This is an exercise from Conway p.130. I am supposed to use the maximum modulus principle but I can't get the knack of how to solve this problem. I can't see where to start... Also I have no idea how log(M/a)/log2 occurs.... Could anyone please help me with this problem?
First observe that, if $|w|<\dfrac{R}{3}$ and $|z|=R$, then $$ \left|\frac{1}{1-\frac{z}{w}}\right|<\frac{1}{2}. $$ Hence $$ a=|g(0)|\le \max_{|z|=R}|g(z)|\le \max_{|z|=R}|\,f(z)|\cdot\prod_{k=1}^n\left|\frac{1}{1-\frac{z}{z_k}}\right|<\frac{M}{2^n}. $$ Thus $$ n\log 2<\log\left(\frac{M}{a}\right). $$