  We consider the $p$-generalized arithmetic-geometric mean inequality for
vectors chosen randomly from the $\ell_p^n$-ball in $\mathbb{R}^n$. In this
setting the inequality can be improved or reversed up to a respective scalar
constant with high probability, and central limit theorems and large deviation
results with respect to this constant have been shown. We sharpen these large
deviation results in the spirit of Bahadur and Ranga Rao, thereby providing
concrete and asymptotically exact estimates on a non-logarithmic scale for the
probability of the inequality being improvable or reversible up to a constant,
respectively.
