$|f(z)| + \ln|z| \le 0$, is $f = 0$?

Question

Let $D = \{z\in \mathbb{C}:0<|z|<1 \}$ , $f:D\rightarrow \mathbb{C}$ holomorphic. Given $$|f(z)| + \ln|z| \le 0$$ for all $z\in D$, where $\ln$ denotes the real natural logarithm, is it true that $f(z) = 0,\forall z\in D$?

I've tried Maximum Modulus to conclude $f(z) = 0$ on the boundary of $D$, but i do not know how to proceed. Any hints?

Answer 1

Hint: From $$ 0 \le |z \, f(z)| \le - |z| \ln |z| $$ one gets $$ \lim_{z \to 0} z \, f(z) = 0 $$ so that $f$ has a removable singularity at $z=0$, i.e. it can be extended to a holomorphic function on the unit disk.

Now apply the maximum principle to the extended function in the closed disks with center $z=0$ and radius $0 < r < 1$, and finally consider what happens for $r \to 1$.