On new singular directions of some Schröder functions

Abstract

In this paper, we study the Hayman T directions and the precise Borel directions of maximal kind of meromorphic solutions f(z) of the Schröder equations f(sz)=R(f(z)), where |s|<1 and R(w) is a rational function with $\deg [R]\geq 2$. We will show that, if $\operatorname {arg}[s]/2\pi \notin Q$, then f(z) has any direction as Hayman T direction and maximus Borel direction as well. This is a continue work of [Ishizaki, K. and Yanaihara, N., Borel and Julia directions of meromorphic Schröder functions, Math. Proc. Camb. Phil. Soc. 139 (2005), 139-147.] and [Yuan, W.J., Qi, J.M. and Seiki Mori. Singular directions of meromorphic solutions of some non-autonomous Schröder equations, Complex Analysis and its Applications Proceedings of the 15th ICFIDCAA held in Osaka (Japan), July 30-August 3, 2007].