On the growth of meromorphic solutions of the Schwarzian differential equations

Abstract

We study the properties of meromorphic solutions of the Schwarzian differential equations in the complex plane by using some techniques from the study of the class W p . We find some upper bounds of the order of meromorphic solutions for some types of the Schwarzian differential equations. We also show that there are no wandering domains nor Baker domains for meromorphic solutions of certain Schwarzian differential equations.  2005 Elsevier Inc. All rights reserved.

All meromorphic solutions of the autonomous Schwarzian differential equations

“…\end{equation*}$$By replacing $$y^{\prime }$$ with $$S(y,z)$$ in (1.2), we obtain the so‐called Schwarzian differential equation $$$$\begin{equation} {S(f,z)}^k = R(z,f), \end{equation}$$$$where $$k$$ is a positive integer, and $$R(z,f)$$ is an irreducible rational function in $$f$$ with meromorphic coefficients. While deficiencies [7] and growth [11] of meromorphic solutions of (1.3) have been studied, Ishizaki [7] derived the Malmquist–Yosida–Steinmetz type result for the autonomous case of (1.3): $$$$\begin{equation} {S(f,z)}^k=R(f)={{P(f)}\over {Q(f)}}, \end{equation}$$$$where $$P$$ and $$ Q$$ are co‐prime polynomials with constant coefficients.Theorem Suppose that the autonomous Schwarzian differential equation (1.4) admits a transcendental meromorphic solution. Then there exists a Möbius transformation …”

“…Lemma 2 [11]. Let 𝑓 be a meromorphic solution of the autonomous Schwarzian differential equation (1.4), then the order 𝜌(𝑓) ⩽ 2.…”

“…where 𝑘 is a positive integer, and 𝑅(𝑧, 𝑓) is an irreducible rational function in 𝑓 with meromorphic coefficients. While deficiencies [7] and growth [11] of meromorphic solutions of (1.3) have been studied, Ishizaki [7] derived the Malmquist-Yosida-Steinmetz type result for the autonomous case of (1.3):…”