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A Completely Monotonic Function Used in an Inequality of Alzer

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Abstract

The function

$$G(x)={\Biggr(1-{\rm {ln}(x)\over ln(1+x)}\Biggr)}\ x\ \rm{ln}(x)$$

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.

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Authors and Affiliations

  1. Institute of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100, København Ø, Denmark

    Christian Berg

  2. Department of Basic Sciences and Environment, University of Copenhagen, Thorvaldsensvej 40, DK-1871, Frederiksberg C, Denmark

    L. Henrik

Authors
  1. Christian Berg
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  2. L. Henrik
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Correspondence to Christian Berg.

Additional information

Both authors acknowledge support by grant 10-083122 from The Danish Council for Independent Research ∣ Natural Sciences.

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Berg, C., Henrik, L. A Completely Monotonic Function Used in an Inequality of Alzer. Comput. Methods Funct. Theory 12, 329–341 (2012). https://doi.org/10.1007/BF03321830

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  • DOI: https://doi.org/10.1007/BF03321830

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