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

Ludmila Nikolova,Sanja Varošanec,Lars-Erik Persson

doi:10.1186/1029-242x-2012-7

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.

Highlights

Let ψ : [0, 1] → R be a concave function with ψ (0) = ψ (1) = 1
We are pointing out articles of Pečarić and Beesack [4], Petree and Persson [5], Persson [6] and Varošanec [7], where the reader can find related literature about this inequality Here we consider inequalities of Beckenbach-Dresher type in more general structures, namely in ψ-direct sums
Since the proof of Beckenbach-Dresher inequality can be obtained as an application of the Minkowski inequality the Holder inequality and its inverse inequalities in different cases, we are going to see what kind of such inequalities we could prove using some ideas of ψ-direct sums

Summary

Let q

The corresponding norm is ∥.∥p,q,l = max{∥.∥p, l∥.∥q}. ∗ ψ is an absolute normalized norm and the corresponding convex function ψ* Î Ψ is. Since the proof of Beckenbach-Dresher inequality can be obtained as an application of the Minkowski inequality the Holder inequality and its inverse inequalities in different cases, we are going to see what kind of such inequalities we could prove using some ideas of ψ-direct sums. We consider a family of concave functions. We prove some properties of concave functions and the inverse Minkowski inequality Using these results and combining with the known results about the family Ψ and normalized absolute norms we obtain a variant of the BeckenbachDresher inequality related to those norms. In the third section we are considering Ψdirect sums of Banach and ordered spaces.

Dresher inequality

Let now

Minkowski inequality with constant