. B. Lacković: Hermite and convexity, Aequationes Math. 28 (1985) 229–232.https://doi.org/10.1007/BF02189414Google Scholar
  10. C. P. Niculescu, L.-E. Persson: Old and new on the Hermite-Hadamard inequality, Real Analysis Exchange 29 (2003/2004) 619–623.https://doi.org/10.7153/mia-05-62Google Scholar
  11. T. Popoviciu: Les Fonctions Convexes, Hermann, Paris (1944).Google Scholar
  12. A. W. Roberts, D. E. Varberg: Convex Functions, Pure and Applied Mathematics Vol. 57, Academic Press, New York (1973).Google Scholar
  13. G. Szegő: Orthogonal Polynomials, American Mathematical Society, Providence (1939).https://doi.org/10.1090/coll/023Google Scholar
  14. L. Tornheim: On n-parameter families of functions and associated convex functions, Trans. Amer. Math. Soc. 69 (1950) 457–467.https://doi.org/10.1090/S0002-9947-1950-0038383-XGoogle Scholar
  15. Sz. Wąsowicz: Support-type properties of convex functions of higher order and Hadamard-type inequalities, J. Math. Analysis Appl. 332/2 (2007) 1229–1241.https://doi.org/10.1016/j.jmaa.2006.11.011Google Scholar
  16. Sz. Wąsowicz: Support-type properties of generalized convex functions, J. Math. Analysis Appl. 365/1 (2010) 415–427.https://doi.org/10.1016/j.jmaa.2009.10.070Google Scholar

Suggested citation

M. Bessenyei, E. Pénzes. “Support Theorems for Generalized Monotone Functions.” Journal of Convex Analysis 30 (2023), No. 1, 359–369. https://doi.org/10.68381/jca30020

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.