Sharp Smith’s bounds for the gamma function

Among various approximation formulas for the gamma function, Smith showed that [Formula: see text] which is a little-known but accurate and simple one. In this note, we prove that the function [Formula: see text] is strictly increasing and concave on [Formula: see text] , which shows that Smith’s ap...

Descripción completa

Detalles Bibliográficos
Autores principales: Li, Xi-Qiao, Liu, Zhi-Ming, Yang, Zhen-Hang, Zheng, Shen-Zhou
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer International Publishing 2018
Materias:
Research
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5846881/
https://www.ncbi.nlm.nih.gov/pubmed/29568209
http://dx.doi.org/10.1186/s13660-018-1620-3
_version_ 1783305650194874368
author Li, Xi-Qiao
Liu, Zhi-Ming
Yang, Zhen-Hang
Zheng, Shen-Zhou
author_facet Li, Xi-Qiao
Liu, Zhi-Ming
Yang, Zhen-Hang
Zheng, Shen-Zhou
author_sort Li, Xi-Qiao
collection PubMed
description Among various approximation formulas for the gamma function, Smith showed that [Formula: see text] which is a little-known but accurate and simple one. In this note, we prove that the function [Formula: see text] is strictly increasing and concave on [Formula: see text] , which shows that Smith’s approximation is just an upper one.
format Online
Article
Text
id pubmed-5846881
institution National Center for Biotechnology Information
language English
publishDate 2018
publisher Springer International Publishing
record_format MEDLINE/PubMed
spelling pubmed-58468812018-03-20 Sharp Smith’s bounds for the gamma function Li, Xi-Qiao Liu, Zhi-Ming Yang, Zhen-Hang Zheng, Shen-Zhou J Inequal Appl Research Among various approximation formulas for the gamma function, Smith showed that [Formula: see text] which is a little-known but accurate and simple one. In this note, we prove that the function [Formula: see text] is strictly increasing and concave on [Formula: see text] , which shows that Smith’s approximation is just an upper one. Springer International Publishing 2018-01-31 2018 /pmc/articles/PMC5846881/ /pubmed/29568209 http://dx.doi.org/10.1186/s13660-018-1620-3 Text en © The Author(s) 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
spellingShingle Research
Li, Xi-Qiao
Liu, Zhi-Ming
Yang, Zhen-Hang
Zheng, Shen-Zhou
Sharp Smith’s bounds for the gamma function
title Sharp Smith’s bounds for the gamma function
title_full Sharp Smith’s bounds for the gamma function
title_fullStr Sharp Smith’s bounds for the gamma function
title_full_unstemmed Sharp Smith’s bounds for the gamma function
title_short Sharp Smith’s bounds for the gamma function
title_sort sharp smith’s bounds for the gamma function
topic Research
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5846881/
https://www.ncbi.nlm.nih.gov/pubmed/29568209
http://dx.doi.org/10.1186/s13660-018-1620-3
work_keys_str_mv AT lixiqiao sharpsmithsboundsforthegammafunction
AT liuzhiming sharpsmithsboundsforthegammafunction
AT yangzhenhang sharpsmithsboundsforthegammafunction
AT zhengshenzhou sharpsmithsboundsforthegammafunction

Ejemplares similares