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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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