A Schwarz–Pick bound in terms of $f(0)$

Suppose \(f\) is analytic in the open unit disk \(D\) and \(|f(z)|<1\) for all \(z\in D\). Let \(a=f(0)\) and define \[ g(z)=\frac{f(z)-a}{1-\overline a\,f(z)}. \] Using \(g\) and the fact that \(g\) maps \(D\) to \(D\), prove that \[ |f(z)|\le\frac{|f(0)|+|z|}{1-|f(0)||z|} \qquad(z\in D). \]