The signs rule for the Laplace integrals with applications

Zhen-Hang Yang () and Jing-Feng Tian ()
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Zhen-Hang Yang: State Grid Zhejiang Electric Power Company Research Institute
Jing-Feng Tian: North China Electric Power University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 4, 1416-1428

Abstract: Abstract 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.

Keywords: Laplace integral; Signs rule; Modified Bessel functions of the second kind; Gamma function; Inequality; Primary 05C38; 15A15; Secondary 05A15; 15A18 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00447-6

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