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,

and add to get –

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

which re-arranges to the desired question 🙂


Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: