# growth scale for entire functions?

I came across this while reading Ya. Levin's book. Let $f$ be an entire function and $r>0$. Define $$M_f(r)=\mbox{max}_{|z|=r}|f(z)|\;.$$ One can express $f$ in power series as $$f(z)=\displaystyle\sum_{n=0}^{\infty}c_nz^n~,~~\displaystyle\lim_{n\rightarrow\infty}\sqrt[n]{|c_n|}=0\;.$$ He asks the question whether $M_f(r)$ can grow arbitrarily fast. This is a nice question, but where would you begin to try to investigate this?

Take any sequence $z_n$ with $|z_n| = n$, and any sequence of positive numbers $v_n$. Find a sequence of polynomials $p_n(z)$ such that $p_n(z_n) = v_n$, $p_n(z_j) = 0$ for $j < n$, and $|p_n(z)| \le 2^{-n}$ on $\{z: |z| < n-1\}$. Then $f(z) = \sum_n p_n(z)$ is entire and has $M_f(n) \ge v_n$.