If we have two sequences of degree $n$ monic polynomials $f_n(z)$ and $g_n(z)$, with zeros of $f_n, g_n$ lies outside the unit circle, i.e. in $\{z:|z|\ge 1\}$, and for any $r\in(0,1)$ we know $\lim_{n\to\infty}\sup_{|z|\le r}|f_n(z)-g_n(z)|=0$. What can we say about the zeros of $f_n$ and $g_n$? Is there any properties they should share in common?
I know the support and the counting measures of these two zeros sets can be very different, in fact it's possible that one lies on the unit circle and another one does not. But for example, is it necessarily true that their angular projections onto the unit circle are close to each other?
Or is there any other common properties that can be derived from that closeness of two polynomials inside the circle?
