A linearisation technique and well-established linear control theory are used to derive relevant information concerning the extremal functions in a minimum-maximum problem

2. S-divisibility property and a Holmstedt type formula

Carro, Maria J.

et al.

Departamento de Matemàtica, Aplicada i Anàlisi, Universitat de Barcelona.

Ericsson, Stefan

Persson, Lars-Erik

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.

Given a conePof positive functions and an operatorS: →Pwith an additive group, we extend the concept ofK-divisibility to get some new formulas for theK-functional of finite families of lattices. Applications are given in the setting of rearrangement invariant spaces and weighted Lorentz spaces. As a consequence of our results, we also obtain a generalized Holmstedt formula due to Asekritova

We consider certain real interpolation methods for families of Banach spaces. We also define some new such methods obtained by passing to the limit in the constructions of Sparr and Cobos-Peetre. The relations between all these methods are studied. Characterizations of minimal and maximal spaces are obtained. Some concrete examples as well as sharp estimates of the corresponding operator norms are also exhibited.

In this paper we prove a strengthened general inequality of the Hardy–Knopp type and also derive its dual inequality. Furthermore, we apply the obtained results to unify the strengthened classical Hardy and Pólya–Knopp's inequalities deriving them as special cases of the obtained general relations. We discuss Pólya–Knopp's inequality, compare it with Levin–Cochran–Lee's inequalities and point out that these results are mutually equivalent. Finally, we also point out a reversed Pólya–Knopp type inequality.

Exact formulas for theKfunctional of some quasi-Banach couples of the type (X,L∞;) are obtained. In particular we can cover the cases whenX=Lp, q(Lorentzspaces), 0

In connection to the study of the isotonicity of the projection operator onto a closed convex set in an ordered Hilbert space, Isac has recently remarked the importance of an inequality named "the property of four elements" (PFE). In this paper a sharp inequality closely connected to (PFE) is proved in a Banach space setting. The property (PFE)Vfor Lyapunov functionalsVis introduced and studied. Some applications are included.

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.

Öberg, Anders

Department of Mathematics and Statistics, University College of Gävle.

On Carleman's and Knopp's inequalities2002In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 117, no 1, p. 140-151Article in journal (Refereed)

Abstract [en]

A sharpened version of Carleman's inequality is proved. This result unifies and generalizes some recent results of this type. Also the “ordinary” sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman-type inequality with a more general measure is proved and this result may also be seen as a generalization of a continuous variant of Carleman's inequality, which is usually referred to as Knopp's inequality. A new elementary proof of (Carleman–)Knopp's inequality and a new inequality of Hardy–Knopp type is pointed out

10. Notes on noninterpolation spacesMaligranda, Lech

et al.

Mastylo, M.

Department of Mathematics, Adam Mickiewicz University.

Notes on noninterpolation spaces1989In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 55, p. 333-347Article in journal (Refereed)