Suppose \(h\) is entire, \[ h(0)=3+4i, \] and \[ |h(z)|\le5 \qquad\text{whenever }|z|<1. \] What is \(h'(0)\)?
Solution
1. The value of \(h\) at the origin has modulus\[ \abs{h(0)}=5. \]
Since \(h(0)=3+4i\), \[ \abs{h(0)} = \sqrt{3^2+4^2} = 5. \]
2. The function \(h\) is constant on the unit disk.
By hypothesis, \[ \abs{h(z)}\leq5 \] whenever \(\abs z<1\). Step 1 shows that equality holds at the interior point \(z=0\). Thus \(h\) attains its maximum modulus at an interior point of the unit disk. By the maximum modulus principle, \(h\) is constant on the unit disk.
3. The requested derivative is\[ \boxed{h'(0)=0}. \]
By step 2, \(h\) is constant on a neighborhood of \(0\), so its derivative at \(0\) is zero.
4. Q.E.D.
Step 3 gives the requested value.