DOI: 10.54974/fcmathsci.1861947 ISSN: 2717-6185

A KKT-Based Approach to Kantorovich-Type Inequalities

İbrahim Halil Gümüş, Asiye Gezen
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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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