On hyper-order of meromorphic solutions of first-order algebraic differential equations (Q5959684)

Let \(\sum_{k,j\geq 0}\varphi_{k,j} (z)w^k(w')^j=0\) be an algebraic differential equation of first order, where the coefficients \(\varphi_{k,j}\) are entire functions of finite order. The authors prove that if \(h\) is a meromorphic solution of this algebraic differential equation then either \(\nu(h) \leq\max_{k,j} \sigma (\varphi_{k,j})\) or \(\lambda(1/h) =\infty\). Here means \(\nu(h)= \displaystyle \limsup_{r\to\infty}{\log\log T(r,k) \over\log r}\) the hyper-order of \(h\) and \[ \sigma(\varphi): = \limsup_{r\to\infty}{\log T(r,\varphi) \over\log r} \] the order of \(\varphi\) and \(\lambda({1\over h})\) the exponent of convergence of the zeros of \(h\). In the case of that \(\varphi_{k,j}\) has finitely many zeros for all \(k\) and \(j\) they show \(\nu(h)\leq \max\sigma (\varphi_{k,j})\).