Suppose \(f\) is analytic in an open set that contains \(\overline{B_r(z_0)}\), and \(\left|\, f(z) \,\right| \le m\) holds on \(\partial B_r(z_0)\) for some constant m. Then for all \(k \ge 0\),

\[ \begin{aligned} \left|\, f^{(k)}(z_0) \,\right| \le \frac{k!\,m}{r^k} \\ \end{aligned} \]

which turns out to be remarkably not hard to prove based on the Cauchy's Integral Formula. If I understand the notation correctly, \(\overline{B_r(z_0)}\) is the disk with radius \(r\) centered at \(z_0\), whereas \(\partial B_r(z_0)\) is the boundary of that disk.

More at Analysis of a Complex Kind.

# posted by rot13(Unafba Pune) @ 2:20 PM