Result: Bounding Zeros of H2 Functions via Concentrations: Bounding zeros of \(H^ 2\) functions via concentrations

Let \(f(z)= \sum a_ j z^ j\in H^ 2(D)\), \(D\) being the open unit disk. Let \(1\leq p\leq 2\), and assume that \(\{a_ j\}\in \ell_ p\). Assume, for some \(0< d\leq 1\) and some non-negative integer \(k\) that \(d^ p \sum^ \infty_{j= 0} | a_ j|^ p\leq \sum^ k_{j= 0} | a_ j|^ p\). Under these assumptions, if \(\{z_ j\}\) is the sequence of zeros of \(f\) in \(| z|< 1\), the author shows that \(d\Phi_{d,k}\leq \prod_{j> k} | z_ j|\), where \[ \Phi_{d,k}:= \max_{0< r< 1} \left((1- r) r^ k d\exp\left( {-1+ r)\over 2(1- r)}\right) \right)^{(1+ r)/(1- r)}. \] This generalizes and improves a result of \textit{B. Beauzamy} and \textit{S. Chou} [J. Math. Anal. Appl. 175, No. 2, 360-379 (1993; Zbl 0790.30004)], who treat the case \(p= 1\).