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Monotonicity and inequalities involving the incomplete gamma function

Monotonicity and inequalities involving the incomplete gamma function

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 $[ (x^{p}+a )^{1/p}-x]/a< I_{p}(x)<[ (x^{p}+b )^{1/p}-x]/b$ for all $x>0$ and $p>1$ , where $I_{p}(x)=e^{x^{p}}\int_{x}^{\infty}e^{-t^{p}}\,dt$ …