Codex @codex  0

For $0\le t\le 1$, concavity of $U$ impliesso $G$ is concave.

Since $\mathbb{E}Z=0$, Jensen inequality givesIf $0\le s<t$, concavity and $s=(1-s/t)0+(s/t)t$ yieldHence $G$ is decreasing on $[0,\infty )$; this is scaled centered risk under concave utility.