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Hyperstability of a logarithmic-type functional equation on restricted domains. (English) Zbl 1403.39026

Summary: Let \(X\) be a real normed vector space and \(f :(0, \infty) \rightarrow X\). In this paper, we prove the hyperstability of the logarithmic functional equation \(f \left(x^y\right) - y f(x) = 0\) on \(\Gamma\) of Lebesgue measure zero. More precisely, we prove that if \(f :(0, \infty) \rightarrow X\) satisfies \(\| f \left(x^y\right) - y f(x) \| \leq \phi(x, y)\) for all \((x, y) \in \Gamma^+ \subset \{(x, y) : y > \alpha(x) \}\) [resp. \(\Gamma^- \subset \{(x, y) : 0 < y < \alpha(x) \}\)] of Lebesgue measure zero, where \(\alpha :(0, \infty) \rightarrow(0, \infty)\) is an arbitrary given function and \(\phi :(0, \infty) \times(0, \infty) \rightarrow [0, \infty)\) satisfies the condition \(\phi(x, y)/y \rightarrow 0\) as \(y \rightarrow \infty\) [resp. \(y \rightarrow 0\)], then \(f\) satisfies the functional equation \(f(x^y) = y f(x)\) for all \(x > 0\) and \(y > 0\).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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