Suppose \(h\) is entire, \(h(0)=3+4i\), and \(|h(z)|\le5\) whenever \(|z|<1\). What is \(h'(0)\)?
Solution
We have \(| h ( 0 ) | = \sqrt { 9 + 1 6 } = 5\) , so \(| h ( 0 ) | \geq | h ( z ) |\) for \(z \in D = \{ | z | < 1 \}\) . By the maximum modulus principle this is only possible if \(h ( z )\) is constant on \(D\) , which implies that \(h ^ { \prime } ( 0 ) = 0\)