Quasi-normal Family of Meromorphic Functions Whose Certain Type of Differential Polynomials Have No Zeros

Define the differential operators ϕ n for n ∈ ℕ inductively by ϕ 1 [ f ]( z )= f ( z ) and ϕ n + 1 [ f ] ( z ) = f ( z ) ϕ n [ f ] ( z ) + d d z ϕ n [ f ] ( z ) . For a positive integer k ≥ 2 and a positive number δ , let ℱ be the family of functions f meromorphic on domain D ⊂ ℂ such that ϕ k [ f ]...

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Bibliographic Details
Published in:Acta mathematica Sinica. English series 2021-08, Vol.37 (8), p.1267-1277
Main Author: Chang, Jian Ming
Format: Article
Language:English
Subjects:
Differential equations
Mathematical analysis
Mathematics
Mathematics and Statistics
Meromorphic functions
Numbers
Operators (mathematics)
Polynomials
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Summary:Define the differential operators ϕ n for n ∈ ℕ inductively by ϕ 1 [ f ]( z )= f ( z ) and ϕ n + 1 [ f ] ( z ) = f ( z ) ϕ n [ f ] ( z ) + d d z ϕ n [ f ] ( z ) . For a positive integer k ≥ 2 and a positive number δ , let ℱ be the family of functions f meromorphic on domain D ⊂ ℂ such that ϕ k [ f ]( z ) ≠ 0 and ∣Res( f, a ) − j ∣ ≥ δ for all j ∈{0, 1,…, k − 1} and all simple poles a of f in D . Then ℱ is quasi-normal on D of order 1.
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-021-0328-3