Generalized Schwarzians and normal families

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 \(F⊆ HD\) of locally univalent functions that satisfy\(S_k(f)(z)≠ b(z)\) for some \(b∈ M(D)\) and all \(f∈ F\), \(z\)and for entire functions \(g\) with \(S_k(g)(z)≠ 0\) and \(S_k(g)(z)≠∞\) for all \(z\). The classical Schwarzian derivative \(S_f\) is contained as the case \(k=2\).