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Sincov’s inequalities on topological spaces. (English) Zbl 1463.39059
Summary: Assume that \(X\) is a non-empty set, and \(T\) and \(S\) are real or complex mappings defined on the product \(X\times X\). Additive and multiplicative Sincov’s equations are: \[T(x,z) = T(x,y) + T(y,z),\;\; x,y,z\in X\] and \[S(x,z) = Sx,y) \cdot S(y,z),\;\; x,y,z\in X,\] respectively. In the present paper, we study three related inequalities. We begin with functional inequality \[G(x,z) \leq G(x,y) \cdot G(y,z),\;\; x,y,z\in X, \] and assume that \(X\) is a topological space and \(G: X \times X\to \mathbb{R}\) is a continuous mapping. In some our statements a considerably weaker regularity than continuity of \(G\) is needed. Next, we study the reverse inequality: \[F(x,z) \geq F(x,y) \cdot F(y,z),\;\; x,y,z\in X, \] as well as the additive inequality: \[H(x,z) \geq H(x,y)+ H(y,z),\;\; x,y,z\in X. \] A corollary for generalized metric is derived.

MSC:
39B62 Functional inequalities, including subadditivity, convexity, etc.
39B82 Stability, separation, extension, and related topics for functional equations
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
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