Monotonicity properties of a function involving the psi function with applications

Abstract

In this paper, we present the best possible parameter a∈(1/15,∞) such that the functions ψ′ (x+1)−Lx(x,a) and ψ″ (x+1)−Lxx(x,a) are strictly increasing or decreasing with respect to x∈(0,∞), where L(x,a)=190 a2 +2log( x2 +x+3a+13)+45 a2 90 a2 +2log( x2 +x+15a−145a) and ψ(x)$\psi(x)$ is the classical psi function. As applications, we get several new sharp bounds for the psi function and its derivatives.