Uniform convergence of the series $\sum\limits_{n=1}^\infty \left| \frac {\ln(n)^d} {n^z} - \frac {\ln(n)^d} {n^{z_0}} \right|^2$

Question

Let $\forall n \in \mathbb{N} f_n : A \to \mathbb{R}$ be a sequence of function where $A$ is a closed disk of the complex plane $\mathbb{C}$ such that $\forall z \in A: \Re(z)>\frac{1}{2}$ and $$ f_n(z)= \left| \frac {\ln(n)^d} {n^z} - \frac {\ln(n)^d} {n^{z_0}} \right|^2 $$ and $z_0 \in A$ and $d\geq0$ is an integer

I would like to know if the series $$ \sum_{n=1}^\infty f_n(z) $$ converges uniformly.