[Paper Review] Reverses of the Schwarz Inequality in Inner Product Spaces Generalising a Klamkin-McLenaghan Result

Sever S Dragomir|ArXiv.org|Aug 1, 2005

Mathematical Inequalities and Applications6 references20 citations

TL;DR

This paper presents new reverse inequalities for the Schwarz (Cauchy-Schwarz) inequality in inner product spaces, generalizing classical results by Klamkin and McLenaghan. It derives sharp bounds involving norms and inner products under distance constraints, with applications to Lebesgue integrals and weighted sums, extending known inequalities for positive tuples to complex and real inner product spaces with optimal constants.

ABSTRACT

New reverses of the Schwarz inequality in inner product spaces that incorporate the classical Klamkin-McLenaghan result for the case of positive n-tuples are given. Applications for Lebesgue integrals are also provided.

Motivation & Objective

• To generalize the Klamkin-McLenaghan reverse inequality for positive n-tuples to complex and real inner product spaces.
• To establish new reverse inequalities for the Schwarz inequality under norm-distance constraints between vectors.
• To provide sharp bounds involving the real part of inner products and vector norms, improving on existing reverse inequalities.
• To extend these results to Lebesgue integrals with weight functions, yielding new integral reverse inequalities.
• To identify optimal constants and characterize equality cases in the derived inequalities.

Proposed method

• Derives a new reverse Schwarz inequality using the condition $\|x - a\| \leq r < \|a\|$, leading to bounds on $\|x\|^2 / |\langle x,a\rangle| - |\langle x,a\rangle| / \|a\|^2$.
• Applies the triangle inequality and algebraic manipulation to bound the difference between normalized norms and inner products.
• Introduces a parameterization via $a = \frac{\Gamma + \gamma}{2} y$ and $r = \frac{1}{2}|\Gamma - \gamma|\|y\|$ to connect vector constraints to complex parameters.
• Translates the vector-space results into integral forms using weighted Lebesgue integrals on measure spaces.
• Uses the condition $\operatorname{Re}(\Gamma \bar{\gamma}) > 0$ to ensure positivity and validity of the reverse bounds.
• Establishes equivalence between geometric vector conditions and integral inequalities via inner product and measure-theoretic formulations.

Experimental results

Research questions

• RQ1 Can the Klamkin-McLenaghan reverse inequality for positive n-tuples be generalized to arbitrary complex or real inner product spaces?
• RQ2 What is the optimal constant in a reverse Schwarz inequality when $\|x - a\| \leq r < \|a\|$?
• RQ3 How can such reverse inequalities be extended to Lebesgue integrals with weight functions?
• RQ4 What are the necessary and sufficient conditions for equality in the derived reverse inequalities?
• RQ5 Can the results be expressed in terms of real and imaginary parts of inner products and complex parameters?

Key findings

• The paper establishes the reverse inequality $\frac{\|x\|^2}{|\langle x,a\rangle|} - \frac{|\langle x,a\rangle|}{\|a\|^2} \leq \frac{2r^2}{\|a\|(\|a\| + \sqrt{\|a\|^2 - r^2})}$ under $\|x - a\| \leq r < \|a\|$.
• The constant 2 in the bound is optimal and cannot be replaced by a smaller value.
• Equality holds if and only if $\|x - a\| = r$ and $\operatorname{Re}\langle x,a\rangle = |\langle x,a\rangle| = \|a\|\sqrt{\|a\|^2 - r^2}$.
• For Lebesgue integrals, the inequality $\int \rho |f|^2 \int \rho |g|^2 - |\int \rho f\bar{g}|^2 \leq (\sqrt{M} - \sqrt{m})^2 |\int \rho f\bar{g}| \int \rho |g|^2$ holds under $m \leq f/g \leq M$ a.e.
• The result generalizes the Klamkin-McLenaghan inequality $\frac{\sum w_k x_k^2}{\sum w_k x_k y_k} - \frac{\sum w_k x_k y_k}{\sum w_k y_k^2} \leq (\sqrt{M} - \sqrt{m})^2$ for positive sequences.
• A sufficient condition for the integral inequality is $\operatorname{Re}[(Mg - f)(\bar{f} - m\bar{g})] \geq 0$ a.e., which reduces to $M\operatorname{Re}g \geq \operatorname{Re}f \geq m\operatorname{Re}g$ and $M\operatorname{Im}g \geq \operatorname{Im}f \geq m\operatorname{Im}g$ a.e.