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Archiv der Mathematik

, Volume 80, Issue 2, pp 201–215 | Cite as

Inequalities for Means in Two Variables

  • Horst Alzer
  • Song-liang Qiu
Original paper

Abstract.

We present various new inequalities involving the logarithmic mean \( L(x,y)=(x-y)/(\log{x}-\log{y}) \), the identric mean \( I(x,y)=(1/e)(x^x/y^y)^{1/(x-y)} \), and the classical arithmetic and geometric means, \( A(x,y)=(x+y)/2 \) and \( G(x,y)=\sqrt{xy} \). In particular, we prove the following conjecture, which was published in 1986 in this journal. If \( M_r(x,y)= (x^r/2+y^r/2)^{1/r}(r\neq{0}) \) denotes the power mean of order r, then \( M_c(x,y)(\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})} \) with the best possible parameter \( c=(\log{2})/(1+\log{2}) \).

Mathematics Subject Classification (2000): 26D07, 26E60. 

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Copyright information

© Birkhäuser Verlag, Basel, 2003

Authors and Affiliations

  • Horst Alzer
    • 1
  • Song-liang Qiu
    • 2
  1. 1.Morsbacher Str. 10, D-51545 Waldbröl, Germany, e-mail: alzer@wmax03.mathematik.uni-wuerzburg.de DE
  2. 2.President's Office, Hangzhou Institute of Electronics Engineering, Hangzhou, Zhejiang 310037, People's Republic of China, e-mail: sl_qiu@hziee.edu.cnCN

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