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February 7, 2020

Power Mean Inequality for Math Olympiad

What is Power Mean in Inequality?


The simplest example of power mean inequality is the arithmetic mean - geometric mean inequality. It says the following:

Arithmetic Mean is greater than or equal to Geometric Mean

Caution: All numbers must be non-negative.

Suppose $ a_1, a_2, ... , a_n $ be non-negative numbers. Then the two means are defined as follows:

Arithmetic Mean: $ \displaystyle{\frac{a_1 + a_2 + ... + a_n}{n}} $

Geometric Mean: $ \displaystyle{(a_1 \cdot a_2 \cdots a_n)^{\frac{1}{n}}} $

Try a problem


This problem is from Regional Math Olympiad, India.

Suppose a, b, c, d are positive numbers. Then show that $$ \displaystyle { \frac{a}{b} + \frac {b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4 } $$

Regional Math Olympiad, India

Inequality (AM-GM)

6 out of 10

Secrets in Inequalities.

Use some hints


Notice that product of the fractions is 1. Can you use this fact to compute the geometric mean of the fractions?

The geometric mean of the fractions is $$ \displaystyle{(\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{a} )^{\frac{1}{4} }}$$

This is equal to $ 1^{\frac{1}{4}} = 1$

Hence the geometric mean of the fractions is 1!

Can you now finish the problem using Arithmetic Mean - Geometric Mean inequality?

Lets use the arithmetic mean - geometric mean inequality on the fractions.

$$ \displaystyle { \frac{\frac{a}{b} + \frac {b}{c} + \frac{c}{d} + \frac{d}{a}}{4} \\ \geq (\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{a} )^{\frac{1}{4} } } $$

But the geometric mean is 1 (right hand side is 1). Hence by cross multiplying we have the final result.

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