  Let $X$ denote a nonnegative random variable with $\mathsf{E} X<\infty$.
Upper and lower bounds on $\mathsf{E} X-\exp\mathsf{E}\ln X$ are obtained,
which are exact, in terms of $V_X$ and $E_X$ for the upper bound and in terms
of $V_X$ and $F_X$ for the lower bound, where $V_X:=\mathsf{Var}\sqrt X$,
$E_X:=\mathsf{E}\big(\sqrt X-\sqrt{m_X}\,\big)^2$,
$F_X:=\mathsf{E}\big(\sqrt{M_X}-\sqrt X\,\big)^2$, $m_X:=\inf S_X$, $M_X:=\sup
S_X$, and $S_X$ is the support set of the distribution of $X$. Note that, if
$X$ takes each of distinct real values $x_1,\dots,x_n$ with probability $1/n$,
then $\mathsf{E} X$ and $\exp\mathsf{E}\ln X$ are, respectively, the arithmetic
and geometric means of $x_1,\dots,x_n$.
