Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T15:11:06.879Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized monotonicity in terms of differential inequalities

Part of: General theory in ordinary differential equations Real functions

Published online by Cambridge University Press:  16 January 2019

Mihály Bessenyei
Mihály Bessenyei*
Affiliation:
Institute of Mathematics, University of Debrecen, H-4002 Debrecen, Pf. 400, Hungary (besse@science.unideb.hu)
Get access Rights & Permissions [Opens in a new window]

Abstract

The classical notions of monotonicity and convexity can be characterized via the nonnegativity of the first and the second derivative, respectively. These notions can be extended applying Chebyshev systems. The aim of this note is to characterize generalized monotonicity in terms of differential inequalities, yielding analogous results to the classical derivative tests. Applications in the fields of convexity and differential inequalities are also discussed.

Keywords

Chebyshev systemgeneralized monotonicitymean value theoremdifferential inequality

MSC classification

Primary: 34A40: Differential inequalities 26A51: Convexity, generalizations 39B62: Functional inequalities, including subadditivity, convexity, etc.
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1473 - 1479
Copyright
Copyright © Royal Society of Edinburgh 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Dedicated to the 80th birthday of professor Zoltán Daróczy

References

1 Beckenbach, E. F.. Generalized convex functions. Bull. Amer. Math. Soc. 43 (1937), 363–371.Google Scholar
2Bessenyei, M., Páles, Zs.. Hadamard-type inequalities for generalized convex functions. Math. Inequal. Appl. 6 (2003), 379–392.Google Scholar
3Bonsall, F. F.. The characterization of generalized convex functions. Quart. J. Math., Oxford Ser. (2) 1 (1950), 100–111.Google Scholar
4Hardy, G. H., Littlewood, J. E., and Pólya, G.. Inequalities, 2nd edn (Cambridge, Cambridge University Press, 1934 (first edition), 1952.Google Scholar
5Hopf, E.. Uber die Zusammenhänge zwischen gewissen höheren Differenzenquotienten reeller Funktionen einer reellen Variablen und deren Differenzierbarkeitseigenschaften, Dissertation, Friedrich–Wilhelms–Universität Berlin, 1926.Google Scholar
6Karlin, S., Studden, W. J.. Tchebycheff systems: with applications in analysis and statistics, Vol. XV Pure and Applied Mathematics (New York-London-Sydney: Interscience Publishers John Wiley & Sons, 1966).Google Scholar
7Páles, Zs.. A unified form of the classical mean value theorems, inequalities and applications, World Sci. Ser. Appl. Anal., 3 (1994), 493–500.Google Scholar
8Páles, Zs.. A general mean value theorem. Publ. Math. Debrecen, 89 (2016), 161–172.10.5486/PMD.2016.7443Google Scholar
9Peixoto, M. M.. Generalized convex functions and second order differential inequalities. Bull. Amer. Math. Soc. 55 (1949), 563–572.Google Scholar
10Popoviciu, T.. Les fonctions convexes (Paris: Hermann et Cie, 1944).Google Scholar
11Tornheim, L.. On n-parameter families of functions and associated convex functions. Trans. Amer. Math. Soc., 69 (1950), 457–467.Google Scholar