Annals of Functional Analysis

Further Refinements of Zhan's Inequality for Unitarily Invariant Norms

Masatoshi Fujii, Yuki Seo, and Hongliang Zuo

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$\newcommand{\UIN}{|\kern-1pt|kern-1pt|}$ In this paper, we show a further improvement of the integral Heinz mean inequality and prove \begin{eqnarray*} \frac{1}{2} \UIN{A^{2}X+2AXB+XB^{2}}\leq \frac{2}{t+2}\UIN{A^{2}X+tAXB+XB^{2}} \ \ \mbox{for all}\ \ t\in (-2, 2].\end{eqnarray*} Then we show some refinements of unitarily invariant norm inequalities, in particular we proved that: If $A, B, X \in M_{n}$ with $A$ and $B$ positive definite, and $f$, $g$ are two continuous functions on $(0,\infty)$ such that $h(x)= \frac{f(x)}{g(x)}$ is Kwong, then \begin{eqnarray*}\UIN{A^{\frac{1}{2}}(f(A)Xg(B)+g(A)Xf(B))B^{\frac{1}{2}}}\leq \frac{k}{2}\UIN{ A^{2}X+2AXB+XB^{2}}\end{eqnarray*} holds for any unitarily invariant norm, where $k=\max{\{\frac{f(\lambda)g(\lambda)}{\lambda}| \lambda \in \sigma(A)\cup \sigma(B)\}}$.

Article information

Ann. Funct. Anal., Volume 6, Number 2 (2015), 234-241.

First available in Project Euclid: 19 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
Secondary: 47A63: Operator inequalities 47A30: Norms (inequalities, more than one norm, etc.)

Heinz inequality Zhan's inequality Hermite-Hadamard inequality unitarily invariant norm


Zuo, Hongliang; Seo, Yuki; Fujii, Masatoshi. Further Refinements of Zhan's Inequality for Unitarily Invariant Norms. Ann. Funct. Anal. 6 (2015), no. 2, 234--241. doi:10.15352/afa/06-2-20.

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