  In this note, we consider Jensen's inequality for the nonlinear expectation
associated with backward SDEs driven by $G$-Brownian motion ($G$-BSDEs for
short). At first, we give a necessary and sufficient condition for $G$-BSDEs
under which one-dimensional Jensen inequality holds. Second, we prove that for
$n>1$, the $n$-dimensional Jensen inequality holds for any nonlinear
expectation if and only if the nonlinear expectation is linear, which is
essentially due to Jia (Arch. Math. 94 (2010), 489-499). As a consequence, we
give a necessary and sufficient condition for $G$-BSDEs under which the
$n$-dimensional Jensen inequality holds.
