Generalized Schwarz lemmas for meromorphic functions (Q2880482)

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.