Mean-value inequalities for the polygamma functions

Let k ≥ 1 be an integer and let ψ(k) be the k-th derivative of the psi function, ψ = Γ′/Γ. Further, let (equation Presented) be the power mean of order t of x1,...,xn with positive weights p1,...,pn, and ∑vnpv = 1. We determine all real numbers α, r, and s such that the inequalities (equation Presented) and (equation Presented) hold for all xv > 0 and pv > 0 (v - 1,...,n) with ∑v=1npv = 1. Our results sharpen and generalize known inequalities. © Birkhäuser Verlag, 2001.