A bound on a sum of five sines

Question

If $0<x_{1},\ldots,x_{5}<\pi$ and $x_{1}+\ldots+x_{5}=\pi$, is it true that $\sin x_{1}+\ldots+\sin x_{5}\le5\sin\frac{\pi}{5}$?

Have you tried Lagrange's method? – Raskolnikov Apr 29 '13 at 07:32 — Raskolnikov, Apr 29 '13 at 07:32

score 14 · Answer 1 · answered Apr 29 '13 at 07:34

14

Sine is a concave function on $[0, \pi]$ and so this follows from Jensen's inequality (4):

$$ \frac{\sin x_1 + \dotsc + \sin x_5}{5} \leq \sin\left( \frac{x_1 + \dotsc + x_5}{5} \right) = \sin \frac{\pi}{5}. $$

answered Apr 29 '13 at 07:34

WimC

1 Answers1