Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 387-390

NEVANLINNA CLASS CONTAINS FUNCTIONS WHOSE SPHERICAL DERIVATIVES GROW ARBITRARILY FAST

Rauno Aulaskari and Jouni Rättyä

University of Joensuu, Mathematics
P.O. Box 111, 80101 Joensuu, Finland; rauno.aulaskari 'at' joensuu.fi

University of Joensuu, Mathematics
P.O. Box 111, 80101 Joensuu, Finland; jouni.rattya 'at' joensuu.fi

Abstract. It is shown that for any given increasing function \varphi : [0,1) \to (0,\infty) there exists a meromorphic function f_\varphi of bounded Nevanlinna characteristic such that its spherical derivative f^#_\varphi(z) = |f'_\varphi(z)| / (1 + |f_\varphi(z)|²) satisfies limsup_{|z| \to 1-}f^#_\varphi(z) / \varphi(|z|) = \infty. Such a function is constructed by using Blaschke products and the desired property is proved by normal family arguments. This study is inspired by results on non-normal Dirichlet and Blaschke quotients due to Yamashita.

2000 Mathematics Subject Classification: Primary 30D50; Secondary 30D35, 30D45.

Key words: Nevanlinna class, bounded characteristic, spherical derivative, Dirichlet space, Besov space, Q_p-space, normal function, Blaschke product, Blaschke quotient.

Reference to this article: R. Aulaskari and J. Rättyä: Nevanlinna class contains functions whose spherical derivatives grow arbitrarily fast. Ann. Acad. Sci. Fenn. Math. 34 (2009), 387-390.

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