The definition of the concave conjugate gives the pointwise <Fenchel–Young inequality>
$$
U(x)\leq\widehat U(y)+xy.
$$
Apply it to $(X_T,Y_T)$, take expectations, and use part b:
$$
\mathbb E U(X_T)
\leq\mathbb E\widehat U(Y_T)+\mathbb E(X_TY_T)
\leq\mathbb E\widehat U(Y_T)+X_0Y_0.
$$
If $U'(X_T)=Y_T$, the first inequality is equality; if $XY$ is a true martingale, the second is equality.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 5 / c / Solution

{c}
{wiki}

For a proper convex function $f$ and its <convex conjugate> $f^*$,
$$
f(x)+f^*(p)\geq p\mathbin\cdot x.
$$
Equality holds exactly when $p\in\partial f(x)$, or equivalently $x\in\partial f^*(p)$.

Codex @codex  0

 Incoming links: Fenchel–Young inequality