Let $z\in \mathbb{D}, t\in S^1, \beta\in \mathbb{R}$. I was dealing with the following integral arising from some other calculation regarding harmonic extension on $\mathbb{D}$:
$I(z)=\int_{S^1}|t-z|^{\beta}|dt|= \int_0^{2\pi}\bigl({1+r^2-2r\cos(\theta-\phi)}\bigr)^{\beta/2}d\theta,$ $t=e^{i\theta}$, $z= re^{i\phi}$, $|dt| $ denote the arc length measure on $S^1$.
My question is: 1) can we, at best, evaluate this integral?
Or if not, 2) can we get $I(z)\leq K(1-|z|)^{1+\beta}$, $\beta \ne -1$, and $I(z)\leq -K\ln(1-|z|)$ if $\beta = -1$?