Tohoku Mathematical Journal
2020
December
SECOND SERIES VOL. 72, NO. 4

Tohoku Math. J.
72 (2020), 621-629

Title GROWTH ESTIMATES FOR MEROMORPHIC SOLUTIONS OF HIGHER ORDER ALGEBRAIC DIFFERENTIAL EQUATIONS

Author Shamil Makhmutov, Jouni Rättyä and Toni Vesikko

(Received June 14, 2019)
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.

Mathematics Subject Classification. Primary 34M05; Secondary 30D45.
Key words and phrases. Complex differential equations, spherical derivative, normal functions.

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