Sincov's inequalities on topological spaces

Abstract

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: $$T(x,z)=T(x,y)+T(y,z)$$ and $$S(x,z)=S(x,y)\cdot S(y,z),$$ respectively. Both equations play important roles in many areas of mathematics. In the present paper we study related inequalities. We deal with functional inequality $$G(x,z)\le G(x,y)\cdot G(y,z),x,y,z\in X$$ and we 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. We also study the reverse inequality: $$F(x,z)\ge F(x,y)\cdot F(y,z),x,y,z\in X$$ and the additive inequality (the triangle inequality): $$H(x,z)\le H(x,y)+H(y,z),x,y,z\in X.$$ A corollary for generalized (non-symmetric) metric is derived.

Some parts of the text are rewritten and several typos are corrected. Main results still remain unchanged