I've got this question that I've been trying to prove using no Taylor/Laurent series, no use of Residue theorem, only using the Cauchy's integral formula but have not made any progress on. unfortunately
Let f is holomorphic on $D_1(0)$ and is such that $max_{z \in C_r(0)} |f(z)| \to 0$ as $r \to 1$. Prove that $f$ is identically zero.
I would appreciate any helps.