A problem on the maximal modulus

Question

Let $f$ be a transcendental entire function, we know that $\log M(r, f)$, with $M(r,f)=\max_{|z|=r}|f(z)|$, is a convex function with respect to $\log r$ and $\lim\limits_{r\rightarrow\infty}\frac{\log M(r,f)}{\log r}=\infty$.

My question is: for any function $\rho : [1, +\infty)\rightarrow [0, \infty)$ such that $\log \rho(r)$ is a convex function with respect to $\log r$ in $(1, +\infty)$ and $\lim\limits_{r\rightarrow\infty}\frac{\log \rho(r)}{\log r}=\infty$, can we always find a transcendental entire function $f$ such that $M(r, f)=\rho(r)$ for any $r\geq M$, where $M$ is some large constant? Any references and comments would be appreciated.

Answer 1

It is easy to show that $M(r)$ is piece-wise analytic, so it cannot be an arbitrary function whose $\log$ is convex with respect to $\log$. On the other hand, arbitrary such function can be approximated by $M(r)$ of an entire function, in the sense that $\log M(r)\sim\log\rho(r)$:

MR0176075 Clunie, J. On integral functions having prescribed asymptotic growth. Canad. J. Math. 17 1965 396–404.

The accuracy of approximation can be improved to $O(\log r)$, see Yulmukhametov, R. S. Approximation of subharmonic functions. (Russian. English summary) Anal. Math. 11 (1985), no. 3, 257–282.

Remark. For germs of functions analytic near the origin, possible functions $M(r)$ have been completely characterized:

MR0045209 Hayman, W. K. A characterization of the maximum modulus of functions regular at the origin. J. Analyse Math. 1, (1951). 135–154.

Answer 2

No, there are more restrictions. For example, if $\rho(r)=r^N$ on some interval $r_1\le r\le r_2$, then $g(z)=f(z)z^{-N}$ takes its maximum value also at interior points of the annulus $r_1\le |z|\le r_2$, so must be constant.

This means that in your original question, a $\rho$ that equals $r^{N_j}$ on a sequence of intervals $I_j$ is not possible.