Monotonicity of the incomplete gamma function with applications

Zhen Hang Yang; Wen Zhang; Yu Ming Chu

doi:10.1186/s13660-016-1197-7

Monotonicity of the incomplete gamma function with applications

Zhen Hang Yang
, Wen Zhang
, Yu Ming Chu

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

In the article, we discuss the monotonicity properties of the function x→(1−e−axp)1/p/∫0xe−tpdt for a, p> 0 with p≠ 1 on (0 , ∞) and prove that the double inequality Γ(1+1/p)(1−e−axp)1/p<∫0xe−tpdt<Γ(1+1/p)(1−e−bxp)1/p holds for all x> 0 if and only if a≤ min {1 , Γ ⁻ ^p(1 + 1 / p) } and b≥ max { 1 , Γ ⁻ ^p(1 + 1 / p) }.

Original language	English
Article number	251
Journal	Journal of Inequalities and Applications
Volume	2016
Issue number	1
DOIs	https://doi.org/10.1186/s13660-016-1197-7
State	Published - 1 Dec 2016
Externally published	Yes

Keywords

gamma function
incomplete gamma function
psi function

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10.1186/s13660-016-1197-7

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title = "Monotonicity of the incomplete gamma function with applications",

abstract = "In the article, we discuss the monotonicity properties of the function x→(1−e−axp)1/p/∫0xe−tpdt for a, p> 0 with p≠ 1 on (0 , ∞) and prove that the double inequality Γ(1+1/p)(1−e−axp)1/p<∫0xe−tpdt<Γ(1+1/p)(1−e−bxp)1/p holds for all x> 0 if and only if a≤ min \{1 , Γ − p(1 + 1 / p) \} and b≥ max \{ 1 , Γ − p(1 + 1 / p) \}.",

keywords = "gamma function, incomplete gamma function, psi function",

author = "Yang, \{Zhen Hang\} and Wen Zhang and Chu, \{Yu Ming\}",

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T1 - Monotonicity of the incomplete gamma function with applications

AU - Yang, Zhen Hang

AU - Zhang, Wen

AU - Chu, Yu Ming

PY - 2016/12/1

Y1 - 2016/12/1

N2 - In the article, we discuss the monotonicity properties of the function x→(1−e−axp)1/p/∫0xe−tpdt for a, p> 0 with p≠ 1 on (0 , ∞) and prove that the double inequality Γ(1+1/p)(1−e−axp)1/p<∫0xe−tpdt<Γ(1+1/p)(1−e−bxp)1/p holds for all x> 0 if and only if a≤ min {1 , Γ − p(1 + 1 / p) } and b≥ max { 1 , Γ − p(1 + 1 / p) }.

AB - In the article, we discuss the monotonicity properties of the function x→(1−e−axp)1/p/∫0xe−tpdt for a, p> 0 with p≠ 1 on (0 , ∞) and prove that the double inequality Γ(1+1/p)(1−e−axp)1/p<∫0xe−tpdt<Γ(1+1/p)(1−e−bxp)1/p holds for all x> 0 if and only if a≤ min {1 , Γ − p(1 + 1 / p) } and b≥ max { 1 , Γ − p(1 + 1 / p) }.

KW - gamma function

KW - incomplete gamma function

KW - psi function

UR - https://www.scopus.com/pages/publications/84991264180

U2 - 10.1186/s13660-016-1197-7

DO - 10.1186/s13660-016-1197-7

M3 - Article

AN - SCOPUS:84991264180

SN - 1025-5834

VL - 2016

JO - Journal of Inequalities and Applications

JF - Journal of Inequalities and Applications

IS - 1

M1 - 251

ER -

Monotonicity of the incomplete gamma function with applications

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Keywords

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