Optimal constants and equality conditions for a class of quadratic functional inequalities in real inner product spaces

Baofeng Lai; Baofeng Lai

doi:10.3934/math.20261086

AIMS Mathematics

2026, Volume 11, Issue 8: 27116-27130. doi: 10.3934/math.20261086

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Optimal constants and equality conditions for a class of quadratic functional inequalities in real inner product spaces

Baofeng Lai ^,

School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou, Fujian, 362000, China

Received: 09 June 2026 Revised: 19 August 2026 Accepted: 21 August 2026 Published: 27 August 2026
MSC : 46C05, 26D15, 15A18

In this paper, we systematically studied the optimal constants and equality conditions for a class of inequalities involving finitely many linear functionals on a real inner product space $ V $. These inequalities arose naturally when combining several Cauchy-Schwarz inequalities, where the resulting optimal constants could not be obtained by simply adding or multiplying the individual bounds. By performing an orthogonal decomposition of $ V $, we transformed the problem into the extremum of a quadratic functional, which was then solved completely using the Rayleigh quotient theory for real symmetric matrices. The main result, Theorem 3.1, provided a unified framework that reduced the determination of sharp constants to computing the extremal eigenvalues of a matrix associated with the Gram matrix. Complete characterizations of the equality cases were also provided. This framework was then applied to solve several illustrative problems, including Problem 12318 from the April 2022 issue of the American Mathematical Monthly. Furthermore, a brief exploration of a non-quadratic case was also included to demonstrate the broader applicability of the orthogonal decomposition approach. The method highlighted the role of the finite-dimensional reduction technique in infinite-dimensional settings and discussed potential extensions to complex inner product spaces, which may be of independent interest for further research.
- real inner product space,
- orthogonal decomposition,
- Gram matrix,
- Rayleigh quotient,
- optimal constants,
- equality conditions
Citation: Baofeng Lai. Optimal constants and equality conditions for a class of quadratic functional inequalities in real inner product spaces[J]. AIMS Mathematics, 2026, 11(8): 27116-27130. doi: 10.3934/math.20261086

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Abstract

In this paper, we systematically studied the optimal constants and equality conditions for a class of inequalities involving finitely many linear functionals on a real inner product space $ V $. These inequalities arose naturally when combining several Cauchy-Schwarz inequalities, where the resulting optimal constants could not be obtained by simply adding or multiplying the individual bounds. By performing an orthogonal decomposition of $ V $, we transformed the problem into the extremum of a quadratic functional, which was then solved completely using the Rayleigh quotient theory for real symmetric matrices. The main result, Theorem 3.1, provided a unified framework that reduced the determination of sharp constants to computing the extremal eigenvalues of a matrix associated with the Gram matrix. Complete characterizations of the equality cases were also provided. This framework was then applied to solve several illustrative problems, including Problem 12318 from the April 2022 issue of the American Mathematical Monthly. Furthermore, a brief exploration of a non-quadratic case was also included to demonstrate the broader applicability of the orthogonal decomposition approach. The method highlighted the role of the finite-dimensional reduction technique in infinite-dimensional settings and discussed potential extensions to complex inner product spaces, which may be of independent interest for further research.

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