From the characterisation of geometrically convex and geometrically concave functions defined on (0, A] or [A, ∞) with A > 0, by means of their multiplicative conditions, we obtain unified proofs of some known and new inequalities. Functions of class C 2 and strictly increasing on (a, b) fulfil some kind of supermultiplicativity and superadditivity. We have obtained a new constant determining the intervals of sub-and supermultiplicativity for the log function.Mathematics Subject Classification (1991). 26A09, 26A48, 26A51, 26D05, 26D15, 33B15.
For two Banach spaces X and Y , we write dim (X) = dim (Y ) if X embeds into Y and vice versa; then we say that X and Y have the same linear dimension. In this paper, we consider classes of Banach spaces with symmetric bases. We say that such a class F has the Cantor-Bernstein property if for every X, Y ∈ F the condition dim (X) = dim (Y ) implies the respective bases (of X and Y ) are equivalent, and hence the spaces X and Y are isomorphic. We prove (Theorems 3.1, 3.3, 3.5) that the class of Orlicz sequence spaces generated by regularly varying Orlicz functions is of this type. This complements some results in this direction obtained earlier by S. Banach (Proposition 1.1), L. Drewnowski (Proposition 1.2), and M. J. Gonzalez, B. Sari and M. Wójtowicz (Theorem 1.4). Our theorems apply to large families of concrete Orlicz spaces.
Introduction.In what follows, we use the notation from the abstract.The study of the comparison of the linear dimension between Banach spaces goes back to S. Banach himself (see [B, Chap. XII, p. 193]). For example, in [B, Chap. XII,
We prove the following general result: Let (x n ) be a boundedly complete symmetric basis for a Banach space X. Then, for every symmetric basic sequence in X, we have the following alternatives: (a) it is equivalent to a basic sequence generated by a vector with respect to (x n ), or (b) it dominates a normalized block basis of (x n ) having coefficients tending to zero. This is an extension of a similar result obtained in 1973 by Altshuler, Casazza and Lin [1] for Lorentz sequence spaces. As an application, we obtain that, if M is a geometrically convex Orlicz function, then every symmetric basic sequence in the Orlicz sequence space M has the property (a) above, or it is equivalent to the standard basis of an p -space.
Let X be a Banach space with an unconditional basis. If X contains an isomorphic copy Y of 1 , then it contains a complemented copy of 1 located inside Y (Theorem 1). The proof is based on the possibility of constructing a projection onto a copy of 1 in X, or in a Banach function space, when the ranges of the unit vectors of 1 are pairwise disjoint (Lemma 1). The latter result applies also to Orlicz spaces. We also show that if U is a complemented copy of 1 in a Banach space W and Y ⊂ W is a "slightly perturbated" copy of U , then Y is complemented in W (Lemma 2).
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