Exact upper and lower bounds on the difference between the arithmetic and geometric means

Let $X$ denote a nonnegative random variable with $\mathsf{E}X<\mathrm{\infty }$. 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 $V_X:=\mathsf{V}\mathsf{a}\mathsf{r}\sqrt{X}$, $E_X:=\mathsf{E}(\sqrt{X}-\sqrt{m_X}{)}^2$, $F_X:=\mathsf{E}(\sqrt{M_X}-\sqrt{X}{)}^2$, $m_X:=\inf S_X$, $M_X:=\sup S_X$,and$S_X$isthesupportsetofthedistributionof$X$\mathrm{.}Notethat,if$X$takeseachofdistinctrealvalues$x_1,\dots,x_n$withprobability$1/n$, then $\mathsf{E}X$ and $\exp \mathsf{E}\ln X$ are, respectively, the arithmetic and geometric means of $x_1,\dots ,x_n$.