If $P(z)=\sum_{k=0}^na_kz^k, (a_n=1)$ having no zeros in $|z|<1,$ I am trying to prove 
$$\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}\leq \frac{n-1}{2}+\frac{1}{1+|a_0|}.$$
The result is true for $n=1.$ Can I apply induction principle to prove this?