Remarks on submultiplicative functions

• Autorzy: Matkowski Janusz
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

submultiplicativitive function; seminorm; Orlicz function; square property

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2019
• Tom: nr 62
• Strony: 85--91
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Matkowski Janusz

• University of Zielona Góra Faculty of Mathematics Computer Science and Econometrics Szafrana 4a, Pl 65-516 Zielona Góra, Poland

Bibliografia

• [1] BHATT S.J., A seminorm with square property on a Banach algebra is submultiplicative, Proc. Amer. Math. Soc. (2), 117(1993), 435-438.
• [2] HILLE E., PHILLIPS R.S., Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31. American Mathematical Society, Providence, R. I., 1957.
• [3] HUDZIK H., MALIGRANDA L., Some remarks on submultiplicative Orlicz functions, Indag. Mathem., N.S., 3(1992), 313-321.
• [4] HUDZIK H., MASTYLO М., MALIGRANDA L., PERSSON L.E., Extension of submultiplicativity and supermultiplicativity of Orlicz functions, Real Analysis Exchange, 24(2)(1998/9), 567-578.
• [5] KRASNOSELSKIJ M.A., RUTICKIJ YA.B., Convex functions and Orlicz spaces, Groningen, 1961.
• [6] KUCZMA М., An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, Prace Naukowe Uniwersytetu Sląskiego w Katowicach [Scientific Publications of the University of Silesia], 489. Uniwersytet Śląski, Katowice, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985.
• [7] MATKOWSKI J., ŚWIĄTKOWSKI T., On subadditive functions, Proc. Amer. Math. Soc., (1)(119)(1993), 187-197.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).