SavedAdd your journal

Journals / Fundamentals of Contemporary Mathematical Sciences / 2026 / Volume: 7 Issue: 2

A KKT-Based Approach to Kantorovich-Type Inequalities

İbrahim Halil Gümüş(Adiyaman University)0000-0002-3071-1159
Asiye Gezen(ADIYAMAN UNIVERSITY)0009-0009-2601-800X
Pages
—
Publication Date
2026-07-31
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Our results provide a systematic variational framework for deriving and understanding Kantorovichtype inequalities, offering both alternative proofs and a basis for further generalizations in matrix analysis.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We develop a unified optimization-based framework for Kantorovich-type inequalities using Karush–Kuhn–Tucker (KKT) conditions.
📊
Key Findings & Data:For two commuting positive definite matrices A and B, we derive a sharp eigenvalue-based upper bound for the product (Ax, x)(Bx, x), which recovers the classical Kantorovich inequality as a special case when B = A−1 .
💡
Conclusion & Impact:Our results provide a systematic variational framework for deriving and understanding Kantorovichtype inequalities, offering both alternative proofs and a basis for further generalizations in matrix analysis.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Kantorovich inequality · KKT conditions · Quadratic forms

Abstract

We develop a unified optimization-based framework for Kantorovich-type inequalities using Karush–Kuhn–Tucker (KKT) conditions. For two commuting positive definite matrices A and B, we derive a sharp eigenvalue-based upper bound for the product (Ax, x)(Bx, x), which recovers the classical Kantorovich inequality as a special case when B = A−1 . A central feature of our approach is a structural characterization of extremal configurations: We show that every global maximizer is supported on at most two indices. This reveals that the reduction to pairs of eigenvalues arises naturally from first-order optimality conditions. In addition, we establish a weighted inequality involving explicit optimal constants, extending known results related to the generalized Kantorovich constant K(h, p). Our results provide a systematic variational framework for deriving and understanding Kantorovichtype inequalities, offering both alternative proofs and a basis for further generalizations in matrix analysis.

🔖

Cite This Article

Gümüş, İ. H., Gezen, A. (2026). A KKT-Based Approach to Kantorovich-Type Inequalities. Fundamentals of Contemporary Mathematical Sciences, Volume: 7 Issue: 2. https://doi.org/10.54974/fcmathsci.1861947

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (14)

  1. Alpargu G., The Kantorovich Inequality, with Some Extensions and with Some Statistical Applications, M.Sc. Thesis, McGill University, Montreal, Canada, 1996.
  2. Bhatia R., Positive Definite Matrices, Princeton University Press, 2007.
  3. Dannan F.M., Generalized Kantorovich constant, a new formulation and properties, Problemy Analiza-Issues of Analysis, 9(27) (1), 38-51, 2020.
  4. Furuta T., Operator inequalities associated with Hölder–McCarthy and Kantorovich inequalities, Journal of Inequalities and Applications, 2(2), 137-148, 1998.
  5. Furuichi S., Moradi H.R., Advances in Mathematical Inequalities, De Gruyter, 2020.
  6. Güler O., Foundations of Optimization, Graduate Texts in Mathematics, 258, Springer, 2010.
  7. Huang J., Zhou J., A direct proof and a generalization for a Kantorovich type inequality, Linear Algebra and Its Applications, 397, 185-192, 2005.
  8. Kantorovich L.V., Functional analysis and applied mathematics, Uspekhi Matematicheskikh Nauk, 3(6), 89-185, 1948.
  9. Moradi H.R., Gümüş İ.H., Heydarbeygi Z., A glimpse at the operator Kantorovich inequality, Linear and Multilinear Algebra, 67(5), 1031-1036, 2019.
  10. Moslehian M.S., Recent developments of the operator Kantorovich inequality, Expositiones Mathematicae, 30(4), 376-388, 2012.
  11. Nocedal J., Wright S.J., Numerical Optimization, Springer, 1999.
  12. Pečarić J., Furuta T., Mićić Hot J., Seo Y., Mond-Pečarić Method in Operator Inequalities, Element, 2005.
  13. Wolkowicz H., Generating eigenvalue bounds using optimization, Nonlinear Analysis and Variational Problems: In Honor of George Isac, Springer, 465-490, 2010.
  14. Xie Z., Liu J., Zhang M., A Kantorovich type inequality for positive matrices, Linear and Multilinear Algebra, 73(8), 1777-1791, 2025.