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A class of functions involving the divided differences of the psi functionand the polygamma functions and originating from Kershaw's double inequalityare proved to be completely monotonic. As applications of these results, themonotonicity and convexity of a function involving ratio of two gamma functionsand originating from establishment of the best upper and lower bounds inKershaw's double inequality are derived, two sharp double inequalitiesinvolving ratios of double factorials are recovered, the probability integralor error function is estimated, a double inequality for ratio of the volumes ofthe unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively isdeduced, and a symmetrical upper and lower bounds for the gamma function interms of the psi function is generalized.

A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications