A harmonic mean inequality for the polygamma function

Abstract

In this work, we discuss some new inequalities and a concavity property of the polygamma function ψ n (x) = ψ (x), x > 0, where ψ (x) represents the digamma function (i.e. logarithmic derivative of the gamma function Γ(x)). Using these inequalities, minimum value of harmonic mean of (−1) ψ n (x) and (−1) ψ n (1/x) is obtained in terms of the Riemann zeta function and the Bernoulli numbers. Further new characterizations of π and the Apéry’s constant are also presented as a consequence. ( ) dn n ( ) n ( ) dx n