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Dergiler / 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
Sayfa
—
Yayın Tarihi
2026-07-31
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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 Maddede Bu Makale

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Amaç & Hipotez:We develop a unified optimization-based framework for Kantorovich-type inequalities using Karush–Kuhn–Tucker (KKT) conditions.
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Temel Bulgular & Veri: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 .
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Sonuç & Katkı: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.
🔬Tür:Modelleme & Sayısal Simülasyon
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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.

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Bu Makaleye Atıf Yap

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

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