On a conjecture of Clark and Ismail

Authors:
Horst Alzer;Christian Berg;Stamatis Koumandos
Affiliations:
Morsbacher Str. 10, D-51545 Waldbröl, Germany;Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100, Denmark;Department of Mathematics and Statistics, The University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
Venue:
Journal of Approximation Theory
Year:
2005

Citing 2
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Abstract

Let @F"m(x)=-x^m@j^(^m^)(x), where @j denotes the logarithmic derivative of Euler's gamma function. Clark and Ismail prove in a recently published article that if m@?{1,2,...,16}, then @F"m^(^m^) is completely monotonic on (0,~), and they conjecture that this is true for all natural numbers m. We disprove this conjecture by showing that there exists an integer m"0 such that for all m=m"0 the function @F"m^(^m^) is not completely monotonic on (0,~).