Picard values and normal families of meromorphic functions with multiple zeros. (English) Zbl 0909.30025
In this paper the following results were proved:
Theorem: If \(f\) is a transcendental meromorpic function which has only zeros of order at least \(n\) (an integer \(\geq 2\)), then, for \(1\leq k\leq n-1\), \(f(k)\) assumes every finite complex number infinitely often.
Theorem: If \(f\) is a transcendental meromorpic function which has only zeros of order \(\geq 3\), then, for any \(k+\geq 1\), \(f(k)\) assumes every finite complex number infinitely often.
Also the authors obtained the corresponding criteria for normality. These results generalized Hayman’s conjectures (which have been completely proved independently and simultaneously by Bergweiler and Eremenko, Chen and Fang, and Zalcman): (1) If \(f\) is a transcendental meromorphic function, then \(f(n)'\) assumes every finite complex number infinitely often. (2) A family of meromorphic functions is normal, if every function \(f\) in the family satisfies \(f(n)\neq 1\).
Theorem: If \(f\) is a transcendental meromorpic function which has only zeros of order at least \(n\) (an integer \(\geq 2\)), then, for \(1\leq k\leq n-1\), \(f(k)\) assumes every finite complex number infinitely often.
Theorem: If \(f\) is a transcendental meromorpic function which has only zeros of order \(\geq 3\), then, for any \(k+\geq 1\), \(f(k)\) assumes every finite complex number infinitely often.
Also the authors obtained the corresponding criteria for normality. These results generalized Hayman’s conjectures (which have been completely proved independently and simultaneously by Bergweiler and Eremenko, Chen and Fang, and Zalcman): (1) If \(f\) is a transcendental meromorphic function, then \(f(n)'\) assumes every finite complex number infinitely often. (2) A family of meromorphic functions is normal, if every function \(f\) in the family satisfies \(f(n)\neq 1\).
Reviewer: Huaihui Chen (Nanjing)
MSC:
| 30D45 | Normal functions of one complex variable, normal families |
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
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