Prove the following sum inequality

Question

If $x_1,x_2,\ldots,x_n$ are positive real numbers such that $\displaystyle \sum_{i = 1}^n x_i = 1$, prove that $$\displaystyle \sum_{i = 1}^n \dfrac{x_i}{\sqrt{1-x_i}} \geq \dfrac{\displaystyle \sum_{i = 1}^n \sqrt{x_i}}{\sqrt{n-1}}.$$

Seeing the $\displaystyle \sum_{i = 1}^n \sqrt{x_i}$ makes me think of Cauchy-Schwarz. But it might get bad as we have a square root. Therefore using a substitution might work. We can say $y_i = x_1+x_2+\cdots+x_n$ and then $1-x_i = y_i-x_i$. I am not sure how to make this substitution work.

Answer 1

First, recall that by Jensen's inequality $$\sqrt {\frac{x_1+...+x_n}{n} } \ge \frac{\sqrt x_1+...+\sqrt x_n } { n }$$ that using $x_1+...+x_n=1$ reduced to $$\sqrt n\ge \sqrt x_1+...+\sqrt x_n \tag{1}. $$

To reach the desired inequality we can use Jensen with $f(x)=\frac{x}{\sqrt {1-x}}$:

$$ \frac{1}{n}\sum_i\frac{x_i}{\sqrt {1-x_i} } \ge \frac{\frac{\sum_i x_i}{n}}{\sqrt {1-\frac{\sum_i x_i}{n}}}=\frac{\frac{1}{n}}{\sqrt {1-\frac{1}{n}}} $$ then multiplying by $n$ on both sides $$ \sum_i\frac{x_i}{\sqrt {1-x_i} } \ge \frac{\sqrt n}{\sqrt{n-1}}. $$ At this point we use $(1)$ and the inequality follows.