Further results on the concavity of operator means. (English) Zbl 07835165
Summary: In this work, we provide some further refinements for the following concavity inequality
\[
A \sigma B + C \sigma D \leq (A+C) \sigma (B+D)
\]
for positive invertible bounded linear operators \(A, B, C, D\) on a complex Hilbert space and for an operator mean \(\sigma\). To this end, we use the operator majorization intended for comparing two \(n + 1\)-tuples of pairs of operators, and prove a Hardy-Littlewood-Pólya-Karamata (HLPK) type theorem for the triangle map induced by \(\sigma\). Next, we apply a specification of the HLPK Theorem for \(n = 2\) in order to obtain the required refinements.
MSC:
| 47A63 | Linear operator inequalities |
| 47A50 | Equations and inequalities involving linear operators, with vector unknowns |
Keywords:
Hilbert space; positive invertible operator; mean of operators; operator majorization; concave mapReferences:
| [1] | Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl., 26, 203-241, 1979 · Zbl 0495.15018 |
| [2] | Ando, T., Majorizations and inequalities in matrix theory, Linear Algebra Appl., 199, 17-67, 1994 · Zbl 0798.15024 |
| [3] | Hardy, G. M.; Littlewood, J. E.; Pólya, G., Inequalities, 1952, Cambridge University Press: Cambridge University Press Cambridge · Zbl 0047.05302 |
| [4] | Hiai, F., Matrix analysis: matrix monotone functions, matrix means, and majorization, Interdiscip. Inf. Sci., 16, 2, 139-248, 2010 · Zbl 1206.15019 |
| [5] | Karamata, J., Sur une inégalité rélative aux fonctions convexes, Publ. Math. Univ. Belgrade, 1, 145-148, 1932 · JFM 58.0211.01 |
| [6] | Kubo, F.; Ando, T., Means of positive linear operators, Math. Ann., 246, 205-224, 1980 · Zbl 0412.47013 |
| [7] | Marshall, A. W.; Olkin, I.; Arnold, B. C., Inequalities: Theory of Majorization and Its Applications, 2011, Springer: Springer New York · Zbl 1219.26003 |
| [8] | Mićić, J.; Pečarić, J.; Seo, Y., Complementary inequalities to inequalities of Jensen and Ando based on the Mond-Pečarić method, Linear Algebra Appl., 318, 87-107, 2000 · Zbl 0971.47014 |
| [9] | Mićić, J.; Moradi, H. R., Some inequalities involving operator means and monotone convex functions, J. Math. Inequal., 14, 135-145, 2020 · Zbl 07199917 |
| [10] | Niezgoda, M., Locally strong majorization and commutativity in \(C^\ast \)-algebras with applications, Results Math., 74, 1, Article 46 pp., 2019, 16 p. · Zbl 1506.47023 |
| [11] | Niezgoda, M., Refinements of triangle like inequalities in Lie’s framework, Bull. Malays. Math. Sci. Soc., 44, 243-250, 2021 · Zbl 1469.22004 |
| [12] | Niezgoda, M., On triangle inequality for Miranda-Thompson’s majorization and gradients of increasing functions, Adv. Oper. Theory, 5, 647-656, 2020 · Zbl 1440.15013 |
| [13] | Niezgoda, M., Refinements of Ky Fan’s eigenvalue inequality for simple Euclidean Jordan algebras by using gradients of K-increasing functions, Math. Inequal. Appl., 24, 1065-1073, 2021 · Zbl 1510.17057 |
| [14] | Niezgoda, M., G-majorization and Fejér and Hermite-Hadamard like inequalities for G-symmetrized convex functions, J. Convex Anal., 29, 1, 231-242, 2022 · Zbl 1497.26034 |
| [15] | Niezgoda, M., A complementary result to the superadditivity of operator means, Commun. Algebra, 50, 8, 3338-3345, 2022 · Zbl 1507.47040 |
| [16] | Niezgoda, M.; Pečarić, J., Hardy-Littlewood-Pólya-type theorems for invex functions, Comput. Math. Appl., 64, 4, 518-526, 2012 · Zbl 1252.26011 |
| [17] | Ng, C. T., Functions generating Schur-convex sums, Int. Ser. Numer. Math., 80, 433-438, 1985 · Zbl 0634.39014 |
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