Bounds and monotonicity results for means involving the q-polygamma functions

Let ?q,n = (?1)n?1 ?(n)q for n ? N? {0} and q ? (0, 1), where ?(n) q is the q-polygamma functions. In this paper, by means of the monotonicity of means and two classes of completely monotonicity functions, we establish lower and upper bounds for the means I?q,n,w (a, b) defined, for b > a > 0 and w being positive and integrable on [a, b], by I?q,n,w (a, b) = ??1 q,n (?ba w(x)?q,n (x) dx/?ba w(x) dx); and prove that the sequence {I?q,n,w (a,b)} n?0 is decreasing with lim n?? I?q,n,w (a, b) = a. Moreover, we show that, for a, b ? R with a ? b, the function x 7? ??1 q,n (?ba w(t)?q,n (x + t) dt/?ba w(t) dt)?x is increasing from (?min {a, b} ,?) onto ( min {a, b}, ?2), where ?2 = logq (?ba w(t) qtdt/?ba w(t) dt). These generalize some known results.