  It is a well known fact that the union of the Reverse H\"{o}lder classes
coincides with the union of the Muckenhoupt classes $A_p$, but the $A_\infty$
constant of the weight $w$, which is a limit of its $A_p$ constants, is not a
natural characterization for the weight in Reverse H\"{o}lder classes. We
introduce the $RH_1$ condition as a limiting case of the $RH_p$ inequalities as
$p$ tends to 1. Then we show sharp bound on $RH_1$ constant of the weight $w$
in terms of its $A_\infty$ constant (from above and from below). We also prove
the sharp version of the Gehring theorem for the case $p=1$, completing the
answer to the famous question of Bojarski in dimension one.
  We illustrate our results by two straight-forward applications: to the
Dirichlet problem for elliptic PDE's.
