2015-10 Product of sine functions

Let $$w_1,w_2,\ldots,w_n$$ be positive real numbers such that $$\sum_{i=1}^n w_i=1$$. Prove that if $$x_1,x_2,\ldots,x_n\in [0,\pi]$$, then $\sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.$

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