Is there any holomorphic function in a unit ball such that $f(1/n)=\frac{1}{n \ln{n}}$

Question

Is there any holomorphic function in a unit ball such that $f(1/n)=\frac{1}{n \ln{n}}$ for $n=2,3,\dots$

Can somebody point me in the right direction? Only thing I know is that $f(0)=0$ if such function exists...

Answer 1

Assume that $f(1/n)=1/(n\log n)$. Then $f(0)=0$, by continuity of $f$, and thus $f(z)=zg(z)$, where $g$ is also holomorphic in the unit disc. Now $$ g\left(\frac{1}{n}\right)=\frac{1}{\log n}. $$ But this means that $g(0)=0$, due to continuity of $g$ at $z=0$, and thus $g(z)=zh(z)$, where $h$ is holomorphic in the unit ball. Now we have that $$ g\left(\frac{1}{n}\right)=\frac{1}{n}h\left(\frac{1}{n}\right)=\frac{1}{\log n}, $$ and hence $$ h\left(\frac{1}{n}\right)=\frac{n}{\log n}\to\infty, $$ as $n\to \infty$, while $h(1/n)\to h(0)\in\mathbb C$. Contradiction. Thus no such $f$ exists!