TITLE: relationship between sum of squares and sum
QUESTION [9 upvotes]: I have to admit I am not good at math since it's been a while since I did the last math problem. I am working on a project where there is a problem that can be summarized like this:
if $\sum_{i=1}^{n}a_i^2 = x$, can we determine a upper(and/or lower) bound for $\sum_{i=1}^{n}|a_i|$ ?
Sorry if this is too simple for you, I appreciate your help!
PS: Thank you for all the helps I got! I just realize that I need to know the upper bounds for both $\sum_{i=1}^{n}|a_i|$ and the individual $|a_i|$. So what is the finest upper bound for $|a_i|$ I can get? Thank you again.

REPLY [9 votes]: You can use the AQM inequality:
$$\frac{\displaystyle\sum\nolimits_{i=1}^n\lvert a_i\rvert}n \le \sqrt{\frac{\displaystyle\sum\nolimits_{i=1}^n a_i^2}n}=\sqrt{\frac xn}, \enspace\text{whence}\enspace \displaystyle\sum_{i=1}^n\lvert a_i\rvert\le\sqrt{nx\mathstrut}.$$