Uniform convergence of $g_n(z) = \log(|f_n(z)|)/\log(n+1)$ when $f_n(z) \to f(z)$ uniformly.

Question

If

(1) $f_n: E \mapsto \mathbb{C}$ be a set of non vanishing continuous functions on open region $E\subset \mathbb{C}$.

(2) $f_n \to f$ uniformly and $f$ is non vanishing on open region $E$ (Note: but limit point of $E$ may have $f(z) = 0$).

(3) $g_n: E \mapsto \mathbb{C}$ be defined as $$g_n(z) = \frac{\log|f_n(z)|}{\log(n+1)}$$

Then can we prove $g_n \to g$ uniformly?

Can anyone provide any hints or counter example? Thank you!

Answer 1

Let $f_n(z):=z+\frac{1}{n}$, let $E=\{z:\text{Im}(z)>0\}$. Then $g_n=\frac{\ln(|z+\frac{1}{n}|)}{\ln(n+1)}\to 0$. This convergence, however, is non uniform, since for every $n$, there exists a $z\in E; z=\frac{1}{n}-\frac{1}{n+1}$ such that $g_n(z)=-1$