Inequalities for Means in Two Variables

@article{Alzer2003InequalitiesFM,
  title={Inequalities for Means in Two Variables},
  author={H. Alzer and S. Qiu},
  journal={Archiv der Mathematik},
  year={2003},
  volume={80},
  pages={201-215}
}
  • H. Alzer, S. Qiu
  • Published 2003
  • Mathematics
  • Archiv der Mathematik
  • 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… CONTINUE READING
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