ON A SUM OF THE PSI FUNCTION WITH A LOGARITHM

Let ψ be 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 ψ(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 ψ(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).