Sharp Estimates for Integral Means for Three Classes of Domains

In this paper, the following sharp estimate is proved:
$$ \int_0^{2\pi}|F'(e^{i\theta})|^p\,d\theta \le\sqrt\pi2^{1+p}\frac{\Gamma(1/2+p/2)}{\Gamma(1+p/2)}, \qquad p>-1, $$
where $F$ is the conformal mapping of the domain $D^-=\{\zeta\colon |\zeta|>1\}$ onto the exterior of a convex curve, with $F'(\infty)=1$. For $p=1$ this result is due to Pólya and Shiffer. We also obtain several generalizations of this estimate under other geometric assumptions about the structure of the domain $F(D^-)$.