SCHWEITZER 2008 Problem 9

For a given $\alpha >0$ let us consider the regular, non-vanishing $f(z)$ maps on the unit disc $\{ |z|<1 \}$ for which $f(0)=1$ and $\mathrm{Re} z\frac{f'(z)}{f(z)}>-\alpha$ ( $|z|<1$ ). Show that the range of $$g(z)=\frac{1}{(1-z)^{2\alpha}}$$ contains the range of all other such functions. Here we consider that regular branch of $g(z)$ for which $g(0)=1$ .

(translated by Miklós Maróti)