Monotonicity of the incomplete gamma function with applications

In the article, we discuss the monotonicity properties of the function x -> (1 - e(-axp))(1/p) / integral(x)(0) e(-tp) dt for a, p > 0 with p not equal 1 on (0, infinity) and prove that the double inequality Gamma(1 + 1/p)(1 - e(-axp))(1/p) < integral(x)(0) e(-tp) dt < Gamma (1 + 1/p)(1 - e(-bxp))(1/p) holds for all x > 0 if and only if a <= min{1, Gamma(-p)(1 + 1/p)} and b >= max{1, Gamma(-p)(1 + 1/p)}.