MathDB

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)

Problem Statement