A characterization of random analytic functions satisfying Blaschke-type conditions

Duan, Yongjiang  ; Fang, Xiang  ; Zhan, Na 

Canadian mathematical bulletin, Tome 67 (2024) no. 3, pp. 670-679

Résumé

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

Détail

DOI : 10.4153/S0008439524000079

Mots-clés : Random analytic function, zero sets, Blaschke condition