Ostrowski-type inequalities in abstract distance spaces
Vladyslav Babenko, Vira Babenko, Oleg Kovalenko
TL;DR
The paper develops a unified Ostrowski-type inequality framework in abstract distance spaces where distances take values in a partially ordered set $M$ with smallest element $\theta$. By introducing $M$-distance spaces, monotone operator pairs $(\Lambda,\lambda)$, and the notions of $H(T,X)$ and $H^\omega(T,X)$, it provides sharp deviation bounds of the form $h_Y(\Lambda f(\tau), \Lambda f(t))\le\lambda(h_T(\tau,t))$ and its modulus-of-continuity variant $h_Y(\Lambda f(\tau), \Lambda f(t))\le\lambda(\omega(h_T(\tau,t)))$ that subsume classical Ostrowski-type inequalities. The results include precise extremal conditions, commutative diagrams yielding equality cases, and extensions to integral and convexifying operators, thereby connecting numeric, Banach-space-valued, and non-numeric-valued Ostrowski-type results under a single abstract theory. Additionally, the paper analyzes when $M$-distances agree with underlying $M$-metrics, providing constructive guidance via an $e$-function framework. Overall, this work offers a versatile, highly general toolkit for approximation and deviation estimation in abstract metric-like spaces.
Abstract
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 $θ$. 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 $Λ\colon H (T,X)\to Y$, we obtain a sharp inequality that estimates the deviation $h_Y(Λf(\cdot),Λf(t))$ 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.