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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2011 / Cilt: 40 - Sayı: 6

On a sum of the psi function with a logarithm

Cristinel MORTİCİ
Sayfa
775–780
Yayın Tarihi
2022-10-18
DOI
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Stat. 39 (2), 219–231, 2010).

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Letbe the psi function, that is the logarithmic derivative of the Euler gamma function.
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Temel Bulgular & Veri: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.
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Sonuç & Katkı:Stat. 39 (2), 219–231, 2010).
🔬Tür:Araştırma Makalesi
📍Kapsam:2005 Verileri
⚙️Değişkenler:Alan Parametreleri

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).

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Bu Makaleye Atıf Yap

MORTİCİ, C. (2022). On a sum of the psi function with a logarithm. Hacettepe Journal of Mathematics and Statistics, Cilt: 40 - Sayı: 6, 775–780. https://ajindex.com/articles/530578

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Kaynakça (8)

  1. Abramowitz, M. and Stegun, I.A. (Eds.) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1965)
  2. Alzer, H. On some inequalities for the gamma and psi functions, Math. Comp. 66 (217), 373–389, 1997
  3. Alzer, H. Sharp inequalities for the harmonic numbers, Expo. Math. 24, 385–388, 2006
  4. Batir, N. Some new inequalities for gamma and polygamma functions, J Inequal. Pure Appl. Math. 6 (4), Art. 103 (electronic), 2005
  5. Guo, B. -N. and Qi, F. Some properties of the psi and polygamma functions, Hacet. J. Math. Stat. 39 (2), 219–231, 2010
  6. Mortici, C. A quicker convergence toward the gamma constant with the logarithm term involving the constant e, Carpathian J. Math. 26 (1), 86–91, 2010
  7. Pölya, G. and Szeg¨o, C. Problems and Theorems in Analysis, vol. I-II (Springer Verlag, Berlin, Heidelberg, 1972)
  8. Widder, D.V. The Laplace Transform (Princeton Univ. Press, Princeton, 1981).