I´m working on a book and in a proof it says from
$\max_{0\leq\theta\leq 2\pi}Re(F(Re^{i\theta}))\ll R(\log(R))^2$
follows with
$\sup_{n\ge1}\frac{R^n}{n!}\vert F^{(n)}(0)\vert\leq 2A$ (Borel-Caratheodory)
that F is an linear function. But hwo do you consider that? I only get, that $F^{(n)}\ll\log(R)^2$. What is the idea to get it
For notice :
$F(s)$ is a ,for $\vert s\vert\leq R,~R\in\mathbb{R}_{>0}$, holomorph function with $F(0)=0$.
$A:=\max_{\vert s\vert=R}Re(F(s))$
Since $A(R)=R(\log(R))^2$ we get from the Borel-Caratheodry relations $\vert F^{(n)}(0)\vert\leq 2n!\frac{(\log(R))^2}{R^{n-1}}$
If $n \ge 2$ RHS converges to $0$ as $R \to \infty$ so we get the result