# Growth estimates for meromorphic solutions of higher order algebraic differential equations

## TL;DR

This paper derives growth estimates for solutions of higher order algebraic differential equations, extending previous results for first order cases, and explores conditions under which certain classes of meromorphic functions are closed under taking roots.

## Contribution

It generalizes pointwise growth estimates for solutions of algebraic differential equations from first to higher order cases and investigates function class properties related to spherical derivatives.

## Key findings

- Established growth estimates for solutions of higher order algebraic differential equations.
- Extended the results to classes of meromorphic functions defined via spherical derivatives.
- Provided conditions under which function classes are closed under m-th roots.

## 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 $\alpha\ge1$ and certain (sufficiently large) Dirichlet type classes.