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.

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2 comments

  1. Aravind Srinivas · · Reply

    Ram, this is the Hugen’s inequality. Also try for a proof by Jensen’s inequality. Darij posted that proof in http://www.mathlinks.ro

  2. cool…so does it mean i proved a theorem? .. 😀

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