Let $\mathbb{B}$ be the closed unit ball in $\mathbb{C}^n$ and let $g:\mathbb{B}\rightarrow \mathbb{C}$ be a real-analytic radial function such that $g(0)=1$ and $|g(z)|<1\, \forall\, \text{non-zero}\, z\in \mathbb{B}$. It is claimed that there is a small neighbourhood $B$ of $0$ such that $|g(z)|<|g(w)|$ if $z,w\in B$, $z,w\neq 0$ and $|z|>|w|$.
Now from the definition of radial function, we have that $g(z)=g(w)$ whenever $|z|=|w|$. Combining it with real-analyticity I get that $|g|^2$ has a power series expansion (with real coefficients) near zero in the even powers of $|z|$, i.e., in $|z|^{2\alpha}$ where $\alpha$ is a multi-index. But I could not show the monotonicity.
Please help.
This has nothing to do with the complex variables. You can do the same thing in ${\mathbb R}^n$. In fact, it is enough to show this in $n=1$. So we have a function $f(x)$ such that $f(x) = f(-x)$, $f(0)=1$ and $f(x) < 1$ if $x \not= 0$. The function $f$ is real-analytic and nonconstant, so the zeros of $f'$ do not accumulate. So there is an interval around zero where $f'$ is zero only at the origin. Now prove that this means that $f'$ is strictly negative for an interval to the left of 0, and strictly positive to the right of zero. Monotonicity then follows easily.