On some generalized Painlevé and Hayman type equations with meromorphic solutions in a bounded domain (Q1637527)

For a given meromorphic function \(w\) in a domain \(D\), and a domain \(g\), the Alhfors simple islands are domains \(g_k\), \(k=1, 2, \dots, n(D,g,w)\), on the Riemann surface \(\{w(z): z\in \overline{D}\}\) that are projected one-to-one onto the complex plane and the projection coincides with \(g\). The authors consider meromorphic solutions for the equation of the form \[ f(z,w)w''=g_0(z,w)(w')^2+g_1(z,w)w'+g_2(z,w) \eqno(F), \] where the coefficients are meromorphic in \(w\in w(\overline{D})\) and analytic in \(z\in \overline{D}\) provided that \(w(z)\) is finite. Moreover, \(\min_{\overline{D}} |g_0(z,a)|>0\) for any \(a\in \mathbb{C}\). In some special cases of equation \((F)\) it is estimated above a number of simple islands \(n(D,\{\zeta:|\zeta-a|<\rho\},w)\) in terms of coefficients, where \(w\) is a meromorphic solution in \(\overline{D}\).