Mathematics – Classical Analysis and ODEs
Scientific paper
2009-03-08
Bai-Ni Guo and Feng Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma funct
Mathematics
Classical Analysis and ODEs
11 pages
Scientific paper
10.4134/JKMS.2011.48.3.655
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 $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.
Guo Bai-Ni
Qi Feng
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