On zeros of Bloch functions and related spaces of analytic functions (Q2769228)

Let \(A^p\), \(0<p<\infty\), denote the Bergman space of functions \(f\) analytic in the unit \(\mathbb D\) satisfying NEWLINE\[NEWLINE \|f\|_p=\left(\frac{1}{\pi}\iint_{\mathbb D}|f(z)|^p dx dy\right)^{1/p}<\infty. NEWLINE\]NEWLINE A function \(f\) analytic in \(\mathbb D\) is said to be a Bloch function if NEWLINE\[NEWLINE \|f\|_{\mathcal B}=|f(0)|+\sup_{z\in\mathbb D}(1-|z|^2)|f'(z)|<\infty. NEWLINE\]NEWLINE The space of all Bloch functions is denoted by \(\mathcal B\). For \(0<r<1\), set NEWLINE\[NEWLINE M_{\infty}(r,f)=\max_{|z|=r}|f(z)|NEWLINE\]NEWLINE and let us define \(A^0\) as the space of all functions \(f\) analytic in \(\mathbb D\) such that NEWLINE\[NEWLINE M_{\infty}(r,f)= O\left(\log\frac{1}{1-r}\right)\quad\text{as}\quad r\to 1. NEWLINE\]NEWLINE The following inclusions are well known NEWLINE\[NEWLINE \mathcal B\subset A^0 \subset\bigcap_{0<p<\infty}A^p. NEWLINE\]NEWLINE If \(f\) is an analytic function in \(\mathbb D\), \(f(z)\neq 0\), and \(\{z_k\}_{k=1}^{\infty}\) is the sequence of its zeros, repeated according to multiplicity and ordered so that \(|z_1|\leq|z_2|\leq|z_3|\leq\ldots\), then \(\{z_k\}\) is said to be the sequence of ordered zeros of \(f\). NEWLINENEWLINENEWLINEIn this paper the author improves Horowitz' theorem [\textit{Ch. Horowitz}, Duke Math J. 41, 693-710 (1974; Zbl 0293.30035)] on the ordered zeros of \(A^p\) functions and the result in [\textit{D. Girela, M. Nowak}, and \textit{P. Waniurski}, Math. Proc. Camb. Philos. Soc. 129, No. 1, 117-128 (2000; Zbl 0958.30022)] for the ordered zeros of \(A^0\) functions. NEWLINENEWLINENEWLINEIt is known that for \(f\in A^p\) and the ordered zeros \(\{z_k\}\) of \(f\), NEWLINE\[NEWLINE \limsup_{n\to \infty}\frac{n(1-|z_n|)}{\log n}\leq\frac{1}{p} NEWLINE\]NEWLINE and the constant \(\frac{1}{p}\) is best possible. The author of this paper gets the analogous inequalities for \(A^0\) and the Bloch space, that is, NEWLINE\[NEWLINE \limsup_{n\to \infty}\frac{n(1-|z_n|)}{\log\log n}\leq 1 \quad \text{if}\quad f\in A^0 NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \limsup_{n\to \infty}\frac{n(1-|z_n|)}{\log\log n}\leq\frac{1}{2}\quad \text{if}\quad f\in\mathcal B. NEWLINE\]NEWLINE Further, an example of the function \(f\in A^0\) for which \(\limsup_{n\to \infty}\frac{n(1-|z_n|)}{\log\log n}\geq\frac{1}{4}\) is given.