TITLE: Find bound for sum of square roots
QUESTION [14 upvotes]: Let $a_1,...,a_n$ be real numbers, such that $a_1+...+a_n=A$.
What can we say about $\sqrt{a_1}+...+\sqrt{a_n}$?
I would like to bound from above thus sum in terms of $A$.

REPLY [7 votes]: Since we are taking the square root of each $a_i$, we will assume that $a_i\ge0$.
Using Jensen's Inequality yields
$$
\left(\frac1n\sum_{i=1}^n\sqrt{a_i}\right)^2\le\frac1n\sum_{i=1}^n\left(\sqrt{a_i}\right)^2=\frac1n\sum_{i=1}^na_i\tag{1}
$$
Rearranging $(1)$ gives
$$
\sum_{i=1}^n\sqrt{a_i}\le\sqrt{n\sum_{i=1}^na_i}\tag{2}
$$