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More supplements to a class of logarithmically completely monotonic functions associated with the gamma function
In this article, a necessary and sufficient condition and a necessary
condition are established for a function involving the gamma function to be
logarithmically completely monotonic on . As applications of the
necessary and sufficient condition, some inequalities for bounding the psi and
polygamma functions and the ratio of two gamma functions are derived.Comment: 8 page
A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications
A class of functions involving the divided differences of the psi function
and the polygamma functions and originating from Kershaw's double inequality
are proved to be completely monotonic. As applications of these results, the
monotonicity and convexity of a function involving ratio of two gamma functions
and originating from establishment of the best upper and lower bounds in
Kershaw's double inequality are derived, two sharp double inequalities
involving ratios of double factorials are recovered, the probability integral
or error function is estimated, a double inequality for ratio of the volumes of
the unit balls in and respectively is
deduced, and a symmetrical upper and lower bounds for the gamma function in
terms of the psi function is generalized.Comment: 11 page
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