Inequalities for Comparison of Means

Abstract

In the present note we investigate the general comparison inequality of means, i.e. the inequality

$$ {\text{C}}\left( {{\text{M}}\left( {{\text{x}}_1 \,, \ldots ,\,\,{\text{x}}_{\text{n}} } \right),\,{\text{N}}\left( {{\text{x}}_1 ,\, \ldots ,{\text{x}}_{\text{n}} } \right)} \right) \leqslant 0, $$

((0))

where C is a comparative function, M and N are given symmetric means, and n and x₁,..., x_n run over the set of positive integers and over a real interval I, respectively. The main result of this paper states that if M and N are upper and lower semiintern repetition invariant means, respectively, and if one of them is infinitesimal, then (0) holds if and only if

$$ {\text{C(M(}}\underbrace {{\text{x}},\,\, \ldots ,\,\,{\text{x}}}_{\text{k}},\,\underbrace {{\text{y}},\, \ldots ,\,{\text{y}}}_{\text{m}}{\text{),}}\,{\text{N(}}\underbrace {{\text{x}},\,\, \ldots ,\,\,{\text{x}}}_{\text{k}},\,\underbrace {{\text{y}},\, \ldots ,\,{\text{y}}}_{\text{m}}{\text{)}} \leqslant {\text{0}} $$

is satisfied for any nonnegative integers k, m with k+m > 0 and x, y in I. The most important special cases of this result, the problems of the comparison and complementary comparison, are discussed in detail in this paper.