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Remarks on submultiplicative functions

Matkowski Janusz

Fasciculi Mathematici

2019

nr 62

85--91

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.