In the article, we discuss the monotonicity properties of the function
$x\rightarrow (1-e^{-ax^{p}} )^{1/p}/\int_{0}^{x}e^{-t^{p}}\,dt$
for
$a, p>0$
with
$p\neq1$
on
$(0, \infty)$
and prove that the double inequality
$\Gamma(1+1/p) (1-e^{-a x^{p}} )^{1/p}
