Further refinements of the Heinz inequality

Abstract

The celebrated Heinz inequality asserts that $2|||A^{1/2}X{B}^{1/2}|||\le |||A^{\nu }X{B}^{1-\nu }+A^{1-\nu }X{B}^{\nu }|||\le |||AX+XB|||$ for $X\in \mathbb{B}(\mathcal{H})$, $A,B\in \+$, every unitarily invariant norm $|||\cdot |||$ and $\nu \in [0,1]$. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function $F(\nu )=|||A^{\nu }X{B}^{1-\nu }+A^{1-\nu }X{B}^{\nu }|||$, some integration techniques and various refinements of the Hermite--Hadamard inequality. In the setting of matrices we prove that \begin{eqnarray*} &&\hspace{-0.5cm}\left|\left|\left|A^{\frac{α+β}{2}}XB^{1-\frac{α+β}{2}}+A^{1-\frac{α+β}{2}}XB^{\frac{α+β}{2}}\right|\right|\right|\leq\frac{1}{|β-α|} \left|\left|\left|\int_α^β\left(A^νXB^{1-ν}+A^{1-ν}XB^ν\right)dν\right|\right|\right|\nonumber\\ &&\qquad\qquad\leq \frac{1}{2}\left|\left|\left|A^αXB^{1-α}+A^{1-α}XB^α+A^βXB^{1-β}+A^{1-β}XB^β\right|\right|\right|\,, \end{eqnarray*} for real numbers $\alpha ,\beta$.

15 pages, to appear in Linear Algebra Appl. (LAA)

References in corpus (1)

• Matrix Hermite-Hadamard type inequalities

Cited by in corpus (3)

• Quadratic interpolation of the Heinz means
• Norm inequalities related to the Heron and Heinz means
• Some generalized numerical radius inequalities for Hilbert space operators