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has been cited by the following article:

• 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.