Let $f: {\bf C} \rightarrow {\bf C}^n$ be a holomorphic function. For any $R>0$, there exists 
a point $z\_R$ such that $|f(z\_R)|^2 = sup\_{|z|\leq R}(|f(z)|^2)$.
This defines a function $g : {\bf R}^+ \rightarrow {\bf C}$ by $g(R) = z\_R$.


Of course for a given $R$ there are many choices for $z\_R$. 


I was wondering whether it is possible to choose a function $g$ as above 
(choose a $z\_R$ for any $R$) in such a way that $g$ is real analytic - at least on an unbounded interval of ${\bf R}^+$.



maximum modulus function
This function is only piecewise real-analytic. It is not difficult to construct examples when it is discontinuous, no matter how you choose it, and each interval of analyticity is bounded.
All details are somewhat long and inconvenient to explain here, 
but the simple idea is that
you have two disjoint unbounded domains in which your function grows, and is bounded outside these domains, but the rate of growth in each domain oscillates, so that
they overtake each other infinitely many times. It is easy to construct such a subharmonic function and then approximate by $\log|f|$, where $f$ is entire.
More elementary method would be to construct two separate functions in these domains, and then make an entire function by Cauchy integral. 


