Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Densely algebraic bounds for the exponential function

Author: Seon-Hong Kim
Journal: Proc. Amer. Math. Soc. 135 (2007), 237-241
MSC (2000): Primary 33B10; Secondary 11A99
Published electronically: June 30, 2006
MathSciNet review: 2280192
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An upper bound for $ e^{x}$ that implies the inequality between the arithmetic and geometric means is generalized with the introduction of a new parameter $ n$. The new upper bound is smoothly and densely algebraic in $ n$, and valid for $ -b<x<1$ for arbitrarily large positive $ b$ provided that $ n$ ($ >1$) is sufficiently close to $ 1$. The range of its validity for negative $ x$ is investigated through the study of a certain family of quadrinomials.

References [Enhancements On Off] (What's this?)

  • [1] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge, 1975. MR 0944909 (89d:26016)
  • [2] J. Karamata, Sur l’approximation de 𝑒^{𝑥} par des fonctions rationnelles, Bull. Soc. Math. Phys. Serbie 1 (1949), 7–19 (Serbian, with Russian and French summaries). MR 0031124
  • [3] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR 0274686
  • [4] W. E. Sewell, Some inequalities connected with exponential function (in Spanish), Rev. Ci (Lima) 40 (1938), 453-456.
  • [5] J. E. Wetzel, On the functional inequality 𝑓(𝑥+𝑦)≥𝑓(𝑥)𝑓(𝑦), Amer. Math. Monthly 74 (1967), 1065–1068. MR 0228865,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33B10, 11A99

Retrieve articles in all journals with MSC (2000): 33B10, 11A99

Additional Information

Seon-Hong Kim
Affiliation: Department of Mathematics, College of Natural Science, Chosun University, 375 Susuk-dong, Dong-gu, Gwangju, 501-759 Korea

Keywords: Algebraic bounds, exponential function, polynomials
Received by editor(s): January 15, 2005
Received by editor(s) in revised form: July 5, 2005, and August 5, 2005
Published electronically: June 30, 2006
Additional Notes: This study was supported (in part) by research funds from Chosun University, 2004
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2006 American Mathematical Society