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Computational Methods and Function Theory 12 (2012), No. 1, 329--341
Copyright Heldermann Verlag 2012

A Completely Monotonic Function Used in an Inequality of Alzer

Christian Berg
berg@math.ku.dk , University of Copenhagen, Institute of Mathematical Sciences, Universitetsparken 5, DK-2100 København Ø, Denmark.

Henrik L. Pedersen
henrikp@life.ku.dk , University of Copenhagen, Department of Basic Sciences and Environment, Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark.

[Abstract-pdf] [Abstract-ps]

The function
G(x)=(1-ln(x)/ln(1+x))xln(x)
has been considered by Alzer and by Qi and Guo. We prove that G' is completely monotonic by finding an integral representation of the holomorphic extension of G to the cut plane. A main difficulty is caused by the fact that G' is not a Stieltjes transform.

Keywords: Completely monotonic function, Stieltjes transform.

MSC 2000: Primary 33B99; Secondary 30E20.

[FullText-pdf (280 K)] [FullText-ps (592 K)]