I need to prove the following: $$\int_{S^1}|x-y|^{-2}dS_y \leq c(1-|x|)^{-1}$$ where $x\in B_1(0)$ and $y \in S^1$.
I tried a fwe things that came to my mind, but I just cannot get there. For example, I cannot get anywhere with the triangle inequality. My ansatz was to seperate the problem in 2 cases:
Case 1: $ |x| \leq \frac{1}{2}$
Case 2: $\frac{1}{2} < |x| < 1$.
I tried the triangle inequality in case 1 for $\frac{1}{|x-y|^2}$, but then I get $\frac{1}{|x-y|^2} \geq \frac{1}{(|x|+|y|)^2} $ which is the wrong way. Am I missing something or does my ansatz not work?