In the introductory chapter of S. G. Krantz's "Complex Analysis: The Geometric Viewpoint" (second edition), he writes the Cauchy estimate theorem in the following form:
Let $F$ be a holomorphic function on a domain $U$ that contains the closed disc $\bar{D}(P, R)$. Let $M$ be the supremum of $|F|$ on $\bar{D}(P, R)$. Then the derivatives of F satisfy the estimates
$|(\frac{\partial^j}{\partial z^j}) F(P)| \leq \frac{j! \cdot M}{R^j}$.
He then goes onto comment that "notice that the Cauchy estimates tell us that if $F$ is bounded on a large disc, then its derivatives are relatively small at the center of the disc".
Could someone explain please why the derivatives are small only at the center of the disc, as opposed to the boundary (for a large enough disc). If this is to do with how the value of $M$ changes from the center of the disc outwards, could there also be a small discussion about the phenomenology of $M$ as this may be where the trouble lies.
Thanks and best.
The Cauchy estimates are inversely proportional to the distance from the point to the boundary, so the general statement follows as the center is farthest, so it should have best estimates apriori without anything else known- this can be made precise for various classes of functions.
For example, assume we have a holomoprhic function $f: \mathbb D \to \mathbb D, f(0)=0$, then $|f'(0)| \le 1$ but $|f'(a)| \le \frac{1-|f(a)|^2}{1-|a|^2}$ and for example if we take the function:
$f(z)=z\frac{z-a}{1-\bar a z}=\frac{z}{\bar a}(-1+\frac{1-|a|^2}{1-\bar a z}), 0< |a| <1$,
then $f'(a)=\frac{a}{1-|a|^2}$ so in particular we can make it quite big when we choose $a$ close to the unit circle, although $|f(z)| < 1, |z|<1$