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For $n\in\mathbb{N}$ let $r_n,\ s_n$ be two polynomials of $O(n)$ degrees with real positive coefficients and set $f_n=r_n/s_n$. Suppose there exists $c>0$ such that

$\bullet$ if $z\in\mathbb{C}$ is a zero of a $p_n$, then $|z^2+c|\leq c$ (note that in particular the $f_n$'s have no real positive zeros)

$\bullet$ $s_n(x)>0$ and $f_n(x)\xrightarrow[n\to\infty]{}f(x)$ for all $x>0$, where $f$ is an analytic function

Can we conclude that $f$ is never zero on $\mathbb{R^+}$?

I think Vitali's theorem should play a role, but I don't see precisely how.

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No. If you take $s_n = n f_n^2$, it satisfies your hypothesis and it is easy to see that in this case $f=0$ everywhere.

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