Exact converses to a reverse AM-GM inequality, with applications to sums of independent random variables and (super)martingales

For every given real value of the ratio $\mu:=A_X/G_X>1$ arithmetic and geometric means a positive random variable $X$ $v>0$, exact upper bounds on right- left-tail probabilities $\mathsf{P}(X/G_X\ge v)$ $\mathsf{P}(X/G_X\le are obtained, in terms $\mu$ $v$. In particular, these imply that $X/G_X\to1$ probability as $A_X/G_X\downarrow1$. Such result may be viewed converse to reverse Jensen inequality for strictly concave function $f=\ln$, whereas well-known Cantelli Chebyshev inequalities converses quadratic $f(x) \equiv -x^2$. As applications mentioned new results, improvements Markov, Bernstein--Chernoff, sub-Gaussian, Bennett--Hoeffding given.