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The property of supermultiplicity for some classical inequalities and applications. (English) Zbl 0909.26012

Let \(C\neq\emptyset\) be endowed with an inner operation “\(+\)”; \(A:C\to (0,\infty)\) additive and \(L: C\to[0,\infty)\) supermultiplicative (i.e., \(L(f+ g)\geq L(f)+ L(g)\) for \(f,g\in C\)) mappings; and put \(F(f)= L(f)/A(f)\), \(H(f)= [F(f)]^{A(f)}\), \(f\in C\). The authors prove two theorems for certain expressions involving \(F\) and \(H\), and deduce many corollaries and applications to the theory of classical inequalities (e.g., Jensen’s, Hölder’s, Minkowski’s, etc.). We quote only the first theorem, which asserts that assuming the above conditions on \(A\) and \(L\), the mapping \(H\) is supermultiplicative, i.e., \(H(f+ g)\geq H(f)\cdot H(g)\) for all \(f,g\in C\). The given interesting applications strongly demonstrate the importance of such simple, but general properties.

MSC:

26D15 Inequalities for sums, series and integrals
26D20 Other analytical inequalities
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