The metric on $E^{\vee}$ is given, for $p\in\mathbb{P}^{n}(\mathbb{C})$ and $q_{0},\dots,q_{r}\in\mathbb{C}$, by

$$||q_{0}s_{0}(p)\oplus\dots\oplus q_{r}s_{r}(p)||_{\infty}^{2}=|q_{0}|^{2}||s_{0}(p)||^{2}+\dots+|q_{r}|^{2}||s_{r}(p)||^{2},$$

where $||s_{j}(p)||$ is the norm of $s_{j}(p)$ with respect to the Fubini-Study metric on ${\mathcal{O}}(-a_{j})^{{\text{\rm an}}}$. By Example 2.2,

$$||s_{j}(p)||^{2}=\bigg(\frac{|p_{0}|^{2}}{|p_{0}|^{2}+\dots+|p_{n}|^{2}}\bigg)^{-a_{j}}.$$

Let $s^{\otimes-1}$ be the monomial section of the tautological line bundle defined by $s$. Then

(8.18)

$$\|s^{\otimes-1}\circ\theta_{0}\circ{\operatorname{\mathbf{e}}}_{\mathbb{C}}(u,v)\|^{2}=\sum_{j=0}^{r}{\operatorname{e}}^{-2v_{j}}\left(\sum_{i=0}^{n}{\operatorname{e}}^{-2u_{i}}\right)^{a_{j}}.$$

By Proposition 5.19(2), $\psi_{\infty}$ is $-1/2$ times the logarithm of the above expression.

For the last statement, observe that the functions ${\operatorname{e}}^{-2v_{j}}(\sum_{i=0}^{n}{\operatorname{e}}^{-2u_{i}})^{a_{j}}$ are log-strictly convex, because $-1/2$ times their logarithm is the function associated to the Fubini-Study metric on ${\mathcal{O}}(a_{j})^{{\text{\rm an}}}$, which is a strictly concave function. Their sum is also log-strictly convex [BV04, §3.5.2]. Hence, $\psi_{\infty}$ is strictly concave. ∎