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) ⇒ {pipe}f(k)(z){pipe} ≤ c, where c is a constant. Then F is normal on D.
| Original language | English |
|---|---|
| Pages (from-to) | 141-154 |
| Number of pages | 14 |
| Journal | Acta Mathematica Sinica, English Series |
| Volume | 27 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 2011 |
Bibliographical note
Funding Information:Received July 8, 2009, accepted January 18, 2010 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)
Funding
Received July 8, 2009, accepted January 18, 2010 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)
| Funders | Funder number |
|---|---|
| Gelbart Research Institute for Mathematical Sciences | |
| National Natural Science Foundation of China | 10671067 |
| Israel Science Foundation | 395107 |
Keywords
- Holomorphic functions
- Normal family
- Zero points