Periodic Points and Normality Concerning Meromorphic Functions with Multiplicity
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY∗
"℄j)
Bingmao DENG (
School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521, China E-mail : [email protected]
²
Mingliang FANG (
§
)†
§
Department of Mathematics Hangzhou Dianzi University Hangzhou 310012, China E-mail : [email protected]
)
Yuefei WANG (
School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China; Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China E-mail : [email protected] Abstract In this article, two results concerning the periodic points and normality of meromorphic functions are obtained: (i) the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by letting R(z) be a nonpolynomial rational function, and if all zeros and poles of R(z) − z are multiple, then Rk (z) has at least k + 1 fixed points in the complex plane for each integer k ≥ 2; (ii) a complete solution to the problem of normality of meromorphic functions with periodic points is given by letting F be a family of meromorphic functions in a domain D, and letting k ≥ 2 be a positive integer. If, for each f ∈ F, all zeros and poles of f (z) − z are multiple, and its iteration f k has at most k distinct fixed points in D, then F is normal in D. Examples show that all of the conditions are the best possible. Key words
normality; iteration; periodic points
2010 MR Subject Classification
1
30D45; 30D35
Introduction
Let F be a family of meromorphic functions in a plane domain D ⊂ C. Then F is said to be normal in D (in the sense of Montel) if each sequence {fn } ⊂ F has a subsequence {fnj } which converges spherically locally uniformly on D to a meromorphic function or ∞ (see [16, 20, 21]). ∗ Received
December 4, 2018; revised January 4, 2020. The first author was supported by the NNSF of China (11901119, 11701188); The third author was supported by the NNSF of China (11688101). † Corresponding author: Mingliang FANG.
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
Let f : D → C be a meromorphic function, and let n be a positive integer. Then the iterates f n : Dn → C of f are defined inductively by f 1 (z) = f (z) and f n (z) = f (f n−1 (z)) for n ≥ 2. To avoid confusion, we use (f (z))n to define the n-power of f (z). Let z0 ∈ Dn . If there exists a smallest positive integer n such that f n (z0 ) = z0 , then z0 is called a periodic point of period n of f , and the corresponding cycle {z0 , f (z0 ), · · · , f n−1 (z0 )} is said to be a periodic cycle of period n of f in D. A periodic point of period 1 is called a fixed point. For z0 6= ∞, define the multiplier of the periodic point z0 by λ = (f n )′ (z0 ), while for z0 = ∞, the multiplier is λ = lim (f n1)′ (z) . A periodic point z0 is said to be attracting, z→∞
neutral, or repelling, according to | λ |< 1, | λ |
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