Prove that the distance of the nearest zero of the function $f(z)={\sum}_{s=0}^{\infty}\: c_nz^n$ to the point $z=0$ is not less than $\frac{r|c_0|}{M+|c_0|}$, where r is any number not exceeding the radius of convergence of the series, and $M(r)= \max\limits_{|z|=r}|f(z)|$.
I followed the hint to first establish that the function $f(z)$ has no zero in the domain where $|f(z)-c_0|<|c_0|$. The next hint is to estimate $|f(z)-c_0|$ using Cauchy's inequality. But, I am getting $|c_0|<M$ using that inequality and unable to proceed further. Please suggest the way forward. Thanks.
$g(z)=\frac{f(z)-c_0}{z}$ has the same radius of convergence as $f$. By the maximums principle, $$ g(z_0)< \sup_{|z|=r}|g(z)|\le \frac{M+|c_0|}{r} $$ for all $|z_0|<r$. For the Rouché theorem to show no roots by comparing $f$ to the constant function $c_0$ we need $$ |f(z)-c_0|< |c_0|\quad\forall |z|=\rho $$ or $$ |g(z)|<\frac{|c_0|}{\rho}. $$ This is satisfied if $\rho<r$ and $$ \frac{M+|c_0|}{r}\le \frac{|c_0|}{\rho} \iff \rho\le \frac{|c_0|r}{M+|c_0|}. $$