Recall that we are assuming that $\omega_{0}=f^{*}\omega_{Z}$ where $f:X\to Z$ is a holomorphic map. We can then use the Chern-Lu formula that appears in Yau’s Schwarz lemma computation [Y2, To1] and get

$$\Delta_{\tilde{\omega}_{t}}\log\textrm{tr}_{\tilde{\omega}_{t}}\omega_{0}\geq-A\textrm{tr}_{\tilde{\omega}_{t}}\omega_{0},$$

for a uniform constant $A$. Noticing that

$$\Delta_{\tilde{\omega}_{t}}\varphi_{t}=n-\textrm{tr}_{\tilde{\omega}_{t}}\omega_{t}\leq n-\textrm{tr}_{\tilde{\omega}_{t}}\omega_{0},$$

we see that

(3.2)

$$\Delta_{\tilde{\omega}_{t}}(\log\textrm{tr}_{\tilde{\omega}_{t}}\omega_{0}-(A+1)\varphi_{t})\geq\textrm{tr}_{\tilde{\omega}_{t}}\omega_{0}-n(A+1).$$

Then the maximum principle applied to (3.2), together with the estimate (2.8), gives (3.1). ∎