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The signs rule for the Laplace integrals with applications

Author

Listed:
  • Zhen-Hang Yang

    (State Grid Zhejiang Electric Power Company Research Institute)

  • Jing-Feng Tian

    (North China Electric Power University)

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.

Suggested Citation

  • Zhen-Hang Yang & Jing-Feng Tian, 2024. "The signs rule for the Laplace integrals with applications," Indian Journal of Pure and Applied Mathematics, Springer, vol. 55(4), pages 1416-1428, December.
  • Handle: RePEc:spr:indpam:v:55:y:2024:i:4:d:10.1007_s13226-023-00447-6
    DOI: 10.1007/s13226-023-00447-6
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    References listed on IDEAS

    as
    1. Yang, Zhen-Hang & Chu, Yu-Ming & Zhang, Wen, 2019. "High accuracy asymptotic bounds for the complete elliptic integral of the second kind," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 552-564.
    2. Belzunce, Felix & Ortega, Eva-Maria & Ruiz, Jose M., 2007. "On non-monotonic ageing properties from the Laplace transform, with actuarial applications," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 1-14, January.
    3. Zhen-Hang Yang & Yu-Ming Chu & Xiao-Jing Tao, 2014. "A Double Inequality for the Trigamma Function and Its Applications," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, July.
    Full references (including those not matched with items on IDEAS)

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