  We prove that the best so far known constant $c_p=\frac{p^{-p}}{1-p},\,
p\in(0,1)$ of a domination inequality, which originates to Lenglart, is sharp.
In particular, we solve an open question posed by Revuz and Yor. Motivated by
the application to maximal inequalities, like e.g. the Burkholder-Davis-Gundy
inequality, we also study the domination inequality under an additional
monotonicity assumption. In this special case, a constant which stays bounded
for $p$ near $1$ was proven by Pratelli and Lenglart. We provide the sharp
constant for this case.
