proof that sin (a + b + c)/3 ≥ 1/3 (sin a + sin b + sin c)

Question

click here for the question

I tried to prove it using the convexity

Answer 1

Let $$x=a+b+c$$ and consider $x$ as fixed, then we want to find the largest value of $$\sin a+\sin b+\sin c$$. This value occurs when $a=b=c$. That follows from the inequality

$$\sin a+\sin b=2\sin \frac{a+b}{2}\cos \frac{a-b}{2}\leq 2\sin\frac{a+b}{2}$$ which means that if two are unequal we can increase the value by replacing them with their means and keeping the sum constant. From this the inequality follows.