Inequalities for Means Regarding the Trigamma Function

Abstract

Let G(α, β), A(α, β) and H(α, β), respectively, be the geometric mean, arithmetic mean and harmonic mean of α and β. In this paper, we prove that G(ψ′ (z), ψ′ (1/z)) ≥ π²/6, A(ψ′ (z), ψ′ (1/z)) ≥ π²/6 and H(ψ′ (z), ψ′ (1/z)) ≤ π²/6. This extends the previous results of Alzer and Jameson regarding the digamma function ψ. The mathematical tools used to prove the results include convexity, concavity and monotonicity properties of certain functions as well as the convolution theorem for Laplace transforms.