On holomorphic functions satisfying $f(z)(1-z^{2})\leq 1$ in the unit disc

Abstract:

Let $f$ be holomorphic in $D = \{ \left . z \right |\left | z \right | < 1\}$, $\left | {f(z)} \right |(1 - {\left | z \right |^2}) \leqslant 1$ in $D$, ${\overline {\lim } _{\left | z \right | \to 1}}\left | {f(z)} \right |(1 - {\left | z \right |^2}) < 1$ and $L(f): = \{ \left . z \right |\left | {f(z)} \right |(1 - {\left | z \right |^2}) = 1\}$. It is shown that the set $L(f)$ consists of one simple closed curve $\gamma$ and a finite number of points in the bounded component of ${\mathbf {C}}\backslash \gamma$ if $L(f)$ is an infinite set.