MONOTONICITY RESULTS FOR FUNCTIONS INVOLVING THE q-POLYGAMMA FUNCTIONS

Let ψq,n=(−1)n−1ψq(n) for n∈ℕ, where ψq(n) are the q-polygamma functions. In this paper, by the monotonicity rules for the ratio of two power series, it is proved that, for q∈(0,1) and n∈ℕ, the function x↦Fq,n(x;α)=qx+α−1ln qψq,n+1(x)ψq,n(x), is decreasing (increasing) on (0,∞) if and only if α≤log q(2n∕(q+1)) (α≥0). The conditions for which several relevant functions are monotonic or completely monotonic on (0,∞) are obtained. Moreover, several relations involving the q-polygamma functions are established.