Assume again $X=\mathbb{C}\mathbb{P}^{n}$ and $\omega$ is the Fubini-Study Kähler form. Consider the totally real subspace $\mathbb{R}\mathbb{P}^{n}$ of points with real coordinates (the closure of $\mathbb{R}^{n}\subset\mathbb{C}^{n}$ in $\mathbb{C}\mathbb{P}^{n}$). Then

$$\frac{1}{2(1+\sqrt{2})}\leq T_{\omega}(\mathbb{R}\mathbb{P}^{n})\leq 1.$$

Indeed set $B_{\mathbb{R}^{n}}:=\mathbb{R}^{n}\cap\mathbb{B}^{n}$. It follows from the discussion above that

$$T_{\omega}(\mathbb{R}\mathbb{P}^{n})\geq\frac{1}{2}T_{\mathbb{B}^{n}}(\mathbb{B}_{\mathbb{R}^{n}}).$$

Now there is an explicit formula for $L_{B_{\mathbb{R}^{n}}}^{*}$ (Lundin’s formula, see [27]),

$$L_{B_{\mathbb{R}^{n}}}^{*}(z)=\sup\{\log^{+}|h(<z,\xi>)|\,/||\xi||=1\},\;z\in\mathbb{C}^{n},$$

where $h(\zeta)=\zeta+\sqrt{\zeta^{2}-1}$. A simple computation yields $|h(\zeta)|\leq\log[|z|+\sqrt{|z|^{2}+1}]$ for $\zeta=<z,\xi>$ with $||\xi||=1$. We infer

$$L_{\mathbb{B}_{\mathbb{R}^{n}}}(z)\leq\log\left[|z|+\sqrt{|z|^{2}+1}\right]\leq\log[1+\sqrt{2}]\;\text{ in }\mathbb{B}^{n},$$

which yields the desired inequality.

Observe that the minorant is independent of the dimension $n$. This has been used recently in complex dynamics by Dinh and Sibony [18].