On weighted mean matrices whose l^p norms are determined on decreasing sequences

Abstract

Abstract. We give a condition on weighted mean matrices so that their l p norms are determined on decreasing sequences when the condition is satisfied. We apply our result to give a proof of a conjecture of Bennett and discuss some related results. Mathematics subject classification

On a Result of Levin and Stečkin

“…As we have explained in Section 1, the study of (1.7) is motivated by (1.8). As (1.7) implies (1.8) and the constant (αp/(αp − 1)) p there is best possible (see [6]), we see that the constant (αp/(αp − 1)) p in (1.7) is also best possible. More generally, we note that inequality (4.7) in [6] proposes to determine the best possible constant U p (α, β) in the following inequality (a ∈ l p , p > 1, β ≥ α ≥ 1):…”

“…Inequality (1.8) was first suggested by Bennett [2, p. 40-41], see [6] and the references therein for recent progress on this. We point out here that it is easy to see that inequality (1.7) implies (1.8) when α > 1, hence it is interesting to know for what α's, inequality (1.7) is valid.…”

On Hardy q-inequalities

“…This implies the strict inequality in (14). In order to obtain a lower estimate we consider the test functions f β (t) = t…”

“…The last strict inequalities give the validity of strict inequality (14). The best constant in (14) can be found by using the test functions f β (t) = t β if 0 < t < 1, where β > −1/p.…”

“…The best constant in (14) can be found by using the test functions f β (t) = t β if 0 < t < 1, where β > −1/p. In the case when p < 0 the proof of estimate (14) can be done by use of the same method as in Theorem 2.1 for F .…”