Revisiting Ostrowski's Inequality

TL;DR

This paper broadens Ostrowski's inequality to non-differentiable functions, introduces a refined version, and provides new bounds that accommodate functions with points of non-differentiability.

Contribution

It extends Ostrowski's inequality to continuous functions with finitely many non-differentiable points and proposes a sharper inequality refinement.

Findings

Extended Ostrowski's inequality to non-differentiable functions.

Proposed a refined inequality with tighter bounds.

Provided explicit bounds involving function derivatives and points of non-differentiability.

Abstract

The main objective of this paper is to present Ostrowski's inequality for a broader class of functions and to propose a refinement to the classical version of it. The original Ostrowski's inequality can be stated as follows "If $f:[a,b]\to\mathbb{R}$ is differentiable and $f'\in L^{\infty}[a, b]$, then for any $p\in\,]a,b[\,$, the following functional inequality holds: \begin{equation*} \Bigg|f(p)-\dfrac{1}{b-a}\int_{a}^{b}f(t)\,dt\Bigg|\leq \dfrac{(p-a)^2+(b-p)^2}{2(b-a)}\Big\| f'\Big\|_a^b\,.\,^{^{^{^{^"}}}} \end{equation*} We relax the condition of differentiability and show that even if $f\in C[a,b]$ is non-differentiable at the points $p_{_{1}},\cdots,p_{_{n}}$, then for any $p\in\,]a,b[\,\setminus\overset{n}{\underset{i=1}{\cup}}\{p_{_{i}}\}$, the following Ostrowski-type inequality holds: \begin{align*} \left| f(p) - \frac{1}{b-a} \int_{a}^{b} f(t)\,dt \right| \leq \frac{1}{2} \max \Bigg\{ & \left\| f' \right\|_a^{p_1}(p_1 - a),\,\ldots,\, \left\| f' \right\|_{p_{i-1}}^p (p - p_{i-1}),\, \left\| f' \right\|_p^{p_i} (p_i - p),\, \\ & \ldots,\, \left\| f' \right\|_{p_n}^{b} (b - p_n) \Bigg\} + \max \left\{ f(a) + \sum_{i=1}^n f(p_i),\, -\sum_{i=1}^n f(p_i) - f(b) \right\}. \end{align*} Also, we investigate the possibility of proposing a refinement for Ostrowski inequality. We prove that if $f'\in L^{\infty}[a, b]$, then for any $p\in\,]a,b[$, we can restructure the inequality as follows: \begin{align*} \left| f(p) - \frac{1}{b-a} \int_{a}^{b} f(t)\,dt \right| \leq \min \Bigg\{ & \left[ \frac{1}{4} + \left( \frac{p - \frac{a + b}{2}}{b - a} \right)^2 \right](b - a) \left\| f' \right\|_{a}^{b}, \\ &\quad + \frac{1}{2} \max \left\{ (p - a) \left\| f' \right\|_{a}^{p},\, (b - p) \left\| f' \right\|_{p}^{b} \right\} \Bigg\}. \end{align*}