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    Uniformly bounded components of normality

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    Suppose that f(z)f(z) is a transcendental entire function and that the Fatou set F(f)≠∅F(f)\neq\emptyset. Set B1(f):=sup⁡Usup⁡z∈Ulog⁡(∣z∣+3)inf⁡w∈Ulog⁡(∣w∣+3)B_1(f):=\sup_{U}\frac{\sup_{z\in U}\log(|z|+3)}{\inf_{w\in U}\log(|w|+3)} and B2(f):=sup⁡Usup⁡z∈Ulog⁡log⁡(∣z∣+30)inf⁡w∈Ulog⁡(∣w∣+3),B_2(f):=\sup_{U}\frac{\sup_{z\in U}\log\log(|z|+30)}{\inf_{w\in U}\log(|w|+3)}, where the supremum sup⁡U\sup_{U} is taken over all components of F(f)F(f). If B1(f)<∞B_1(f)<\infty or B2(f)<∞B_2(f)<\infty, then we say F(f)F(f) is strongly uniformly bounded or uniformly bounded respectively. In this article, we will show that, under some conditions, F(f)F(f) is (strongly) uniformly bounded.Comment: 17 pages, a revised version, to appear in Mathematical Proceedings Cambridge Philosophical Societ
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