Mean-value inequalities for the polygamma functions

被引：0
作者：
Alzer H. [1 ]
机构：
[1] D-51545 Waldbröl
来源：
aequationes mathematicae | 2001年 / 61卷 / 1期
关键词：
Real Number; Positive Weight; Polygamma Function;
D O I：
10.1007/s000100050167
中图分类号：
学科分类号：
摘要：
Let k ≥ 1 be an integer and let ψ(k) be the k-th derivative of the psi function, ψ = Γ′/Γ. Further, let (equation Presented) be the power mean of order t of x1,...,xn with positive weights p1,...,pn, and ∑vnpv = 1. We determine all real numbers α, r, and s such that the inequalities (equation Presented) and (equation Presented) hold for all xv > 0 and pv > 0 (v - 1,...,n) with ∑v=1npv = 1. Our results sharpen and generalize known inequalities. © Birkhäuser Verlag, 2001.
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页码：151 / 161
页数：10
相关论文
共 19 条
  • [1] Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, (1965)
  • [2] Alzer H., On some inequalities for the gamma and psi functions, Math. Comp., 66, pp. 373-389, (1997)
  • [3] Alzer H., Inequalities for the gamma and polygamma functions, Abhandl. Math. Sem. Univ. Hamburg, 68, pp. 363-372, (1998)
  • [4] Alzer H., Ruehr O.G., A submultiplicative property of the psi function, J. Comp. Appl. Math., 101, pp. 53-60, (1999)
  • [5] Alzer H., Wells J., Inequalities for the polygamma functions, SIAM J. Math. Anal., 29, pp. 1459-1466, (1998)
  • [6] Anderson G.D., Barnard R.W., Richards K.C., Vamanamurthy M.K., Vuorinen M., Inequalities for the zero-balanced hypergeometric functions, Trans. Amer. Math. Soc., 347, pp. 1713-1723, (1995)
  • [7] Anderson G.D., Qiu S.-L., A monotoneity property of the gamma function, Proc. Amer. Math. Soc., 125, pp. 3355-3362, (1997)
  • [8] Artin E., The Gamma Function, (1964)
  • [9] Bullen P.S., Mitrinovic D.S., Vasic P.M., Means and Their Inequalities, (1988)
  • [10] Bustoz J., Ismail M.E.H., On gamma function inequalities, Math. Comp., 47, pp. 659-667, (1986)
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