On a class of analytic functions

Abstract:

We show that the class ${\mathfrak {E}_0}$ of analytic functions $f$ in a plane region $\Omega \notin {O_{AB}}$ vanishing at ${z_0} \in \Omega$ and such that $1/f$ omits a set of values of area $\geqq \pi$ is not compact. Here ${O_{AB}}$ denotes the class of Riemann surfaces which have no nonconstant bounded analytic functions. We remark that the extremal functions maximizing $|f’({z_0})|$ in ${\mathfrak {E}_0}$ coincide with linear transformations $w/(1 - cw)$ of those for the same problem in the class ${\mathfrak {B}_0}$ consisting of functions $f$ such that $f({z_0}) = 0$ and $|f(z)| \leqq 1$, i.e. so-called Ahlfors functions. Here $1/c$ is an omitted value of the Ahlfors function.