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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 $\frac1p+ \frac1q=1$. 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 $t\ne 0$, $$|A-B|^2+ \tfrac 1t|tA+B|^2= (1+t)|A|^2+ \big(1+\tfrac1t\big) |B|^2.$$

47A63Operator inequalities
47B15Hermitian and normal operators
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