The signs rule for the Laplace integrals with applications

Let the function $$f\left( t\right) $$ f t have a unique zero on $$\left( 0,\infty \right) $$ 0 , ∞ . Then the signs of $$F\left( x\right) =\int _{0}^{\infty }f\left( t\right) e^{-xt}dt$$ F x = ∫ 0 ∞ f t e - x t d t on $$\left( 0,\infty \right) $$ 0 , ∞ depends on the sign of $$ F\left( 0^{+}\right) $$ F 0 + . As applications, a double inequality involving the modified Bessel functions of the second kind is extended, and a new proof of Alzer’s inequalities for the gamma function is presented. Finally, we introduce a notion of incompletely monotonic functions of order n, which reduces to the usual notion of completely monotonic functions when the order is infinite.