A harmonic mean inequality for the digamma function and related results

Abstract

We present some inequalities and a concavity property of the digamma function ψ = Γ /Γ, where Γ denotes Euler's gamma function. In particular, we offer a new characterization of Euler's constant γ = 0.57721.... We prove that −γ is the minimum of the harmonic mean of ψ(x) and ψ(1/x) for x > 0.

A harmonic mean inequality for the q-gamma and q-digamma functions

“…Proposition 2.1. The function ψ q (0 < q) has a uniquely determined positive zero on (1, 3 2 ), which we denote by x q . In [6], Alzer proved for x > 0 the interesting inequality…”

“…Besides, in [3], Alzer proved an interesting harmonic mean inequality for the psi-function, i.e. where γ = 0, 57721.... is the Euler's constant.…”

Complete monotonicity and inequalities related to generalized k-gamma and k-polygamma functions

“…-e -kt e -3t/2 e -xt dt, (3.6) where η k (t) = 24kt 1 -e kt + 8(kt) 2 1 -e kt -(kt) 3 1 -e kt + 48e 3kt/2 -48e kt/2 -48(kt)e kt/2 .…”

A Harmonic Mean Inequality for the Exponential Integral Function

“…Specifically, our objective is to establish a harmonic mean inequality for the function. For harmonic mean inequalities involving other special functions, the interested reader may refer to [2], [4], [6], [9], [10], [11], [18], [19].…”