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.
 B. C. Carlson and M. D. Tobey, “A Property of the Hypergeometric Mean Value,” Proceedings of the American Mathematical Society, Vol. 19, No. 2, 1968, pp. 255-262. doi:10.1090/S0002-9939-1968-0222349-X