Logarithmic derevative and angular density of zeros for the Blaschke product

Анотація

Let $z_0=1$ be the only boundary point of zeros $\left(a_n\right)$ of the Blaschke product $B\left(z\right)$,

${\Gamma }_m=\underset{j=1}{\overset{m}{\cup }}\left\{z: \left|z\right|<1, arg(1-z)={\theta }_j\right\}=\underset{j=1}{\overset{m}{\cup }}{I}_{{\theta }_j},$
$\raisebox{1ex}{$-\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\varsigma <{\theta }_1<{\theta }_2<\cdots <{\theta }_m<\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\varsigma ,$
be a finite system of rays, $0<\varsigma <1$. We found asymptotics of the logarithmic derivative of $B\left(z\right)$ as $z=1-r{e}^{-i\phi }\to 1$,  $\raisebox{1ex}{$-\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.<\phi <\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, under the condition of existing the angular density of its zeros related to the comparison function ${\left(1-r\right)}^{-\rho }$, $0<\rho <1$. We also considered the inverse problem for $B\left(Z\right)$, whose zeros lie on ${\Gamma }_m$.