Find positive integers $n$ such that the derivative of $f(z)=(z-1)^nB(z)$ is bounded in $U$, where $B$ is a Blaschke product with zeros on $(0,1)$.

Question

Suppose that $B$ is a Blaschke product with zeros on $(0,1)$. Find all $n \in \mathbb{N}$ such that the derivative of $f(z)=(z-1)^nB(z)$ is bounded in $U$. My thoughts: My guess is that it is true for all $n \ge 1$. I take the derivative directly \begin{equation*} f'(z)=n(z-1)^{n-1}B(z)+(z-1)^nB'(z). \end{equation*} Now $n(z-1)^{n-1}B(z)$ is clearly bounded in $U$, but I cannot show that $B'(z)$ is also bounded in $U$. Any hints?

Answer 1

You need $n \ge 2$ but I do not know an easy proof of necessity without some general Hardy space theory; for sufficiency, it is easy since if $B=\Pi{\frac{|z_n|(z_n-z)}{z_n(1-\bar z_n z)}}$, and $B_n$ is the Blaschke product where we omit the $n$ term, we immediately get the formula (with normal convergence on the open unit disc):

$B'(z)=-\sum{\frac{|z_n|(1-|z_n|^2)}{z_n(1-\bar z_n z)^2}B_n(z)}$

But now $|B_n(z)| <1$ and if $z_n=r_n>0$ real we have $|1-r_nz|^2 \ge C^2|1-z|^2, (C=\min (|r_n| \ne 0 >0)$ for any $|z|<1$, hence $|B'(z)| \le \frac{1}{C^2}\frac{\sum(1-|z_n|^2)}{|1-z|^2}$ and the Blaschke condition is precisely that the numerator is convergent so we get $|1-z|^2|B'(z)| \le M$

On the other hand it is a theorem that if we call the individual factors $b_n(z)$ and we define as usual, (for $p>0$), $||B'||_p=\sup_{r<1}(\frac{1}{2\pi}\int_0^{2\pi}{|B'(re^{it})|^pdt})^{\frac{1}{p}}$, we have $||B'||_p \ge \sum ||b_n'||_p$ (which is non-trivial obviously only when $||B'||_p < \infty$ and actually with a little more work we get equality so LHS and RHS are finite or infinite at the same time)

But now it is an easy exercise to show that $||b_n'||_{\frac{1}{2}} \ge C(1-r_n)\log^2(1-r_n)$, so in particular if $\sum{(1-r_n)\log^2(1-r_n)} =\infty$ we cannot have $||B'||_{\frac{1}{2}} <\infty$. However $||\frac{1}{1-z}||_{\frac{1}{2}}$ is clearly finite so $(1-z)B'$ cannot be bounded.

Taking $r_n=1-\frac{1}{n \log^2 n}, n \ge 2$ we obviously satisfy the Blaschke convergence condition, but $(1-r_n)\log^2(1-r_n)$ ~$\frac{1}{n}$, so for the corresponding $||B'||_{\frac{1}{2}} =\infty$, hence $(z-1)B'$ cannot be bounded