Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 325-337

ON THE HAUSDORFF MEASURE OF THE JULIA SET AND THE ESCAPING SET OF ENTIRE FUNCTIONS WITH REGULARLY GROWING MAXIMUM MODULUS

Jie Ding

Taiyuan University of Technology, School of Mathematics, Taiyuan, Shanxi, 030024, P.R. China
and Christian-Albrechts-Universität zu Kiel, Mathematisches Seminar
Kiel, D-24098, Germany; dingjie 'at' tyut.edu.cn

Abstract. We prove that the Hausdorff measure of the escaping set and the Julia set of an entire function f is infinite with respect to certain gauge functions, provided that f is outside of the Eremenko–Lyubich class, and that the maximum modulus M(r,f) of f satisfies a certain regularity condition.

2010 Mathematics Subject Classification: Primary 37F10, 30D05.

Key words: Entire function, escaping set, Julia set, Hausdorff measure, gauge function.

Reference to this article: J. Ding: On the Hausdorff measure of the Julia set and the escaping set of entire functions with regularly growing maximum modulus. Ann. Acad. Sci. Fenn. Math. 42 (2017), 325-337.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4215