Proposition 11.39.1675DA440E964

Proof. Fix $G$, and let $u_\varphi ,u_\psi $ solve the equations in the ordinary types of $\varphi $ and $\psi $, with constants $c_\varphi ,c_\psi $. Then $u_\varphi \preceq u_\psi $. If $c_\varphi >c_\psi $, the singular strict domination principle applied to $u_\psi $ and $u_\varphi +t$ would give $u_\psi \geq u_\varphi +t$ for every real $t$, which is impossible at a point where both are finite. This proves the monotonicity of $B$, and taking the defining supremum gives the monotonicity of $H$. Since $\varphi \preceq 0$ and the zero potential is itself a model reference, this also proves both upper bounds.

For the lower bound, let $v$ be a bounded solution for $G$ with the zero reference, with constant $c_0$, and put

The two summands are $\varepsilon \theta $- and $(1-\varepsilon )\theta $-plurisubharmonic, respectively, so positivity and homogeneity give

As in the proof of lemma 15.12.2, this inequality is justified first on finite truncation levels and then off the pole set of $P_{\varepsilon \theta }(\varphi )$, which has zero smooth volume. Also $h\preceq u_\varphi $. If $c_\varphi <(1-\varepsilon )^ nc_0$, strict domination would force $u_\varphi \geq h+t$ for every $t$. Thus $c_\varphi \geq (1-\varepsilon )^ nc_0$. Taking logarithms proves the scalar bound, and taking the defining suprema proves the dual bound. $\square$