Admissible solutions of delay Schwarzian differential equations

2025/10/16 by Wu, Shijian
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.14736

Abstract

In this paper, we study delay differential equations involving the Schwarzian derivative S(f,z), expressed in the form f(z+1)f(z-1) + a(z)S(f,z) =R(z,f(z))= (P(z,f(z)))/(Q(z,f(z))) where a(z) is rational, P(z,f) and Q(z,f) are coprime polynomials in f with rational coefficients. Our main result shows that if a subnormal transcendental meromorphic solution exists, then the rational function R(z,f)=P(z,f)/Q(z,f) satisfies °_fR≤ 7 and °_fP≤ °_fQ +2, where °_fR =max\°_fP, °_fQ\. Furthermore, for any rational root b₁ of Q(z,f) in f with multiplicity k, we show that k ≤ 2. Finally, a classification of such equations is provided according to the multiplicity structure of the roots of Q(z,f). Some examples are given to support these results.

Citations

• Studies of Differences from the point of view of Nevanlinna Theory
• Growth of meromorphic solutions of delay differential equations
• Finite-order meromorphic solutions and the discrete Painleve equations
• Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes

Related

• A Malmquist-Yosida type theorem for Schwarzian differential equations
• Subnormal transcendental meromorphic solutions of difference equations with Schwarzian derivative
• Generalized Schwarzian derivatives and higher order differential equations
• Transcendental meromorphic solutions and the complex Schrödinger equation with delay
• Growth of meromorphic solutions of delay differential equations