Relative asymptotics for polynomials orthogonal on the real axis

Given a positive Borel measure $\rho$ on the real line $\mathbf R$ and a function $g$ on $\mathbf R$, the main purpose of the paper is to prove (under certain assumptions on $\rho$) relative asymptotic formulas of the type
$$ \frac{h_n(gd\rho,z)}{h_n(d\rho,z)}\underset{n\to\infty}\rightrightarrows S(g,\Omega;z),\qquad z\in\Omega, $$
where $\Omega=\{z:\operatorname{Im}z>0\}$, $S(g,\Omega;z)$ is Szegö's function corresponding to $\Omega$ and the function $g$, $h_n(gd\rho,z)$ and $h_n(d\rho,z)$ are polynomials orthonormal relative to the measures $gd\rho$ and $d\rho$ respectively.
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