Generalized Schwarzians and normal families

Abstract

We study families of analytic and meromorphic functions with bounded generalized Schwarzian derivative \(S_k(f)\). We show that these families are quasi-normal. Further, we investigate associated families, such as those formed by derivatives and logarithmic derivatives, and prove several (quasi-)normality results. Moreover, we derive a new formula for \(S_k(f)\), which yields a result for families \(F\subseteq H\mathbb{D}\) of locally univalent functions that satisfy

\(S_k(f)(z)\neq b(z)\) for some \(b\in M(\mathbb{D})\) and all \(f\in F\), \(z\in\mathbb{C}\)

and for entire functions \(g\) with \(S_k(g)(z)\neq 0\) and \(S_k(g)(z)\neq\infty\) for all \(z\in\mathbb{C}\). The classical Schwarzian derivative \(S_f\) is contained as the case \(k=2\).