Further refinements of the Heinz inequality

The celebrated Heinz inequality asserts that $2|||A^{1/2}XB^{1/2}|||\leq |||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||\leq |||AX+XB|||$ for $X \in \mathbb{B}(\mathscr{H})$, 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 $F(\nu)=|||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||$, 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{\alpha+\beta}{2}}XB^{1-\frac{\alpha+\beta}{2}}+A^{1-\frac{\alpha+\beta}{2}}XB^{\frac{\alpha+\beta}{2}}\right|\right|\right|\leq\frac{1}{|\beta-\alpha|} \left|\left|\left|\int_{\alpha}^{\beta}\left(A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\right)d\nu\right|\right|\right|\nonumber\\ &&\qquad\qquad\leq \frac{1}{2}\left|\left|\left|A^{\alpha}XB^{1-\alpha}+A^{1-\alpha}XB^{\alpha}+A^{\beta}XB^{1-\beta}+A^{1-\beta}XB^{\beta}\right|\right|\right|\,, \end{eqnarray*} for real numbers $\alpha, \beta$.Comment: 15 pages, to appear in Linear Algebra Appl. (LAA

Detection of the onset of nanocrystallization by calorimetric and magnetic measurements

It is generally accepted that measurements of the magnetic properties are more sensitive than measurements of the enthalpy changes in the detection of the onset of crystallization of ferromagnetic phases emerging from a paramagnetic amorphous alloy. In this work, it is shown that the formation of a very fine nanocrystalline microstructure can make this assumption incorrect. Under some circumstances, the nanocrystallization onset temperature obtained from magnetic techniques is higher than the one obtained from enthalpy changes. The phenomenon is explained in terms of the superparamagnetic behavior of the uncoupled nanocrystals at the very early stages of nanocrystallizatio

Thermomagnetic detection of recrystallization in FeCoNbBCu nanocrystalline alloys

The recrystallization process in FeCoNbBCu nanocrystallinealloys is evidenced from thermomagnetic results as a significant decrease in magnetization at the second crystallization stage. The lowering in the volume fraction of α-FeCo crystals indicates that some of these crystals contribute to the boride phases formed. Electron microscopy images reveal that the final microstructure consists of large crystals (∼500 nm) of a fcc (FeCo)23B6(FeCo)23B6 phase and small crystals (∼20 nm) of bcc α-FeCo and of some boride phases as such (FeCo)2B

Argon assisted chemical vapor deposition of CrO2_2: an efficient process leading to high quality epitaxial films

A comparative study of the structural, microstructural and magnetic properties of CrO$_2$ thin films grown onto (110) and (100) TiO$_2$ rutile single crystal substrates by chemical vapor deposition (CVD), using CrO$_3$ as chromium precursor and either oxygen or argon as carrier gas is presented. Our results show that growth under argon carrier gas leads to high quality CrO$_2$ epilayers with structural and magnetic properties similar to those obtained using the more standard oxygen carrier gas. Furthermore, we interpret the larger magnetic coercivity observed for the (110) oriented films in terms of their microstructure, in particular of the highest strain and edge roughness of the building structures of the CrO$_2$ epilayers, which are settled by the substrate crystallographic orientation.Comment: 27 pages, 2 tables, 8 figure