Generalized Schwarzians and Normal Families
Matthias Grätsch
TL;DR
The paper studies families of analytic and meromorphic functions with bounded generalized Schwarzian derivative $S_k(f)$ and proves these families are (quasi-)normal; it also analyzes associated derivative and logarithmic-derivative families. A new representation of $S_k(f)$ as a differential polynomial in $f"/f'$ is derived, enabling normality conclusions via Grahl's framework when $S_k(f)$ omits a meromorphic function. The authors establish local boundedness and normality for several derived families, bound poles through disconjugacy arguments, and prove a self-improving result: if $S_k(\mathcal{F})$ omits a function, then $\mathcal{F}"/\mathcal{F}'$ is normal and derivatives inherit normality properties. They also discuss limitations via counterexamples and relate the general theory to the classical case $k=2$, connecting to the Schwarzian differential equation and broader value-distribution phenomena.
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 $\mathcal{F}\subseteq\mathcal{H}(\mathbb{D})$ of locally univalent functions that satisfy $$S_k(f)(z)\neq b(z)\qquad \text{for some }b\in\mathcal{M}(\mathbb{D})\text{ and all } f\in\mathcal{F},\,z\in\mathbb{C}$$ and for entire functions $f$ with $S_k(f)(z)\neq0$ and $S_k(f)(z)\neq\infty$ for all $z\in\mathbb{C}$.\\ The classical Schwarzian derivative $S_f$ is contained as the case $k=2$.