Sincov's inequalities on topological spaces

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: $$ T(x,z)=T(x,y)+T(y,z),\qquad x,y,z\in X $$ and $$ S(x,z)=S(x,y)\cdot S(y,z),\qquad x,y,z\in X, $$ respectively. In the present paper, we study three related inequalities. We begin with functional inequality $$ G(x,z)\leq G(x,y)\cdot G(y,z),\qquad x,y,z\in X, $$ and assume that $X$ is a topological space and $G\colon X\times X\to\mathbb{R}$ is a continuous mapping. In some our statements a considerably weaker regularity than continuity of $G$ is needed. Next, we study the reverse inequality: $$ F(x,z)\geq F(x,y)\cdot F(y,z),\qquad x,y,z\in X, $$ as well as the additive inequality: $$ H(x,z)\leq H(x,y)+H(y,z),\qquad x,y,z \in X. $$ A corollary for generalized metric is derived.