Complete Monotonicities of Functions Involving the Gamma and Digamma Functions

By Feng Qi (祁锋), Bai-Ni Guo (郭白妮). 2004

139 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/811f84bc18bd6a1d61e9a053bdb535c935233765

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Mills' ratio: Monotonicity patterns and functional inequalities (2008, background): “A positive function f is called strictly logarithmically completely monotonic [26] on an interval I if f has derivatives of all orders on I and its logarithm logf satisfies (−1)n[logf (x)](n) > 0, for all x ∈ I and n 1.”
The Best Bounds in Gautschi-Kershaw Inequalities (2006, unknown): “([ 2, 14, 15, 16 ]) A logarithmically completely monotonic function is also completely monotonic.”
Some completely monotonic functions involving polygamma functions and an application (2005, background): “Recently, some completely monotonic properties of functions involving the gamma, psi, and polygamma functions are verified in [6, 7, 8, 9].”
TWO NEW PROOFS OF THE COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE PSI FUNCTION (2009, background): “Recall also [4, 5, 8, 15, 17] that a positive function f is said to be logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and (−1)[ln f(x)] ≥ 0 (3) for all x ∈ I and n ∈ N.”
Logarithmically completely monotonic functions relating to the gamma function (2006, background): “The notion “logarithmically completely monotonic function” was posed explicitly in [25] and published formally in [24] and a much useful and meaningful relation L ⊂ C between the completely monotonic functions and the logarithmically completely monotonic functions was proved in [24,25].”
Some completely monotonic functions involving the gamma and polygamma functions (2006, background): “Among other things, the following results were obtained in [6]: For a < 0, the function x/[T{x + l)] l /x is strictly logarithmically completely monotonic in (0, oo).”
A PROPERTY OF LOGARITHMICALLY ABSOLUTELY MONOTONIC FUNCTIONS AND THE LOGARITHMICALLY COMPLETE MONOTONICITY OF A POWER-EXPONENTIAL FUNCTION (2009, background): “In [7, 38, 41, 43, 44, 52] and many other references, the inclusions CL[I] ⊂ C[I] and S ⊂ CL[(0,∞)] were revealed implicitly or explicitly, where S denotes the class of Stieltjes transforms.”
Refinements of lower bounds for polygamma functions (2009, background): “Recall also [3, 18] that a function f is said to be logarithmically completely monotonic on an interval I ⊆ R if it has derivatives of all orders on I and its logarithm ln f satisfies 0 ≤ (−1)[ln f(x)] <∞ (15) for k ∈ N on I.”
Completely monotonic functions involving the gamma and q-gamma functions (2005, background): “We call such functions f(x) logarithmically completely monotonic [8, 24, 25].”
Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity (2007, background): “4(ii) in that paper can be stated as the inclusion L ⊂ C, a fact also established in [23].”
Some properties of functions related to the gamma and psi functions (2010, methodology): “We begin by listing some necessary definitions and preliminaries which can be found in [ 9 , Chapter XIII; 11 ; 13 ; 21 ; 24 , Chapter IV].”
The Dagum family of isotropic correlation functions (2007, background): “4(ii) in that paper can be stated as the inclusion L⊂ C, a fact also established in [9].”
Three classes of logarithmically completely monotonic functions involving gamma and psi functions (2007, background): “Stimulated by the papers [7, 8], among other things, it was further revealed in ref.”
A Completely Monotonic Function Involving Divided Differences of Psi and Polygamma Functions and an Application (2006, unknown): “Recall also [5, 46, 49] that a positive function f is called logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and its logarithm ln f satisfies (−1)[ln f(x)] ≥ 0 (3) for all k ∈ N on I.”
Bounds for the Ratio of Two Gamma Functions (2009, background): “It has been proved once and again in [10,23,66,67,83,87,88,89,103] that a logarithmically completely monotonic function on an interval I must also be completely monotonic on I .”
Integral Representation of Some Functions Related to the Gamma Function (2004, background): “4(ii) in that paper can be stated as the inclusion L ⊂ C, a fact also established in [11].”