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

Abstract

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 $V_X:=\mathsf{Var}\sqrt{X}$, $E_X:=\mathsf{E}(\sqrt{X}-\sqrt{m_X}{)}^2$, $F_X:=\mathsf{E}(\sqrt{M_X}-\sqrt{X}{)}^2$, $m_X:=\inf S_X$, $M_X:=\sup S_X$, and $S_X$ 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 $x_1,\dots ,x_n$.

8 pages; to appear in the Bulletin of the Australian Mathematical Society. Version 2: the condition that the random variable X is nonnegative was missing in the abstract

References in corpus (1)

• Tchebycheff systems and extremal problems for generalized moments: a brief survey