Radial balanced metrics on the unit disk

Let ΦΦ be a strictly plurisubharmonic and radial function on the unit disk D⊂CD⊂C and let gg be the Kähler metric associated to the Kähler form ω=i2∂∂̄Φ. We prove that if gg is geuclgeucl-balanced of height 3 (where geuclgeucl is the standard Euclidean metric on C=R2C=R2), and the function h(x)=e−Φ(z), x=|z|2x=|z|2, extends to an entire analytic function on RR, then gg equals the hyperbolic metric. The proof of our result is based on a interesting characterization of the function f(x)=1−xf(x)=1−x.