Harnack-type bounds for $|f(z)|$ for self-maps of the disk

Let f be a non-constant analytic function on D with \(f ( \mathbb { D } ) \subseteq \mathbb { D } . { \mathrm { ~ U s e ~ } } \psi _ { a } ( f ( z ) )\) (where \(a = f ( 0 )\) \(\psi _ { a } ( z ) = \frac { a - z } { 1 - \bar { a } z } )\) to prove that \({ \frac { | f ( 0 ) | - | z | } { 1 + | f ( 0 ) | | z | } } \leq | f ( z ) | \leq { \frac { | f ( 0 ) | + | z | } { 1 - | f ( 0 ) | | z | } } ,\)