How to construct a complex function series that converges uniformly on a closed region but its derivative does not converge uniformly on the boundary?

Question

I'm looking for an example of a series of complex functions $\{f_n(z)\}$ with the following properties:

1. The series $\sum_{n=1}^{\infty} f_n(z)$ converges uniformly on a closed region $D$ in the complex plane, including its boundary.

2. The derivative series $\sum_{n=1}^{\infty} f_n'(z)$converges uniformly on any closed subregion within (D), but not on the boundary of $D$.

To clarify, $D$ could be, for instance, the closed unit disk. I am interested in an explicit construction of such a series where the uniform convergence of the derivative series breaks down precisely at the boundary of $D$, while maintaining uniform convergence on any closed subregion inside $D$.

I have been pondering for hours how to construct such a series and whether it might involve some delicate balancing of the growth rates of the terms in the series or some other non-trivial feature of complex analysis. Any insights or specific examples that satisfy these conditions would be greatly appreciated.

Answer 1

On the unit disk, take a continous function for which it's Fourier series does not converge uniformly, say $\sum a_n e^n$. Then integrate it, e.i. look at $\sum a_n/n e ^n$. By the Cauchy-Schwartz inequality Fourier coefficients are in $l_1$ so it converges uniformly on the whole closed disk.