Growth estimates for meromorphic solutions of higher order algebraic differential equations

Abstract

We establish pointwise growth estimates for the spherical derivative of solutions of the first order algebraic differential equations. A generalization of this result to higher order equations is also given. We discuss the related question of when for a given class $X$ of meromorphic functions in the unit disc, defined by means of the spherical derivative, and $m\in \mathbb{N}\setminus\{1\}$, $f^m\in X$ implies $f\in X$. An affirmative answer to this is given for example in the case of $\mathord{\rm UBC}$, the $\alpha$-normal functions with $\alpha\ge 1$ and certain (sufficiently large) Dirichlet type classes.