## Continuous Newton method for starlike functions

### Summary

Summary: We study a continuous analogue of Newton method for solving the nonlinear equation $$\varphi (z) =0,$$ where $\varphi(z)$ holomorphic function and $0\in\overline{\varphi ( D)}$. It is proved that this method converges, to the solution for each initial data $z\in D$, if and only if $\varphi(z)$ is a star-like function with respect to either an interior or a boundary point. Our study is based on the theory of one parameter continuous semigroups. It enables us to consider convergence in the case of an interior as well as a boundary location of the solution by the same approach.

### Mathematics Subject Classification

49M15, 46T25, 47H25

### Keywords/Phrases

Newton method, star-like functions, continuous semigroup