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Abstract
For any unitarily invariant norm �| · �| , the Heinz inequalities for operators assert that 2�| A 1 2 XB 1 2 �| ≤ �| A ν XB 1-ν + A 1-ν XB ν �| ≤ �| AX + XB�| ,f orA, B ,a ndX any operators on a complex separable Hilbert space such that A, B are positive and ν ∈ (0, 1). In this paper, we obtain a family of refinements of these norm inequalities by using the convexity of the function f (ν )= �| A ν XB 1-ν + A 1-ν XB ν �| and the Hermite-Hadamard inequality.
Highlights
Let Mn(C) be the algebra of n × n complex matrices
We denote by Hn(C) the set of all Hermitian matrices in Mn(C)
The arithmetic-geometric mean inequality for two nonnegative real numbers a and b is which has been generalized to the context of matrices as follows:
Summary
Let Mn(C) be the algebra of n × n complex matrices. We denote by Hn(C) the set of all Hermitian matrices in Mn(C). For any unitarily invariant norm | · |, the Heinz inequalities for operators assert that We obtain a family of refinements of these norm inequalities by using the convexity of the function f (ν) = |Aν XB1–ν + A1–ν XBν | and the Hermite-Hadamard inequality. The arithmetic-geometric mean inequality for two nonnegative real numbers a and b is which has been generalized to the context of matrices as follows:
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