A harmonic mean inequality for the digamma function and related results
Alzer, Horst1 ; Jameson, Graham2
1 , WALDBRÖL, GERMANY
2 Lancaster University, LANCASTER, UNITED KINGDOM
Rendiconti del Seminario Matematico della Università di Padova, Tome 137 (2017), pp. 203-209.

We present some inequalities and a concavity property of the digamma function ψ=Γ ' /Γ, where Γ denotes Euler's gamma function. In particular, we offer a new characterization of Euler's constant γ=0.57721⋯. We prove that -γ is the minimum of the harmonic mean of ψ(x) and ψ(1/x) for x>0.

DOI : 10.4171/RSMUP/137-10
Classification : 33, 39, 41
Mots clés : Digamma function, inequalities, concavity, harmonic mean, Euler’s constant
Alzer, Horst 1 ; Jameson, Graham 2

1 , WALDBRÖL, GERMANY
2 Lancaster University, LANCASTER, UNITED KINGDOM 
@article{RSMUP_2017__137__203_0,
     author = {Alzer, Horst and Jameson, Graham},
     title = {A harmonic mean inequality for the digamma function and related results},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {203--209},
     publisher = {European Mathematical Society Publishing House},
     address = {Zuerich, Switzerland},
     volume = {137},
     year = {2017},
     doi = {10.4171/RSMUP/137-10},
     mrnumber = {3652875},
     zbl = {1366.33003},
     url = {http://archive.numdam.org/articles/10.4171/RSMUP/137-10/}
}
TY  - JOUR
AU  - Alzer, Horst
AU  - Jameson, Graham
TI  - A harmonic mean inequality for the digamma function and related results
JO  - Rendiconti del Seminario Matematico della Università di Padova
PY  - 2017
SP  - 203
EP  - 209
VL  - 137
PB  - European Mathematical Society Publishing House
PP  - Zuerich, Switzerland
UR  - http://archive.numdam.org/articles/10.4171/RSMUP/137-10/
DO  - 10.4171/RSMUP/137-10
ID  - RSMUP_2017__137__203_0
ER  - 
%0 Journal Article
%A Alzer, Horst
%A Jameson, Graham
%T A harmonic mean inequality for the digamma function and related results
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2017
%P 203-209
%V 137
%I European Mathematical Society Publishing House
%C Zuerich, Switzerland
%U http://archive.numdam.org/articles/10.4171/RSMUP/137-10/
%R 10.4171/RSMUP/137-10
%F RSMUP_2017__137__203_0
Alzer, Horst; Jameson, Graham. A harmonic mean inequality for the digamma function and related results. Rendiconti del Seminario Matematico della Università di Padova, Tome 137 (2017), pp. 203-209. doi : 10.4171/RSMUP/137-10. http://archive.numdam.org/articles/10.4171/RSMUP/137-10/

Cité par Sources :