Taylor Series

[WannaBe22]

Homework Statement

Let $$f(z)=\sum_{n=0}^{\infty} a_n z^n$$ be analytic at {z: |z|<R} and satisfies:
$$|f(z)| \leq M$$ for every |z|<R.
Let's define: d=the distance between the origin and the closest zero of f(z).

Prove: $$d \geq \frac{R|a_0|}{M+|a_0|}$$.

Hope you'll be able to help me


Thanks !

Homework Equations

The Attempt at a Solution

I've tried using Cauchy's Inequality... But it doesn't give anything new for $$a_0$$.
I've also tried isolating $$a_0$$ from this inequality, but it gives me nothing...


Hope someone will be able to help me

Thanks in advance