Let $\Omega$ be a half-plane and $f_n$ a sequence of holomorphic functions on $\Omega$.

Question

I want to solve the second part of this 2-part problem. The first part, which I solved, states the following:

Let $\Omega$ be a bounded region and $f_n$ a sequence of holomorphic functions on $\Omega$ and continuous on $\overline\Omega$. Prove that if $f_n$ converges uniformly in $\partial \Omega$ then it converges in $H(\Omega)$.

The second part asks whether this is true if $\Omega$ is a half plane.

My intuition tells me that this does not hold, but I can't seem to find an "easy" counterexample, or a proof to show that it does hold.

Any help is more than appreciated :)

Answer 1

$e^{-n(1+e^{z})} \to 0$ uniformly on the real line but it does not converge in $H(\Omega)$ where $\Omega$ is the upper half plane. [Put $z=2+i\pi$].

Answer 2

$f_n(z)=\dfrac{e^{-inz}}{n}\to 0$ uniformly on $\mathbb R$ but diverges for each $z$ in the upper half plane.