  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.
