On a sum of the psi function with a logarithm

Abstract

Letbe the psi function, that is the logarithmic derivative of the Euler gamma function. The aim of this paper is to establish an asymptotic formula for the function $psi(x)+log(e^{1/x}-1)$ and to improve some results of Batir (Some new inequalities for gamma and polygamma functions, J. Ineq. Pure Appl. Math. 6 (4), Art 103, 2005) and Alzer (Sharp inequalities for the harmonic numbers, Expo. Math. 24, 385–388, 2006). Finally we give a short proof of, respectively, the monotonicity and concavity of the function $psi(x)+log(e^{1/x}-1)$ , previously stated by Alzer above, and by Guo and Qi (Some properties of the psi and polygamma functions, Hacet. J. Math. Stat. 39 (2), 219–231, 2010).