As far as I understand, Calabi ansatz is (in particular) a way to produce Kähler metrics on total spaces of line bundles (or their disk subbudles) over Kähler manifolds of the following form:

Calabi Ansatz. Let $p:(L,h)\to (M,\omega_M)$ be a Hermitian line bundle over a Kähler manifold $M$. Consider on the total space $L$ the following two-form:

$$\omega_L=p^*(\omega_M)+dd^cf(t).$$

Here $t=t(v)=\log\lvert v\rvert_h$ is the log of the norm function on $L$ defined by $h$ and $f\in C^{\infty}(\mathbb R^1)$.

Question. As far as I understand, $\omega_L$ is Kähler on some disk sub-bundle of the total space $L$ provided $f$ satisfies certain (convexity?) conditions. Are you aware of a good reference on this that would give these conditions on $f$? (I am aware of a few articles, like Hwang–Singer (A momentum construction for circle-invariant Kähler metrics, Transactions of the AMS 2002), but would like something addressing my question more directly.)