TITLE: The Bound of the 8th Derivative of an Analytic Function
QUESTION [8 upvotes]: 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 $|f^{(8)}(\frac{i}{4})|\leq 120$.

REPLY [4 votes]: 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 $a$,
$$|f_a^{(8)}(\frac{i}{4})|=a^8e^{-a}>120.$$