# work – 31 –

Problem
Prove that –
$(a_1+a_2+\hdots +a_n)^n \le (n)^{n-1}(a_1^n +a_2^n +\hdots + a_n^n)$
P.S. It is straightforward holder, but what about a solution by Cauchy or AM-GM ?

Solution

By, AM-GM we have ,

$\frac{a_i^n}{\sum a_i^n} + (n-1)\cdot \frac{1}{n} \ge \frac{a_1\cdot \sqrt[n]{n}}{\sqrt[n]{\sum a_i^n}}$

Write up analogous inequalities,

$n \ge \sqrt[n]{n}\cdot \frac{\sum a_i}{\sqrt[n]{\sum a_i^n}}$

which re-arranges to the desired question 🙂