TITLE:
An Elementary Proof of the Mean Inequalities
AUTHORS:
Ilhan M. Izmirli
KEYWORDS:
Pythagorean Means; Arithmetic Mean; Geometric Mean; Harmonic Mean; Identric Mean; Logarithmic Mean
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.3 No.3,
May
13,
2013
ABSTRACT: In this paper we will extend the well-known chain of inequalities involving the Pythagorean means, namely the harmonic, geometric, and arithmetic means to the more refined chain of inequalities by including the logarithmic and identric means using nothing more than basic calculus. Of course, these results are all well-known and several proofs of them and their generalizations have been given. See [1-6] for more information. Our goal here is to present a unified approach and give the proofs as corollaries of one basic theorem.