Does there exist a function which is holomorphic in $|z|<1,$ continuous in $|z|\leq1$ and such that the series $\sum |a\_n|$ is divergent, where $a\_n$'s coefficients in the Taylor series expansion of $f?$



A holomorphic function in the open unit disk satisfying certain properties
Yes there are lots of classical examples by Hardy and Littlewood like say $f(z)=\sum \_{n \ge 1}e^{in \log n}\frac{z^n}{n^{3/4}}$.


Using the second derivative test for exponential sums with $f(u)=(u\log u+u\theta)/(2\pi)$, so $f''(u) \sim 1/M$ when $u \sim M$, the exponential sum $\sum\_{n=M}^{2M}e^{in\log n}e^{in\theta} =O(\sqrt M)$ uniformly in $\theta$ and an easy dyadic argument shows that $\sum\_{n=1}^Ne^{in\log n}e^{in\theta}=O(\sqrt N)$ also uniformly in $\theta$, so it follows that $f$ converges uniformly on the unit circle and obviously satisfies $\sum |a\_n|=\infty$


