Let Φ be a strictly plurisubharmonic and radial function on the unit disk D ⊂ C and let g be the Kähler metric associated to the Kähler form ω. We prove that if g is g_eucl -balanced of height 3 (where g_eucl is the standard Euclidean metric on C = R^2 ), and the function h(x) = exp(−Φ(z)), x = |z|^2, extends to an entire analytic function on R, then g equals the hyperbolic metric. The proof of our result is based on a interesting characterization of the function f(x) = 1 − x.
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