Some Ostrowski Type Inequalites via Cauchy's Mean Value Theorem

TL;DR

The paper develops a general Ostrowski-type framework by applying the Cauchy mean value theorem to bound deviations $\left|f(x)-\frac{1}{b-a}\int_a^b f(t)\,dt\right|$ in terms of the supremum norm of the derivative ratio $\left\|\frac{f'}{g'}\right\|_{\infty}$ for an auxiliary function $g$. By choosing various $g$ (e.g., $g(t)=t$, $t^p$, $\ln t$, $e^t$) the authors recover classical Ostrowski inequalities and derive a variety of midpoint-type and weighted-integral inequalities. The main contributions include a general inequality (via Cauchy MVT), several concrete specializations yielding midpoint and exponential/cosine/sine-mean variants, and a weighted-mean extension with explicit bounds involving the weight function $w$. This provides a flexible toolkit for error bounds in numerical integration and approximation tasks where the derivative ratio is controlled.

Abstract

Some Ostrowski type inequalities via Cauchy's mean value theorem and applications for certain particular instances of functions are given.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• proof
• Corollary 1
• Remark 1
• Proposition 1
• Proposition 2