Comparison of two variable homogeneous means

Abstract

In this paper we give necessary and sufficient conditions for the inequality

$$\left( * \right)M\left( {x,y} \right) \leq N\left( {x,y} \right),\quad x,y \in \left[ {\alpha ,\beta } \right],$$

(*)

where 0 < α < β < ∞ are fixed values and M: ℝ₊ × ℝ₊ → ℝ₊ and N: ℝ₊ × ℝ₊ → ℝ₊ belong to one of the following classes of means:

$${D_{a,b}}\left( {x,y} \right) = {\left( {\frac{{{x^a} - {y^a}}}{a}\frac{b}{{{x^b} - {y^b}}}} \right)^{\frac{1}{{a - b}}}},\quad {S_{a,b}}\left( {x,y} \right) = {\left( {\frac{{{x^a} + {x^a}}}{{{x^b} + {x^b}}}} \right)^{\frac{1}{{a - b}}}}.$$

The following two inequalities are simple necessary conditions for (*) to hold:

$$- {\partial _1}{\partial _2}M\left( {1,1} \right) \leq - {\partial _1}{\partial _2}N\left( {1,1} \right),\quad M\left( {\alpha ,\beta } \right) \leq N\left( {\alpha ,\beta } \right).$$

In the class of power means, already the first inequality is a sufficient condition. The aim of this paper is to show that these two inequalities together are sufficient conditions for the comparison of the above D and S means as well.