An operator extension of Bohr's inequality

Abstract

We establish an operator extension of the following generalization of Bohr's inequality, due to M.P. Vasić and D.J. Kečkić: $$|\sum _{i=1}^n{z}_i{|}^r\le (\sum _{i=1}^n\hat{I}{\pm }_i^{1/(1-r)}{)}^{r-1}\sum _{i=1}^n\hat{I}{\pm }_i|z_i{|}^r(r>1,z_i\in \mathbb{C},\hat{I}{\pm }_i>0,1\le i\le n).$$ We also present some norm inequalities related to our noncommutative generalization of Bohr's inequality.

7 Pages, to appear in Bull. Iranian Math. Soc