$f$ holomorphic in $\mathbb{D}$. Prove $f$ has a zero in $\mathbb{D}$ Let $f$ be holomorphic in $\mathbb{D}$ and $f$ be continuous on $\overline{\mathbb{D}}$. Assume $f(0)=c$ and $|f(z)|>|c|$ for $|z|=1$. Prove that $f$ has a zero in $\mathbb{D}$.
Since it's dealing with the number of zeros (or existence of), My initial thought is to find another function $g(z)$ such that $|g(z)-f(z)|<|g(z)|$ for all $z\in\partial\mathbb{D}$, then show that $g$ has at least one zero in $\mathbb{D}$ and use Rouche's theorem to complete the proof. Another approach I thought of is using Argument principle, and showing $\displaystyle\int\dfrac{f'}{f}\geq1$. But I maybe in completely wrong track. 

Hint: Suppose $f$ has no zero in $\mathbb D.$ Consider $1/f.$
