On radial distributions of Julia sets of Newton's method of solutions of complex differential equations
By using the classical Newton method for finding the zeros for an entire function and Nevanlinna theory in angular domains, the authors study the radial distribution of Julia sets of certain Riccati equations, and prove the following result: Suppose \(f\) is a transcendental meromorphic solution of the Riccati equation \(f'(z)+A(z)(f(z)-z)^2=0\), where \(A(z)\) is a transcendental entire function of finite order \(\sigma (f)\). Then \(\mathrm{mes\,}\Delta(f)\geq \min\{2\pi, \pi/\sigma (f)\}\). This result improves and extends the corresponding results from the references [\textit{Z.-G. Huang} and \textit{J. Wang}, J. Math. Anal. Appl. 409, No. 1, 478--484 (2014; Zbl 1314.34174); ibid. 387, No. 2, 1106--1113 (2012; Zbl 1238.37012)] given in the present paper. The results of the present paper are very interesting. The analysis and the reasoning of the present paper is new and deep.
scientific article; zbMATH DE number 6586127
scientific article