TITLE: A bound on a sum of five sines
QUESTION [6 upvotes]: If $0<x_{1},\ldots,x_{5}<\pi$ and $x_{1}+\ldots+x_{5}=\pi$,
is it true that $\sin x_{1}+\ldots+\sin x_{5}\le5\sin\frac{\pi}{5}$?

REPLY [14 votes]: Sine is a concave function on $[0, \pi]$ and so this follows from Jensen's inequality (4):
$$
\frac{\sin x_1 + \dotsc + \sin x_5}{5} \leq \sin\left( \frac{x_1 + \dotsc + x_5}{5} \right) = \sin \frac{\pi}{5}.
$$