[Paper Review] A Generalisation of an Ostrowski Inequality in Inner Product Spaces

Sever S Dragomir, Anca C. Goşa|ArXiv.org|May 23, 2003

Mathematical Inequalities and Applications2 references3 citations

TL;DR

This paper generalizes Ostrowski's inequality to inner product spaces by deriving a sharp upper bound for the absolute value of the inner product $|\langle x,b\rangle|$ under the constraints $\langle x,a\rangle = 0$ and $\|x\| = 1$, using Schwarz's inequality. The key result is a norm-based inequality involving the Gram determinant, with explicit characterization of equality conditions via orthogonal projection in Hilbert spaces.

ABSTRACT

A generalisation of inner product spaces of an inequality due to Ostrowski and applications for sequences and integrals are given.

Motivation & Objective

• To extend Ostrowski's classical inequality to abstract inner product spaces, generalizing finite-dimensional and $L^2$-based results.
• To characterize the sharp upper bound of $|\langle x,b\rangle|$ when $x$ is orthogonal to $a$ and normalized.
• To provide a unified framework for inequalities in $\ell^2$ and $L^2$ spaces through the structure of Hilbert spaces.
• To analyze the equality case in the generalized inequality, identifying the precise form of $x$ that achieves the bound.

Proposed method

• Apply Schwarz's inequality to the orthogonal projection of $b$ onto the orthogonal complement of $a$ in the inner product space.
• Derive the inequality $\frac{\|a\|^2\|b\|^2 - |\langle a,b\rangle|^2}{\|a\|^2} \geq |\langle x,b\rangle|^2$ under the constraints $\langle x,a\rangle = 0$ and $\|x\| = 1$.
• Use the projection formula $x = \nu \left( b - \frac{\overline{\langle a,b\rangle}}{\|a\|^2} a \right)$ to characterize equality.
• Verify that the normalization condition $\|x\| = 1$ leads to the specific modulus $|\nu| = \frac{\|a\|}{\left( \|a\|^2\|b\|^2 - |\langle a,b\rangle|^2 \right)^{1/2}}$.
• Extend the result to $\ell^2(\mathbb{K})$ and $L^2(\Omega,m)$ spaces by applying the abstract inequality to sequences and square-integrable functions.
• Confirm that the equality case in the original Ostrowski inequality is recovered as a special case of the general result.

Experimental results

Research questions

• RQ1 What is the optimal upper bound for $|\langle x,b\rangle|$ when $x$ is orthogonal to $a$ and normalized in an inner product space?
• RQ2 Under what conditions does equality hold in the generalized Ostrowski-type inequality?
• RQ3 How can the classical Ostrowski inequality for sequences and integrals be derived as a special case of a more general Hilbert space inequality?
• RQ4 What is the role of the Gram determinant $\|a\|^2\|b\|^2 - |\langle a,b\rangle|^2$ in bounding the inner product $\langle x,b\rangle$?

Key findings

• The inequality $\frac{\|a\|^2\|b\|^2 - |\langle a,b\rangle|^2}{\|a\|^2} \geq |\langle x,b\rangle|^2$ holds for all $x$ with $\langle x,a\rangle = 0$ and $\|x\| = 1$ in a real or complex inner product space.
• Equality holds if and only if $x = \nu \left( b - \frac{\overline{\langle a,b\rangle}}{\|a\|^2} a \right)$ with $|\nu| = \frac{\|a\|}{\left( \|a\|^2\|b\|^2 - |\langle a,b\rangle|^2 \right)^{1/2}}$.
• In $\ell^2(\mathbb{K})$, the inequality becomes $\frac{\sum |a_i|^2 \sum |b_i|^2 - \left|\sum a_i \overline{b_i}\right|^2}{\sum |a_i|^2} \geq \left|\sum x_i \overline{b_i}\right|^2$ under $\sum x_i \overline{a_i} = 0$ and $\sum |x_i|^2 = 1$.
• In $L^2(\Omega,m)$, the inequality takes the form $\frac{\int |f|^2 \int |g|^2 - \left|\int f \overline{g} \right|^2}{\int |f|^2} \geq \left|\int h \overline{g} \right|^2$ under $\int h \overline{f} = 0$ and $\int |h|^2 = 1$.
• The equality condition in the original Ostrowski inequality (Theorem 2) is recovered as a special case of the general equality condition in Theorem 3.