Abstract
Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f (k)(z)| ≤ c, where c is a constant. Then F is normal on D.
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The first author is supported by the Gelbart Research Institute for Mathematical Sciences and by National Natural Science Foundation of China (Grant No. 10671067); the second author is supported by the Israel Science Foundation (Grant No. 395107)
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Liu, X.J., Nevo, S. A criterion of normality based on a single holomorphic function. Acta. Math. Sin.-English Ser. 27, 141–154 (2011). https://doi.org/10.1007/s10114-011-9396-0
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DOI: https://doi.org/10.1007/s10114-011-9396-0