The Bound of the 8th Derivative of an Analytic Function This is another question from a recent qualifying exam that really stumped me.  I was thinking of using something with the Cauchy estimate for derivatives, but was clueless beyond that.
Let $f:[-1,1]\times [0,2]\rightarrow \mathbb{C}$ be real valued on the interval $[-1,1]$ in the $x$-axis.  Show that if $|f|\leq 1$ on its domain, and is analytic on the interior of its domain, then $|f^{(8)}(\frac{i}{4})|\leq 120$.
 A: Based on Pavel M's answer, a simpler counter-example is $f_a(z)=e^{a(z-1)}$, where $a>0$ satisfies that $a^8>120e^a$. The range of $a$ is larger than $[2.5,18.5]$.
Write $z=x+iy$. On the one hand, $|f_a(z)|=f_a(x)=e^{a(x-1)}\le 1$ when $x\le 1$, so $f_a$ satisfies the conditions. On the other hand, since $f_a^{(8)}(z)=a^8f_a(z)$, according to the choice of $a$,
$$|f_a^{(8)}(\frac{i}{4})|=a^8e^{-a}>120.$$
A: By the reflection principle, we can extend $f$ to the larger rectangle $R=[-1,1]\times [-2,2]$, and the upper bound $|f|\le 1$ holds for the extended function as well. The rectangle $R$ contains a disk of radius $1$ centered at $i/4$. Applying the Cauchy integral formula on this disk, we find that $|f^{(n)}(i/4)|\le n!$ for all $n$. In particular, $|f^{(5)}(i/4)|\le 120$, which I believe was the intent of the problem. For the 8th derivative we get $|f^{(8)}(i/4)|\le 40320$. 

Indeed, I think I have a counterexample to the stated bound on $f^{(8)}(i/4)$. The idea is to map $R$ into the unit disk $\mathbb D$. Mapping rectangles is hard (elliptic functions, whatever). Instead I'm going to map a larger domain: the vertical strip $S=\{z:-1<\operatorname{Re}z<1\}$. This is a standard exercise with conformal maps: 
$$\psi(z)= i\,\frac{\exp(\pi i z/2)-1}{\exp(\pi i z/2)+1}$$
maps $S$ onto $\mathbb D$ and is real on the segment $[-1,1]$ (this segment is sent onto a half of the unit circle by the exponential map, and then onto horizontal diameter by the Möbius map). 
No, $\psi$ is not my example. After all, $|\psi^{(8)}(i/4)|<24$ as you can find from your friendly computer algebra system, e.g., evalf(eval(diff(psi,z$8),z=Pi*I/4)) in Maple. When maximizing the 8th derivative, we should take the 8-fold cover: $f(z)=\psi(z)^8$. The same Maple thing now tells me that $|f^{(8)}(i/4)|>1800$.
