An Approximation Formula for Nielsen’s Beta Function Involving the Trigamma Function

Abstract

We prove that the function σ(s) defined by β(s)=6s2+12s+53s2(2s+3)−ψ′(s)2−σ(s)2s5,s>0, is strictly increasing with the sharp bounds 0<σ(s)<49120, where β(s) is Nielsen’s beta function and ψ′(s) is the trigamma function. Furthermore, we prove that the two functions s↦(−1)1+μβ(s)−6s2+12s+53s2(2s+3)+ψ′(s)2+49μ240s5, μ=0,1 are completely monotonic for s>0. As an application, double inequality for β(s) involving ψ′(s) is obtained, which improve some recent results.