I am trying to calculate the $L^p$ norm of the function $f_\alpha(e^{i\theta}) = \frac{1}{1-\overline{\alpha }e^{i\theta}},$ where $\alpha$ is a complex number with $|\alpha| < 1.$ Id est, for every $1\leq p < \infty$ I would like to compute $$ \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1}{|1-\overline{\alpha } e^{i\theta}|^p} d\theta.$$ Observe that, if $p=2$, the function $f_\alpha$ is the evaluation functional on $\alpha$ on the Hardy space $H^2$, so it is easy to deduce that $$ \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1}{|1-\overline{\alpha } e^{i\theta}|^2} d\theta = \langle{f_\alpha,f_\alpha\rangle} = \frac{1}{1-|\alpha|^2}.$$ But, if we consider $1\leq p < \infty$ with $p\neq 2$ I don't know how to compute the integral directly, or maybe there is some kind of trick using the reproducing kernel property...
Any one can help me? Thank you very much!