The real functions satisfying the inequality Ф (uv) ≤ KФ (u) Ф (v) for some positive K which occur among others in [5], [3], [4], and referred there as submultiplicative, are discussed. A simplifying remark that Ф satisfies this inequality iff KФ is submultiplicative in the standard sense, is done. It is shown that, under general conditions, the standard submultiplicativity of Ф and the inequality Ф (u) Ф (1/u) ≤ 1 imply that Ф must be multi-plicative. Applying a result of Bhatt [1], we observe that if p is a nontrivial seminorm on a Banach algebra X such that the set { [formula] .. : ∈ G X, p (x) ≠ 0} is a singleton {λ}, then s = λp is a submultiplicative seminorm on X.
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In this paper we define (...), the sequence spaces on a seminormed complex linear space, using an Orlicz function. We give various properties and some inclusion relations on this space.
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In this paper we introduce the sequence spaces x2m(p, q, u), using an modulus function M and defined over a semi normed space (X, q); semi normed by q. We study some properties of these sequence spaces and obtain some inclusion relations.
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In this article we introduce some vector valued difference paranormed double sequence spaces defined by Orlicz function. We study some of their properties like solidness, symmetric-ity, completeness etc. and prove some inclusion results.
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The purpose of this paper is to introduce the space of sequences those are strongly -summable with respect to an Orlicz function. We give some relations related to these sequence spaces. We also show that the spaces may be represented as a space.
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