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.
Highlights
For a sequence of numbers x x1, x2, xn we will let AM x1, x2, xn x
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
The following are some basic properties of the logarithmic means: 1) Logarithmic mean LM a,b can be thought of as the mean-value of the function f x ln x over the interval a,b
Summary
N n j 1 x j to denote the well known arithmetic, geometric, and harmonic means, called the Pythagorean means. The Pythagorean means have the obvious properties: 1) PM x1, x2 , , xn is independent of order. The harmonic mean satisfies the same relation with reciprocals, that is, it is a solution of the equation. The geometric mean of two numbers x1 and x2 can be visualized as the solution of the equation x1 GM GM x2
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