Monotonicity and inequalities involving the incomplete gamma function

Abstract

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

Keywords

• incomplete gamma function
• gamma function
• psi function