the Cauchy estimates
/ koh-SHEE /
Once you can write each derivative as a contour integral, you can also bound its size, because an integral of something small over a short path is small. The Cauchy estimates turn the integral formula for derivatives into a clean inequality: if you know how big f is on a circle, you know how big its derivatives can possibly be at the center.
The statement: suppose f is holomorphic on and inside the circle of radius R centered at z_0, and |f(z)| is at most M on that circle. Then for every n, |f^(n)(z_0)| is at most n! M / R^n. The proof is just the ML inequality (length times max) applied to f^(n)(z_0) = (n! / (2 pi i)) times the integral over the circle of f(z) / (z - z_0)^(n+1) dz: the circle has length 2 pi R, the kernel has size 1 / R^(n+1), and |f| is at most M, so the modulus of the integral is at most (n! / (2 pi)) times (2 pi R) times M / R^(n+1) = n! M / R^n. That is the whole argument.
These bounds are the lever behind several headline results. Liouville's theorem is the case n = 1: if a bounded entire function has |f| at most M everywhere, you can take R as large as you like, and |f'(z_0)| at most M / R forces f' = 0, so f is constant. The estimates also control how fast Taylor coefficients can grow, which is exactly what you need to prove power series converge. A growing R shrinking a derivative bound is the recurring trick worth memorizing.
If |f| is at most 5 on the circle |z| = 2 and f is holomorphic inside, then |f'(0)| is at most 1! times 5 / 2 = 2.5 and |f''(0)| is at most 2! times 5 / 2^2 = 2.5.
Bounding derivatives at the center from the maximum of |f| on a surrounding circle.
The bound uses the maximum of |f| on the circle, not at the center; a small value at z_0 says nothing — it is the surrounding values that constrain the derivatives.