Limit of a sequence of a series of holomorphic functions

Question

Let $\{f_n\}_{n=0}^{\infty}$ be a sequence of holomorphic functions defined on the right half of the complex plane. The members of the sequence coverges uniformly to $0$ when restricted to the positive reals, i.e. $\lim_{n\rightarrow \infty}f_n|_{\mathbb{R}_{+}}=0$.

Is it true that $\lim_{n\rightarrow \infty}f_n=0$ on the whole right half plane uniformly?

Answer 1

It is not even true pointwise. For instance, consider $g_n(z) = \frac{1}{(z-2i)^n}$. Then for $t>0$ we have that $|g_n(t)| \leq 2^{-n}$. However $|g_n(1+2i)| = 1$ for all $n$.