When I read Part III of this scientific paper in telecommunication, I see that the author use the following result from the Jensen Inequality
$E\left[ {\max \left( {X,Y} \right)} \right] \geqslant \max \left[ {E\left( X \right)E\left( Y \right)} \right]$
Where $E\left[ . \right]$ is the expectation operator.
Does the inequality still true if we conditioned on a variable $T$ like this:
${E_{\left\{ T \right\}}}\left[ {\left. {\max \left( {X,Y} \right)} \right|T = t} \right] \geqslant \max \left[ {{E_{\left\{ T \right\}}}\left( {\left. X \right|T = t} \right){E_{\left\{ T \right\}}}\left( {\left. Y \right|T = t} \right)} \right]$
Thank you for your enthusiasm !
I'm not sure what you mean by $E_{T}$ But normally, you see: $$ \mathbb{E}_{T}[max(X,Y)| T=t] \ge \mathbb{E}_{T}[X| T=t]$$ and $$ \mathbb{E}_{T}[max(X,Y)| T=t] \ge \mathbb{E}_{T}[Y| T=t]$$ Hence, your conclusion