Lifts of logarithmic derivatives
DOI:
https://doi.org/10.7146/math.scand.a-158582Abstract
Consider a sequence of meromorphic functions $(f_n)_n$. This paper presents a technique that enables the transfer of convergence properties from $(f_n^{(m+1)}/f_n^{(m)})_n$ to subsequences of $(f_n^{(m)}/f_n^{(m-1)})_n$. As an application, we will show that the families of functions with bounded Schwarzian derivative are quasi-normal.
References
Chuang, C.-T., Normal families of meromorphic functions, World Scientific, Singapore, 1993. https://doi.org/10.1142/1904
Grahl, J., and Nevo, S., Quasi-normality induced by differential inequalities, Bull. Lond. Math. Soc. 50 (2018), no. 1, 73–84. https://doi.org/10.1112/blms.12111
Lehto, O., Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, 109. Springer-Verlag, New York, 1987. https://doi.org/10.1007/978-1-4613-8652-0
Ma, W., Meija, D., and Minda, D., Bounded Schwarzian and two-point distortion, Comput. Methods Funct. Theory 13 (2013), no. 4, 705–715. https://doi.org/10.1007/s40315-013-0043-x
Nevo, S., and Shem Tov, Z., Differential Marty-type inequalities which lead to quasi-normality, J. Math. Anal. Appl. 527 (2023), no. 1, part 1, Paper No. 127337, 11 pp. https://doi.org/10.1016/j.jmaa.2023.127337
Zalcman Zalcman, L., A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817. https://doi.org/10.2307/2319796
