Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 109-118

JOINTLY MAXIMAL PRODUCTS IN WEIGHTED GROWTH SPACES

Janne Gröhn, José Ángel Peláez and Jouni Rättyä

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; janne.grohn 'at' uef.fi

Universidad de Málaga, Departamento de Análisis Matemático
Campus de Teatinos, 29071 Málaga, Spain; japelaez 'at' uma.es

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; jouni.rattya 'at' uef.fi

Abstract. It is shown that for any positive, non-decreasing, continuous and unbounded doubling function \omega on [0,1), there exist two analytic infinite products f₀ and f₁ such that the asymptotic relation |f₀(z)| + |f₁(z)| \asymp \omega(|z|) is satisfied for all z in the unit disc. It is also shown that both functions f_j for j = 0,1 satisfy T(r,f_j) \asymp log\omega(r), as r \to 1^-, and hence give examples of analytic functions for which the Nevanlinna characteristic admits the regular slow growth induced by \omega.

2010 Mathematics Subject Classification: Primary 30J99.

Key words: Doubling function, infinite product, zero distribution.

Reference to this article: J. Gröhn, J.Á. Peláez and J. Rättyä: Jointly maximal products in weighted growth spaces. Ann. Acad. Sci. Fenn. Math. 39 (2014), 109-118.

Full document as PDF file

doi:10.5186/aasfm.2014.3901