TY - JOUR

T1 - A criterion of normality based on a single holomorphic function II

AU - Liu, Xiaojun

AU - Nevo, Shahar

PY - 2013

Y1 - 2013

N2 - In this paper, we continue to discuss normality based on a single holomorphic function. We obtain the following result. Let F be a family of functions holomorphic on a domain D ⊂ C. Let k ≥ 2 be an integer and let h (⊂ 0) be a holomorphic function on D, such that h(z) has no common zeros with any f ε F. Assume also that the following two conditions hold for every f ε F: (a) f(z) = 0 → f′(z) = h(z), and (b) f′(z) = h(z) → |f(k)| ≤ c, where c is a constant. Then F is normal on D. A geometrical approach is used to arrive at the result that significantly improves a previous result of the authors which had already improved a result of Chang, Fang and Zalcman. We also deal with two other similar criterions of normality. Our results are shown to be sharp.

AB - In this paper, we continue to discuss normality based on a single holomorphic function. We obtain the following result. Let F be a family of functions holomorphic on a domain D ⊂ C. Let k ≥ 2 be an integer and let h (⊂ 0) be a holomorphic function on D, such that h(z) has no common zeros with any f ε F. Assume also that the following two conditions hold for every f ε F: (a) f(z) = 0 → f′(z) = h(z), and (b) f′(z) = h(z) → |f(k)| ≤ c, where c is a constant. Then F is normal on D. A geometrical approach is used to arrive at the result that significantly improves a previous result of the authors which had already improved a result of Chang, Fang and Zalcman. We also deal with two other similar criterions of normality. Our results are shown to be sharp.

KW - Holomorphic functions

KW - Normal family

KW - Zero points

UR - http://www.scopus.com/inward/record.url?scp=84877680128&partnerID=8YFLogxK

U2 - 10.5186/aasfm.2013.3810

DO - 10.5186/aasfm.2013.3810

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AN - SCOPUS:84877680128

SN - 1239-629X

VL - 38

SP - 49

EP - 66

JO - Annales Academiae Scientiarum Fennicae Mathematica

JF - Annales Academiae Scientiarum Fennicae Mathematica

IS - 1

ER -