  Let $\Phi$ be a strictly plurisubharmonic and radial function on the unit
disk ${\cal D}\subset {\complex}$ and let $g$ be the \K metric associated to
the \K form $\omega =\frac{i}{2}\partial\bar\partial\Phi$. We prove that if $g$
is $g_{eucl}$-balanced of height 3 (where $g_{eucl}$ is the standard Euclidean
metric on ${\complex}={\real}^2$), and the function $h(x)=e^{-\Phi (z)}$,
$x=|z|^2$, extends to an entire analytic function on ${\real}$, then $g$ equals
the hyperbolic metric. The proof of our result is based on a interesting
characterization of the function $f(x)=1-x$.
