Let $x=\frac{\sum_{i=1}^nx_i}{n}$ and let the angles $x_i\in(0,\pi)$. Then, is the following inequality true? $$\prod_{i=1}^n\sin(x_i)\le\sin^nx$$
I think yes, by analogy with the AM-GM inequality. But, the application is not direct. Does convexity of the sine function in the interval have a role here? Does Jensen inequality work here? Thanks beforehand.
The sine function is concave on $(0, \pi)$, therefore $$ \sin x \ge \frac 1n \sum_{i=1}^n \sin x_i \, . $$ Using the AM-GM inequality, this expression is $$ \ge \left (\prod_{i=1}^n\sin(x_i)\right) ^{1/n} \, . $$
You can also apply Jensen's inequality to the (concave) function $\log \circ \sin$ to obtain the same result.