Solution
= Solution
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.