On the growth of meromorphic and entire solutions of second-order algebraic differential equation
This paper is devoted to studying the growth of meromorphic solutions to the second-order algebraic differential equation \[ f''=L(z,f)(f')^2+M(z,f)f'+N(z,f) \] with birational coefficients \(L\), \(M\), \(N\). The main results are as follows: (1) If \(L(z,f)=(a_n(z)f^n+\dots+a_s(z)f^s)/ (b_m(z)f^m+\dots+b_r(z)f^r)\) and if \(m-n<1\) or \(r-s>1\), then all meromorphic solutions are of finite order. (2) In the special case \(f''=M(z,f)f'+N(z,f)\), where \(M\), \(N\) are polynomials in \(f\) with rational coefficients, all entire solutions must be of an order which is a positive integral multiple of \(1/2\). (3) If \(f\) is a transcendental meromorphic solution to \(L(z,f)(f')^2+M(z,f)f'+A(z,f)=0\), where \(L\), \(M\), \(N\) are polynomials in \(f\) with rational coefficients, then again \(\rho(f)=\frac{12}n\), \(n\in\mathbb{N} \cup\{0\}\). We remark that this paper may be understood as a contribution towards proving the Malmquist-type conjecture which claims that, whenever \(f\) is a transcendental meromorphic solution to \(f''=R(z,f,f')\), \(R\) rational in \(z,f,f'\), then \(R(z,f,f')= L(z,f)(f')^2+ M(z,f)f'+ N(z,f)\) [see \textit{Ilpo Laine}, Nevanlinna theory and complex differential equations. Berlin: Walter de Gruyter (1992; Zbl 0784.30002)]. As for other recent papers by the same authors of similar type [see \textit{L.-W. Liao} and \textit{Ch.-Ch. Yang}, Ann. Acad. Sci. Fenn., Math. 25, No. 1, 73--84 (2000; Zbl 0942.34077) and J. Differ. Equations 187, No. 1, 63--71 (2003; Zbl 1032.34082)].
scientific article; zbMATH DE number 1993712