On interpolating Blaschke products and Blaschke-oscillatory equations
This research is partially a continuation of the author's paper [``Growth estimates for logarithmic derivatives of Blaschke products and of functions in the Nevanlinna class'', Kodai Math. J. 30, No. 2, 263--279 (2007; Zbl 1134.30025)]. If \(\{z_n\}\) is an infinite sequence of nonzero points in the unit disc \(D = \{z: |z| <1\}\) satisfying \(\sum_{n=1}^\infty(1-|z_n|)<\infty\), then the Blaschke product \(B(z)=\prod_{ n=1}\frac{|z_n|}{z_n}\frac{z_n-z}{1-\overline z_nz}\) represents an analytic function in \(D\) and has zeros precisely at the points \(z_n\). The next result relates the strengthened Blaschke condition given by \[ \sum_{n=1}(1-|z_n|)^\alpha<\infty,\quad \alpha\in(0,1],\tag{1} \] to logarithmic derivatives of Blaschke products. Let \(B\) be a Blaschke product with zeros \(z_n\neq0\) such that \text{(1)} holds for some \(\alpha\in (0,1]\), and let \(k\in\mathbb N\). Suppose that \(\omega : [1,\infty)\to [1,\infty)\) is a continuous and increasing function such that \[ \int_1^\infty\frac{x}{\omega(x)}\,dx<\infty. \] Then there exists a set \(E_1\subset [0,1)\) which satisfies \[ \int_{E_1}\frac{dr}{1-r}<\infty, \] such that for all \(z\in D\) satisfying \(|z|\notin E_1\), we have \[ \left|\frac{B^{(k)}(z)}{B(z)}\right|=O\left(\left(\frac{1}{1-|z|}\right)^{(1+\alpha)k}\omega\left(\log\frac{e}{1-|z|}\right)^k\right). \] A sequence \(\{z_n\}\) of points in \(D\) is called uniformly separated provided that \[ \delta:=\inf_{k}\prod_{ n\neq k}\left|\frac{z_n-z_k}{1-\overline z_nz_k}\right|>0, \] and is called exponential if there is a constant \(q\in (0,1)\) such that \[ 1-|z_{n+1}|\leq q(1-|z_n|). \] Such Blaschke products are known as interpolating Blaschke products which appear naturally in studying the frequency of zeros of solutions of \(f'' + A(z)f = 0\), where \(A(z)\) is analytic in \(D\). In particular, a unit disc analogue of a 1988 result due to S. Bank on prescribed zero sequences for entire solutions is obtained, and a more careful analysis of a 1955 example due to B.~ Schwarz on the case \(A(z) =\frac{1+4\gamma^2}{(1-z^2)^2}\) reveals that an infinite zero sequence is always a union of two exponential sequences. Growth estimates for generalized logarithmic derivatives of Blaschke products are provided under the assumption that the zero sequences are either uniformly separated or exponential. The growth estimates are then proven to be sharp in a rather strong sense. The sharpness discussion yields a solution to an open problem posed by E.~ Fricain and J.~ Mashreghi in 2008, related to the case of an exponential zero sequence.
scientific article; zbMATH DE number 5939413
scientific article