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Matrix order in Bohr inequality for operators. (English) Zbl 1186.47012

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 $\frac{1}{p}+\frac{1}{q}=1$. The equality holds if and only if $\left(p-1\right)a=b$. 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 $t\ne 0$,

${|A-B|}^{2}+\frac{1}{t}{|tA+B|}^{2}={\left(1+t\right)|A|}^{2}+\left(1+\frac{1}{t}\right){|B|}^{2}·$

##### MSC:
 47A63 Operator inequalities 47B15 Hermitian and normal operators