A Malmquist-Yosida type theorem for Schwarzian differential equations

2025/09/23 by X. L. Liu, Liu, Xiong-Feng
Computer Science · Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2509.18649

openalex publication_date 2025/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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)), where m ∈ ℕ⁺, P(z,f) and Q(z,f) are irreducible polynomials in f with rational coefficients. If \eqref1 admits a transcendental meromorphic solution f, then by a suitable M\mathrmobius transformation f → 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 \eqref1 with small meromorphic coefficients when m=1.

Citations

• Exact meromorphic solutions of Schwarzian differential equations
• Existence of meromorphic solutions of first order difference equations

Related

• Exact meromorphic solutions of Schwarzian differential equations
• Admissible solutions of delay Schwarzian differential equations
• Generalized Schwarzian derivatives and higher order differential equations
• Subnormal transcendental meromorphic solutions of difference equations with Schwarzian derivative
• Existence of meromorphic solutions of first order difference equations