Varianten des Schwarzschen Lemma. (German) Zbl 0566.30021

Pfluger, Albert

Elem. Math. 40, 46-47 (1985).

In his solution to problem 901 [ibid. 39, 130-131 (1984)] R. Martini essentially proved the following. Let f(z) be a holomorphic function of the unit disk \({\mathbb{D}}\) into itself such that \(f(0)=0\). Then \(| f(z)+f(-z)| <2| z|\) in \({\mathbb{D}}\setminus \{0\}\) unless \(f(z)=cz^ 2\) and \(| c| =1\). The author points out that this is only the first step of a whole sequence of variants to the Schwarz’ lemma whereof the second step says that \[ | f(z)+f(\epsilon z)+f(\epsilon^ 2z)| <3| z|^ 3\quad in\quad {\mathbb{D}}\setminus \{0\}, \] if \(\epsilon =\exp (2\pi i/3)\), unless \(f(z)=cz^ 3\).

MSC:

30C80

Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination

Keywords:

Schwarz’ lemma