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A characterization of random analytic functions satisfying Blaschke-type conditions

Part of: Series expansions Holomorphic functions of several complex variables

Published online by Cambridge University Press:  17 January 2024

Yongjiang Duan Open the ORCID record for Yongjiang Duan [Opens in a new window] ,
Xiang Fang Open the ORCID record for Xiang Fang [Opens in a new window]  and
Na Zhan Open the ORCID record for Na Zhan [Opens in a new window]
Yongjiang Duan
Affiliation:
School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, P.R. China e-mail: duanyj086@nenu.edu.cn
Xiang Fang
Affiliation:
Department of Mathematics, National Central University, Chungli, Taiwan (R.O.C) e-mail: xfang@math.ncu.edu.tw
Na Zhan*
Affiliation:
School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, P.R. China e-mail: duanyj086@nenu.edu.cn
*
e-mail: zhann153@nenu.edu.cn
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Abstract

Let $f(z)=\sum _{n=0}^{\infty }a_n z^n \in H(\mathbb {D})$ be an analytic function over the unit disk in the complex plane, and let $\mathcal {R} f$ be its randomization:

$$ \begin{align*}(\mathcal{R} f)(z)= \sum_{n=0}^{\infty} a_n X_n z^n \in H(\mathbb{D}),\end{align*} $$

where $(X_n)_{n\ge 0}$ is a standard sequence of independent Bernoulli, Steinhaus, or Gaussian random variables. In this note, we characterize those $f(z) \in H(\mathbb {D})$ such that the zero set of $\mathcal {R} f$ satisfies a Blaschke-type condition almost surely:

$$ \begin{align*}\sum_{n=1}^{\infty}(1-|z_n|)^t<\infty, \quad t>1.\end{align*} $$

Keywords

Random analytic functionzero setsBlaschke condition

MSC classification

Primary: 32A60: Zero sets of holomorphic functions
Secondary: 30B20: Random power series

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 3 , September 2024 , pp. 670 - 679
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Y. Duan is supported by the NNSF of China (Grant No. 12171075) and Science and Technology Research Project of Education Department of Jilin Province (Grant No. JJKH20241406KJ). X. Fang is supported by the NSTC of Taiwan (112-2115-M-008-010-MY2).

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