Applications of Weighted Arithmetic-Geometric Means Inequality to Functional Inequalities

Year 2017, Volume: 2 Issue: 1, 27 - 31, 30.04.2017

Halil İbrahim Çelik

https://doi.org/10.30931/jetas.303624

Abstract

In this paper we  prove  the functional  inequality $f(x)^{f(x)}\leq g(x)^{g(x)}$ for positive real functions $f$ and $g$ satisfying natural conditions and apply it to derive
inequalities  between some of the elementary functions and to prove monotonocity of certain sequences of real numbers.

Keywords

Arithmetic-geometric means inequality , Young inequality , Extremum values , Functional inequalities , Elementary functions , Monotone sequences

References

• [1] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd. ed., Cambridge, New York 1959.
• [2] N. D. Kazarinoff, Analytic Inequalities, Holt, Rinehart and Winston, New York, 1961.
• [3] I. J. Maddox, Elements of Functional Analysis, Second ed., Cambridge Univ. Press, 1988.
• [4] Thomas.J. Mildford, Olympiad Inequalites, 2006, http://www.unl.edu/amc.
• [5] W. Rudin, Real and Complex Analysis , McGraw-Hill Book Company, New York, 1987.