Lifts of Logarithmic Derivatives

Consider a sequence of meromorphic functions $(f_n{)}_n$. This paper presents a technique that enables the transfer of convergence properties from $(f_n^{(m+1)}\mathrm{/}{f}_n^{(m)}{)}_n$ to subsequences of $(f_n^{(m)}\mathrm{/}{f}_n^{(m-1)}{)}_n$. As an application, we will show that the families of functions with bounded Schwarzian derivative are quasi-normal.