Ludmila Nikolova

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

Let ψ̃:[0,1]→R be a concave function with ψ̃(0)=ψ̃(1)=1. There is a corresponding map .ψ̃ 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 ψω,qand ∥.∥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.

Interpolation of Limited and Weakly Compact Operators on Families of Banach Spaces

In this paper, we study how the limited and weakly compact properties of operators are preserved by interpolation of the real method for infinite families of Banach spaces introduced by Carro in Studia Math. 109 (1994). We apply these results to the case of Sparr, Fernández and Cobos–Peetre methods of interpolation for finite families.

Properties of some functionals associated with h-concave and quasilinear functions with applications to inequalities

We consider quasilinearity of the functional (h∘v)⋅(Φ∘gv), where Φ is a monotone h-concave (h-convex) function, v and g are functionals with certain super(sub)additivity properties. Those general results are applied to some special functionals generated with several inequalities such as the Jensen, Jensen-Mercer, Beckenbach, Chebyshev and Milne inequalities.

Generalized quasi-Banach sequence spaces and measures of noncompactness

Given 0 < s ≤ 1 and ψ an s-convex function, s – ψ -sequence spaces are introduced. Several quasi-Banach sequence spaces are thus characterized as a particular case of s – ψ -spaces. For these spaces, new measures of noncompactness are also defined, related to the Hausdorff measure of noncompactness. As an application, compact sets in s – ψ -interpolation spaces of a quasi-Banach couple are studied.

A new look at classical inequalities involving Banach lattice norms

Some classical inequalities are known also in a more general form of Banach lattice norms and/or in continuous forms (i.e., for ‘continuous’ many functions are involved instead of finite many as in the classical situation). The main aim of this paper is to initiate a more consequent study of classical inequalities in this more general frame. We already here contribute by discussing some results of this type and also by deriving some new results related to classical Popoviciu’s, Bellman’s and Beckenbach-Dresher’s inequalities.