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Let $\{f_n\}_{n=1}^{\infty}$ be a uniformly bounded sequence of nonconstant holomorphic functions in a connected open set $\Omega$. Let $f \not \equiv 0 $ be a holomorphic function in $\Omega$. Suppose that the equation

$nf(z) = f_n(z) + \ln{n}$

does not have any roots in $\Omega$ for $n =1,2,\ldots$. Prove that then $f$ does not have any zeros in $\Omega$.

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Suppose $f(z_0)=0$. Pick $r>0$ so that $f\ne 0$ when $|z-z_0|=r$. Apply Rouche's theorem in the disk $D=\{z: |z-z_0|\le r\}$ to the function $$nf(z) - (f_n(z)+\ln n)\tag{1}$$ The theorem applies when $n$ is large enough, because $|nf(z)|>|f_n(z)+\ln n|$ on the boundary of $D$.

Why?

Let $m=\min_{\partial D} |f|$ and $M=\sup_{n,z} |f_n(z)|$; observe that $mn>M+\ln n$ when $n$ is large enough.)

This is a contradiction, since (1) is assumed not to have zeros.

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