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 
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     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://www.numdam.org/articles/10.4171/RSMUP/137-10/}
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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://www.numdam.org/articles/10.4171/RSMUP/137-10/

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