@article{1186.47012,
author="Fujii, Masatoshi and Zuo, Hongliang",
title="{Matrix order in Bohr inequality for operators.}",
language="English",
journal="Banach J. Math. Anal. ",
volume="4",
number="1",
pages="21-27",
year="2010",
abstract="{Summary: The classical Bohr inequality says that $|a+b|^2\le
    p|a|2+q|b|^2$ for all scalars $a,b$ and $p,q>0$ with $\frac1p+ \frac1q=1$.
    The equality holds if and only if $(p-1)a=b$. Several authors discussed
    operator versions of the Bohr inequality. In this paper, we give a unified
    proof to operator generalizations of the Bohr inequality. One viewpoint of
    ours is a matrix inequality, and the other is a generalized parallelogram
    law for the absolute value of operators, i.e., for operators $A$ and $B$ on
    a Hilbert space and $t\ne 0$,
 $$|A-B|^2+ \tfrac 1t|tA+B|^2= (1+t)|A|^2+
    \big(1+\tfrac1t\big) |B|^2.$$}",
keywords="{Bohr inequality for operators; matrix order; parallelogram law for
    operators; absolute value of operators}",
classmath="{*47A63 (Operator inequalities, etc.)
47B15 (Hermitian and normal operators)
}",
}