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| A Big Picard Type Theorem Concerning Derivative and Its Application |
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Citation： |
Shuxian LI,Xiaojun LIU.A Big Picard Type Theorem Concerning Derivative and Its Application[J].Chinese Annals of Mathematics B,2026,(2):359~374 |
| Page view： 82
Net amount： 32 |
Authors： |
Shuxian LI; Xiaojun LIU |
Foundation： |
National Natural Science Foundation of China (No. 11871216) |
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| Abstract： |
In this paper, the authors prove a big Picard type theorem concerning derivative: Let f(z) be meromorphic in for each δ>0, if z0? is an essential singularity of f(z), then either f(z) assumes every finite value infinitely often or f′(z) assumes every finite value except possibly zero infinitely often. As an application of this result, they extend Nevo, Pang and Zalcman’s quasinormal criterion: Let {fn?(z)} be a sequence of meromorphic functions on the plane domain D, all of whose zeros are multiple such that fn′?(z)?1 has zeros of multiplicity at least n for all n on D, then {fn?(z)} is quasinormal of order 1 on D. Then they obtain a corresponding result in value distribution theory: Let f(z) be a meromorphic function on C, all but finitely many of whose zeros are multiple such that limr→+∞?(logr)2T(r,f)?=+∞, then there exist a positive integer M and 0 such that for each, there exists z0?∈C satisfying ∣z0?∣>r such that z0? is a zero of f′(z)?1 with multiplicity at most M. |
Keywords： |
Essential singularity Quasinormal families Value distribution theory |
Classification： |
30D30, 30D35, 30D45 |
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