Norm Inequalities Related to the Heron and Heinz Means

Abstract

In this article, we present several inequalities treating operator means and the Cauchy-Schwarz inequality. In particular, we present some new comparisons between operator Heron and Heinz means, several generalizations of the difference version of the Heinz means and further refinements of the Cauchy-Schwarz inequality. The techniques used to accomplish these results include convexity and Löwner matrices.2010 Mathematics Subject Classification. 47A30, 47A63, 47A64, 15A60.

Matrix inequalities for unitarily invariant norms

“…Refinements and improvements for unitarily invariant norms have been extensively studied. Many researchers, e.g., Bhatia 10 , Fujii 11 , Hu 12, 13 , Kapil 14 , Kittaneh 15 , Kuzma 16 , Matharu 17 , paid attention to the improvement and generalization of inequalities for unitarily invariant norms.…”

Norm inequalities related to Heinz and Heron operator means

“…1 2 , arranged in decreasing order and repeated according to multiplicity. For Hermitian matrices A, B ∈ M n (C), we write that A 0 if A is positive semidefinite, A > 0 if A is positive definite, and A B if A − B 0.…”

Refining and reversing the weighted arithmetic–geometric mean inequality involving convex functionals and application for the functional entropy

“…In fact, according to the Kubo-Ando theory [14], (1.9) and (1.11) can be immediately deduced from (1.3) and (1.6), respectively. For more inequalities related to the Heinz mean involving matrix/operator arguments, we refer the reader to [4,8,9,11,15,24] and the related references cited therein.…”