Generalized Schwarzians and Normal Families

TL;DR

This paper explores families of functions with bounded generalized Schwarzian derivatives, establishing their quasi-normality, deriving new formulas, and extending classical results to broader classes of functions.

Contribution

It introduces new quasi-normality results for families with bounded generalized Schwarzian derivatives and derives a novel formula for $S_k(f)$, extending classical Schwarzian theory.

Findings

Families with bounded $S_k(f)$ are quasi-normal.

Derived a new formula for $S_k(f)$.

Extended classical Schwarzian results to generalized derivatives.

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 $k=2$.