If a complex power series $$f(z)=\sum_{n=0}^{\infty } a_n z^n $$ is uniformly convergent in any closed subset of the open disc $\mathrm{B} (0,R)$, and converges pointwise on $\partial\mathrm{B} $, is it uniformly convergent on $\overline{\mathrm{B} (0,R)} $?
I have tried to prove but have difficulties at some step, so I think of it as wrong. If it is wrong, are there some counterexamples?
On the boundary of the disc of convergence, the power series essentially reduces to a Fourier series. There are Fourier series that converge pointwise but not uniformly. You can find an example here: Pointwise but not uniform convergence of a Fourier series
One of the comments there also references Zygmund (Trigonometric Series I, p. 300). Zygmund mentions a similar result on power series following from the Fourier series one.