On a sum of the psi function with a logarithm

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

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).

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