A Simple Inequality with Many Variables

Problem

A Simple Inequality with Many Variables

Solution

From the AM-GM inequality, $x+y\ge 2\sqrt{xy},$ and its general form

$\displaystyle\begin{align} \sum_{k=1}^n\sqrt{\frac{a_k+a_{k+1}}{a_{k+2}}}&\ge \sum_{k=1}^n\sqrt{\frac{2\sqrt{a_k\cdot a_{k+1}}}{a_{k+2}}}\\ &=\sqrt{2}\sum_{k=1}^n\sqrt[4]{\frac{a_k\cdot a_{k+1}}{a_{k+2}^2}}\\ &\ge\sqrt[2]\cdot n\cdot\sqrt[4n]{\prod_{k=1}^n\frac{a_{k}\cdot a_{k+1}}{a_{k+2}^2}}=n\sqrt{2}. \end{align}$

Generalization

Prove that for positive $a_1,$ $a_2,\ldots,a_n,$ $n\ge m+1,$ $m\ge 2,$

$\displaystyle \sum_{k=1}^n\sqrt[m]{ \frac{\displaystyle \sum_{i=0}^{m-1}a_{k+i}}{a_{k+m}}}\ge n\sqrt{m},$

where the indices are taken modulo $n.$

Acknowledgment

This is problem 1277 from the Kvant magazine (1991, n 4). The problem is by L. Kurlyandchik. The above solution is by N. N. Taleb who also noticed that equality is attained when all the variables are equal.

 

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