Ostrowski-type inequalities in abstract distance spaces

For non-empty sets X we define notions of distance and pseudo metric with values in a partially ordered set that has a smallest element $\theta$. If $h_X$ is a distance in $X$ (respectively, a pseudo metric in $X$), then the pair $(X,h_X)$ is called a distance (respectively, a pseudo metric) space. If $(T,h_T)$ and $(X,h_X)$ are pseudo metric spaces, $(Y,h_Y)$ is a distance space, and $H(T,X)$ is a class of Lipschitz mappings $f\colon T\to X$, for a broad family of mappings $\Lambda\colon H (T,X)\to Y$, we obtain a sharp inequality that estimates the deviation $h_Y(\Lambda f(\cdot),\Lambda f(t))$ in terms of the function $h_T(\cdot, t)$. We also show that many known estimates of such kind are contained in our general result.