A Malmquist-Yosida type theorem for Schwarzian differential equations

In this paper, we study a Malmquist-Yosida type theorem for Schwarzian differential equations S ( f , z ) m = R ( z , f ) = P ( z , f ) Q ( z , f ) , {equation*} S(f,z)ᵐ = R(z,f) = P(z,f)/Q(z,f), {equation*} where m ∈ N + m ∈ N⁺ , P ( z , f ) P(z,f) and Q ( z , f ) Q(z,f) are irreducible polynomials in f f with rational coefficients. If the above equation admits a transcendental meromorphic solution f f , then by a suitable Möbius transformation f → u f → u , u u satisfies a Riccati differential equation with small meromorphic coefficients, or one of the six types of first-order differential equations {1.2}Thm.1.E.6, or u u satisfies one of types {Thm.1.E.7}E.14. In addition, we improve the result of Ishizaki [Kodai Math. J. 20 (1997), pp. 67–78; Theorem 1.1] on Schwarzian differential equations with small meromorphic coefficients when m = 1 m=1 .