## Banach Journal of Mathematical Analysis

### Matrix order in Bohr inequality for operators

#### Abstract

The classical Bohr inequality says that $|a+b|^2\leq p|a|^2+q|b|^2$ for all scalars $a, \ b$ and positive $p,q$ with $\frac 1p + \frac 1q =1.$ The equality holds if and only if $(p-1)a=b.$ Several authors discussed operator version of Bohr inequality. In this paper, we give a unified proof to operator generalizations of Bohr inequality. One viewpoint of ours is a matrix inequality, and the other is a generalized parallelogram law for absolute value of operators, i.e., for operators $A$ and $B$ on a Hilbert space and $t\neq0$, $$|A-B|^2+\frac{1}{t}|tA+B|^2=(1+t)|A|^2+(1+\frac{1}{t})|B|^2.$$

#### Article information

Source
Banach J. Math. Anal., Volume 4, Number 1 (2010), 21-27.

Dates
First available in Project Euclid: 27 April 2010

https://projecteuclid.org/euclid.bjma/1272374669

Digital Object Identifier
doi:10.15352/bjma/1272374669

Mathematical Reviews number (MathSciNet)
MR2593904

Zentralblatt MATH identifier
1186.47012

#### Citation

Fujii, Masatoshi; Zuo, Hongliang. Matrix order in Bohr inequality for operators. Banach J. Math. Anal. 4 (2010), no. 1, 21--27. doi:10.15352/bjma/1272374669. https://projecteuclid.org/euclid.bjma/1272374669

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