Q:

Location of the zeros set of holomorphic function

Recently I proved the following result.
"If a holomorphic function $f$ maps the unit disc $\Delta$ into the unit disk $\Delta $ with $0<|f(0)|$ then $f$ doesn't vanish in the disk $D(0,|f(0)|)$. "
I would like to know whether this result is well-known or how it can be useful. I will appreciate any kind of help. Thanks.

A:

The result is trivial and follows from Schwarz lemma: if $f(z_1)=0$, then $$|f(\frac{z+z_1}{1+z\bar z_1})|\le |z|$$ so $|f(0)|\le |-z_1|$.

