A Malmquist-Yosida type theorem for Schwarzian differential equations

In this paper, we study a Malmquist-Yosida type theorem for Schwarzian differential equations \begin{equation}\label{1} S(f,z)^{m} = R(z,f) = \frac{P(z,f)}{Q(z,f)},\tag{+} \end{equation} where $m \in \mathbb{N}^{+}$, $P(z,f)$ and $Q(z,f)$ are irreducible polynomials in $f$ with rational coefficients. If \eqref{1} admits a transcendental meromorphic solution $f$, then by a suitable M$\mathrm{\ddot{o}}$bius transformation $f \to u$, $u$ satisfies a Riccati differential equation with small meromorphic coefficients, or one of the six types of first-order differential equations (E.2)-(E.7), or $u$ satisfies one of types (E.8)-(E.14). In addition, we improve the result of Ishizaki [6, Theorem~1.1] on Schwarzian differential equations \eqref{1} with small meromorphic coefficients when $m=1$.