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I have the following question:

Let $f_n$ are continuous real-valued functions on the unit circle and $u_n$ are harmonic functions on the open unit disk such and continuous on the closed unit disk. Furthermore, assume that $u_n$ agrees with $f_n$ on the unit circle. Show that if $f_n$ is equicontinuous on the unit circle, then $u_n$ is equicontinuous on the closed unit disk.

I think that applying the maximum principle may be useful, but I do not know how to relate that to equicontinuity. Any help would be appreciated.

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Up to adding some sequence of constant functions $C_n$, we may assume that $f_n$ (hence $u_n$) is bounded. Thus $f_n$ is precompact (by Ascoli). We want to show that $u_n$ is precompact too, and this will conclude by the “easy” part of Ascoli.

Let $u_{\psi(n)}$ be a subsequence of $u_n$, there is a subsequence $f_{\psi(\theta(n))}$ that converges uniformly thus is Cauchy. Now, for any integers $m,n$, $u_{\psi(\theta(m))}-u_{\psi(\theta(n))}$ is harmonic on the open unit disk, continuous on the closed unit disk and equals $f_{\psi(\theta(m))}-f_{\psi(\theta(n))}$ on the boundary. So by the maximum modulus principle, $$\|u_{\psi(\theta(m))}-u_{\psi(\theta(n))}\|_{\infty} \leq \| f_{\psi(\theta(m))}-f_{\psi(\theta(n))}\|_{\infty}. $$

(the first norm is on the unit disk, the second one on the unit circle)

Thus $u_{\psi(\theta(n))}$ is Cauchy thus converges, which concludes.

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