In the article, we deal with the monotonicity of the function
$x\rightarrow[ (x^{p}+a )^{1/p}-x]/I_{p}(x)$
on the interval
$(0, \infty)$
for
$p>1$
and
$a>0$
, and present the necessary and sufficient condition such that the double inequality
$[ (x^{p}+a )^{1/p}-x]/a< I_{p}(x)0$
and
$p>1$
, where
$I_{p}(x)=e^{x^{p}}\int_{x}^{\infty}e^{-t^{p}}\,dt$
is the incomplete gamma function.
