PomeCorollary 16.5.23.555B1CEACB0F The function $E^{\phi }_{\theta }\colon \mathrm{PSH}(X,\theta ;\phi )\rightarrow [-\infty ,\infty )$ is concave.
Corollary 16.5.23555B1CEACB0F
Proof. Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta ;\phi )$ and $t\in (0,1)$. We need to show that
3BD672AA942B
16.5.24
\begin{equation} E^{\phi }_{\theta }\Bigl ( t\varphi +(1-t)\psi \Bigr )\geq tE^{\phi }_{\theta }(\varphi )+(1-t)E^{\phi }_{\theta }(\psi ). \end{equation}
For this purpose, observe that when $C\to \infty $, we have
\[ t\Bigl (\varphi \lor (\phi -C)\Bigr )+(1-t)\Bigl (\psi \lor (\phi -C)\Bigr )\searrow t\varphi +(1-t)\psi . \]
Observe that
\[ \varphi \lor (\phi -C),\psi \lor (\phi -C)\in \mathcal{E}^{\infty }(X,\theta ;\phi ). \]
By 16.5.18, for each $C>0$, we have
\[ E_{\theta }^{\phi }\Biggl ( t\Bigl (\varphi \lor (\phi -C)\Bigr )+(1-t)\Bigl (\psi \lor (\phi -C)\Bigr ) \Biggr )\geq tE_{\theta }^{\phi }\Bigl (\varphi \lor (\phi -C)\Bigr )+(1-t)E_{\theta }^{\phi }\Bigl (\psi \lor (\phi -C)\Bigr ). \]
Letting $C\to \infty $, we conclude (16.5.24) using proposition 16.5.20. $\square$
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