Let $D$ be a domain in $\Bbb C$. $f_n$ be holomorphic function on $\bar D$. Assume there exists a complex valued $f$ such that $f_n$ is uniformly bounded on $\partial D$. Show (1) $f_n$ is locally uniformly convergent to $f$, and $f$ is holomorphic in $D$. (2) If $f_n$ is univalent (just means injective) in $D$, $f$ is not constant, show $f$ is also univalent.
By the maximum pricincle, it follows that $f_n$ converges uniformly on $\bar D$. But how to show it converges to $f$ exactly? No assumption on $f$ are made. Is this problem wrong? Need we add an assumption: $f_n\to f$ in $D$?
Assuming that $f_n$ are holomorphic on the closed unit disc, uniformly bounded there (which by maximum modulus it is, of course, equivalent to uniformly bounded on the unit circle) and converge pointwise to some $f$ inside the disc then:
the convergence is locally uniform inside the disc hence $f$ is analytic (and bounded); however $f$ may not extend continuously to the unit circle (generally it will only have a.e. extension by Fatou) and $f_n$ may not converge on the unit circle; if $f_n$ univalent on the open disc, then $f$ is constant or univalent by Hurwitz theorem of course
Proof: because by Montel the sequence $f_n$ is normal inside the unit disc, pointwise convergence means locally uniform convergence as any locally uniform limit must coincide with $f$, so $f$ is analytic in $D$ and is clearly bounded by the uniform bound of $f_n$ on the disc
For example $f_n(z)=z^n$ converges to zero inside the unit disc, $f_n$ are entire, $||f_n||=1$ but $z^n$ fails to converge on the unit circle except at $1$; if we take $f_n(z)=\exp \frac{z+1}{z-1-1/n}$ which are analytic on the closed unit disc (they clearly extend analytically to the circle of radius $1+1/n$), have norm $1$ there since $\Re \frac{z+1}{z-1-1/n}$ has the sign of $|z|^2-1-1/n-(1/n) \Re z \le 0, |z| \le 1$ and $f_n(-1)=1$, while $f_n \to f(z)=\exp \frac{z+1}{z-1}$ everywhere except at $1$ and $f$ fails to be continuous at $1$ - these two examples show some of the pathologies involved on the unit circle without any extra hypothesis there