Monotonicity Results and Inequalities for the Gamma and Incomplete Gamma Functions

By Feng Qi (祁锋). 2002

66 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/bb66409dfe4a47733629894a7ab0bb3da20d66d5

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In Applied Mathematics

Citations in the corpus

A Completely Monotonic Function Involving Divided Differences of Psi and Polygamma Functions and an Application (2006, unknown): “u e−xudu, (26) ψ(n)(x) = (−1)n+1 Z∞ 0 un 1−e−u e−xudu, (27) ψ(n−1)(x+1) = ψ(n−1)(x) + (−1)n−1(n−1)! xn , (28) lnx− 1 x <ψ(x) <lnx− 1 2x . (29) As x→ ∞, ψ′(x) ∼ 1 x + 1 2x2 +··· . (30) Lemma3 ([37, 38]). For s>r>0, exp[(s−r)ψ(s)] > Γ(s) Γ(r) >exp[(s−r)ψ(r)]. (31) Lemma 4 ([57]). A ”
Generalized Abstracted Mean Values (2000, unknown): “As concrete applications of the monotonicities and properties of the generalized weighted mean values Mp,f (r, s; x, y), some monotonicity results and inequalities of the gamma and incomplete gamma functions are presented in [10].”
Complete Monotonicities of Functions Involving the Gamma and Digamma Functions (2004, unknown): “These are extended and generalized in [16]: The function [Γ(r)]1/(r−1) is increasing in r > 0.”
Bounds for the Ratio of Two Gamma Functions (2009, methodology): “It may be worthwhile mentioning the paper [105] in which some monotonic properties and inequalities for the gamma and incomplete gamma functions were constructed by using properties of extended mean values E(r, s; x, y) or generalized weighted mean values Mf,g(r, s;x, y).”
Inequalities and Bounds for the Incomplete Gamma Function (2013, unknown): “, [1,3,4,8,10,11]).”
The Best Bounds in Gautschi-Kershaw Inequalities (2006, unknown): “The first and second Gautschi-Kershaw inequalities state that and For more information on the background and history of these two inequalities, please refer to [ 3, 4, 11 ] .”