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Semi-inner products and parapreseminorms on groups and a generalization of a theorem of Maksa and Volkmann on additive functions. (English) Zbl 1452.39008

Brzdęk, Janusz (ed.) et al., Ulam type stability. Based on the conferences on Ulam type stability (CUTS), Cluj-Napoca, Romania, July 4–9, 2016 and Timisoara, Romania, October 8–13, 2018. Cham: Springer. 383-408 (2019).
The author introduces and thoroughly studies some functionals defined on groups which resemble the notions of an inner product or a norm (especially the norm generated by an inner product). They are called a semi-inner product and a (para)(pre)seminorm, respectively. One should be aware that the names semi-inner product and seminorm are used here in a different sense than elsewhere in the literature but consistently with author’s previous papers. Having defined the above mentioned functionals and structures, the author considers some functional equations and establishes their connections with the Cauchy functional equation.
In particular the result of G. Maksa and P. Volkmann [Publ. Math. Debrecen 56, 197–200 (2000; Zbl 0991.39015)] is generalized to the following one.
Theorem. Let \(f\colon X\to Y\) be a mapping between two groups \(X,Y\) (not necessarily commutative) and let \(q\colon Y\to [0,\infty)\) be a paraprenorm on \(Y\). If \(f\) satisfies the functional inequality \[ q(f(x)+f(y))\leq q(f(x+y)),\quad x,y\in X, \] then \(f\) has to be additive.
The introduced concepts and results are broadly compared with similar issues appearing in the literature. The list of references is impressive.
For the entire collection see [Zbl 1431.39001].

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B62 Functional inequalities, including subadditivity, convexity, etc.
47A30 Norms (inequalities, more than one norm, etc.) of linear operators

Citations:

Zbl 0991.39015
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Full Text: DOI

References:

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