Prove for all positive integer $n$ and for all $\alpha_1,\alpha_2,...\alpha_n \in \mathbb{R}$ Prove that $\int_{0}^{\pi} \sqrt[n]{\prod_{k=1}^{n}\csc^2(x-\alpha_k)}dx\geq4π$
I have tried by applying GM-HM inequality We know the inequility GM$\geq $ HM
Appliying this on $\csc^2(x-\alpha_1), \csc^2(x-\alpha_2), \cdots \csc^2(x-\alpha_n)$ We will get
$$\sqrt[n]{\prod_{k=1}^{n}\csc^2(x-\alpha_k)}\geq \frac{n}{\sum_{k=1}^{n}\sin^2 (x-\alpha_k)}$$ I am stuck after that. Please help me to proceed further