Power Mean Inequality for Math Olympiad

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

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}}} $

This problem is from Regional Math Olympiad, India.

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

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.