Positivity criterion for a radial Kähler potential ($u^{\prime \prime }(t)>0,{\lim }_{t\to -\infty }{e}^{-t}{u}^{\prime \prime }(t)>0$)

For a potential already smooth on all of $\mathbb{C}$, the associated form is a Kähler form exactly when the displayed two positivity conditions hold. The first gives a positive metric coefficient off the origin; the second is positivity of $f_{z\bar{z}}(0)$. The associated real metric is $2f_{z\bar{z}}(d{x}^2+d{y}^2)$. Positivity away from the origin alone permits a degenerate metric at the origin.