Generalized Schwarz lemmas for meromorphic functions
The authors give two generalized Schwarz lemmas which apply to meromorphic functions in the unit disc \(\mathbb{D}\). In Theorem 1 it is assumed that \(f\) is meromorphic in \(\mathbb{D}\) with a zero of order \(k\) at 0, that there is a number \(r_0\in (0,1)\) such that \(|f(z)|\leq1\) for \(r_0<|z|<1\), and that, if \(f^{(k)}(0)\neq0\), the function \(g(z):=z^k-k!f(z)/f^{(k)}(0)\) satisfies \(Z(g)-P(g)\neq k\), where \(Z(g)\) and \(N(g)\) denote, respectively, the number of zeroes and poles of \(g\) in \(\mathbb{D}\). Under these assumptions it is then proved that NEWLINE\[NEWLINE\frac{1}{k!}|f^{(k)}(0)|\leq1,NEWLINE\]NEWLINE with equality if and only if \(f(z)=cz^k\) for some constant \(c\) of modulus 1.NEWLINENEWLINE In Theorem 2 it is assumed again that \(f\) is meromorphic in \(\mathbb{D}\) with a zero of order \(k\) at 0, that there is a number \(r_0\in (0,1)\) such that \(|f(z)|\leq1\) for \(r_0<|z|<1\), and that, for \(\omega \in \mathbb{D}\) not a pole of or a zero of \(f\), the function \(g_\omega(z):=z^k-\omega^kf(z)/f(\omega)\) satisfies \(Z(g_\omega)-P(g_\omega)\neq k\). In this case, the conclusion is that NEWLINE\[NEWLINE|f(\omega)|\leq|\omega|^k,NEWLINE\]NEWLINE with equality if and only if \(f(z)=cz^k\) for some constant \(c\) of modulus 1.NEWLINENEWLINEExamples are given to show that the last condition in each of the theorems is necessary. The proofs are applications of the argument principle.
scientific article; zbMATH DE number 6023971