0
$\begingroup$

Let $x_1,\dots, x_n\in \mathbb{R}_{++}$ such that $\sum_{i=1}^{n}{x_i}=1$. Prove that $$ \sum_{i=1}^{n}{\frac{x_i}{\sqrt{1-x_i}}} \geq \sqrt{\frac{n}{n-1}} $$

I tried using AM-GM and Cauchy-Schwarz but didn't come to anything useful. Hint could be an help too.

Please advise.

$\endgroup$
6
  • 1
    $\begingroup$ Think convex $\endgroup$
    – Martin R
    May 5 at 14:12
  • 1
    $\begingroup$ Did not try yet, but do you know Jensen inequality? $\endgroup$
    – Thomas
    May 5 at 14:14
  • 1
    $\begingroup$ Apparently several solutions on AoPS: approach0.xyz/search/… $\endgroup$
    – Martin R
    May 5 at 14:16
  • 1
    $\begingroup$ There is one solution on this site here as well... $\endgroup$
    – Momo
    May 5 at 14:16
  • 1
    $\begingroup$ Another one: math.stackexchange.com/q/1600259/42969 $\endgroup$
    – Martin R
    May 5 at 14:18
1
$\begingroup$

The function $f(x)=x/\sqrt{1-x}$ is convex on $(-1,1)$, so by Jensen's inequality, one has \begin{align*} \sum\dfrac{1}{n}\dfrac{x_{i}}{\sqrt{1-x_{i}}}\geq \dfrac{\displaystyle\sum\dfrac{1}{n}x_{i}}{\sqrt{1-\displaystyle\sum\dfrac{1}{n}x_{i}}}. \end{align*}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.