# work 013 – Huygen’s Inequality

Problem

Prove that for non-negative reals $x_1,x_2,.....,x_n$ ,

$(1+x_1)(1+x_2)....(1+x_n) \ge (1+(x_1x_2x_3...x_n)^{\frac{1}{n}})^{n}$

Proof

By AM-GM we have ,

$\sum \frac{x_1}{1+x_1} \ge n(\frac{x_1x_2x_3...x_n}{(1+x_1)(1+x_2)....(1+x_n)})^{\frac{1}{n}}$

and also ,

$\sum \frac{1}{1+x_1} \ge n[\frac{1}{(1+x_1)(1+x_2)...(1+x_n)}]^{\frac{1}{n}}$.

Adding the two inequalities we get the desired.

Equality occurs when all the variables are equal.