Interchanging sums and integrals in a specific instance

Question

Suppose $f_n$ is a sequence of holomorphic functions taking $\mathbb{D} \to \mathbb{C}$ where $\mathbb{D}$ is the unit disk. Further, $f_n$ has a continuous extension to $\overline{\mathbb{D}}$. We can assume $\sum_{n=0}^\infty f_n$ converges normally on compact subsets of $\mathbb{D}$ to a holomorphic function $f$.

Additionally, for each $f_n$ we know $\int_C |f_n| < \infty$ for any contour $C \subset \overline{\mathbb{D}}$. Less strictly, so we don't have enough to use the monotone convergence theorem, we know that $\sum_{n=0}^\infty |\int_C f_n| < \infty$. But additionally, for any closed contour $C^*$ in $\overline{\mathbb{D}}$ we know $\int_{C^*}f_n = 0$.

Must it follow

$$\int_C f = \sum_{n=0}^\infty \int_C f_n$$

I'm asking because all the naive instances where the monotone convergence theorem fail are exempt from these criterion. I think there's something more subtle in this instance.

I've been able to strengthen the condition to a proof that

$$\int_C f = \sum_{n=0}^\infty \int_C f_n \,\,\Leftrightarrow\,\, \sum_{n=0}^\infty \sup_{C\subset \overline{\mathbb{D}}}|\int_Cf_n| < \infty$$

Or rephrase it to

$$\sum_{n=0}^\infty |\int_C f_n| < \infty\,\, \Leftrightarrow \,\,\,f \, \text{can be continuously extended to}\,\overline{\mathbb{D}}$$

Those are the two avenues I've taken. None have really given me an answer.

Any suggestions, comments, questions are greatly appreciated.

Answer 1

Take $f_n(z) = 3z^{3n} - z^n$ and $C = \{ e^{i\theta} \ | \ \theta \in (0,\pi) \}$. It satisfies all of your hypotheses with $$ f(z) = \frac{2+z}{1+z+z^2} $$ and $\int_C f_n =0$ for $n \geq 1$. But $\int_C f$ is not a convergent integral.

Of course the stronger condition $\sum_n \int_C |f_n| < \infty$ is sufficient for your conclusion to hold.