Show that $f$ given by a power series has no roots in $B_r^{\mathbb{C}}(0)$

Question

One has a convergent power series $f(z)= 1+ \sum_{n=1}^\infty a_n z^n$ on $B_R^{\mathbb{C}}(0)$ (open ball) for some $R>0$, $~0<\rho<R$, $~M_\rho(f):=\max\{|f(z)|:|z|=\rho\}$ and $r:=\frac{\rho}{1+M_\rho(f)}$. How can I show that such $f$ does not have any roots in $B_r^\mathbb{C}(0)$?

Answer 1

The function $g(z)={f(z)-1\over z}$ is holomorphic for $|z|<R.$ We have $$|g(z)|\le {1+M_{\rho}(f)\over \rho},\quad |z|=\rho $$ By the maximum modulus principle we get $$|g(z)|\le {1+M_{\rho}(f)\over \rho},\quad |z|\le\rho $$ Thus $$|f(z)-1|\le {1+M_{\rho}(f)\over \rho}\,|z|<1$$ provided that $$|z|<{\rho \over 1+M_{\rho}(f)}$$ The inequality $|f(z)-1|<1$ implies $f(z)\neq 0.$