The Beckenbach-Dresher inequality in the Ψ-direct sums of spaces and related results

Abstract

Abstract Let ψ ̃ : [ 0 , 1 ] → R be a concave function with ψ ̃ ( 0 ) = ψ ̃ ( 1 ) = 1 . There is a corresponding map . ψ ̃ for which the inverse Minkowski inequality holds. Several properties of that map are obtained. Also, we consider the Beckenbach-Dresher type inequality connected with ψ-direct sums of Banach spaces and of ordered spaces. In the last section we investigate the properties of functions ψ ω,q and ∥.∥ ω,q , (0 < ω < 1, q < 1) related to the Lorentz sequence space. Other posibilities for parameters ω and q are considered, the inverse Holder inequalities and more variants of the Beckenbach-Dresher inequalities are obtained. 2000 MSC: Primary 26D15; Secondary 46B99.