Some classes of completely monotonic functions. II
type: Item
instance of: scholarly article
title: Some classes of completely monotonic functions. II
zbMATH Open document ID: 1110.26015
author: Christian Berg
author: Horst Alzer
published in: The Ramanujan Journal
publication date: 2006-11-03T00:00:00Z
review text: [For part I see Ann. Acad. Sci. Fenn., Math. 27, No. 2, 445--460 (2002; Zbl 1021.26002).] A function \(f:(0,\infty )\rightarrow \mathbb{R}\) is said to be completely monotonic if \((-1)^{n}f^{(n)}(x)\geq 0\) for all \(x>0\) and \(n=0,1,2,\dots\) The authors present some new classes of completely monotonic functions. They are defined in terms of the classical gamma function \(\Gamma \) and their logarithmic derivatives (the digamma and polygamma functions). For example, they investigate the complete monotonicity of the functions defined by \[ g_{a}(x)=x^{a}\left( \Gamma (1+1/x)\right) ^{x}, \qquad H_{a,b}(x)=\left[ x^{a}(e/x)^{x}\Gamma (x)\right] ^{b}, \] \[ Q_{a,b}(x)=\left[ \Gamma (x+a+1) \right] ^{1/(x+a)}/\left[ \Gamma (x+b+1)\right] ^{1/(x+b)}, \quad f_{a,b,\alpha ,\beta }(x)=\left[ \Gamma (ax)\right] ^{\alpha }/\left[ \Gamma (bx)\right] ^{\beta }, \] for \(a,b,\alpha ,\beta \in \mathbb{R}.\)
Mathematics Subject Classification ID: 26A48
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 11A41
zbMATH DE Number: 5070736
zbMATH Keywords: comlete monotonicity
zbMATH Keywords: gamma, digamma and polygamma functions
zbMATH Keywords: prime numbers
zbMATH Keywords: inequalities
reviewed by: Gheorghe Toader
MaRDI profile type: Publication
full work available at URL: s11139-006-6510-5
OpenAlex ID: W2006572265
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DOI: 10.1007/S11139-006-6510-5
description: scientific article; zbMATH DE number 5070736