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. Vasi\'c and D.J. Ke\v{c}ki\'{c}: $$|\sum_{i=1}^n z_i|^r \leq (\sum_{i=1}^n \alpha_i^{1/(1-r)})^{r-1}\sum_{i=1}^n \alpha_i|z_i|^r \quad (r>1, z_i \in{\mathbb C}, \alpha_i>0, 1 \leq i \leq n) .$$ We also present some norm inequalities related to our noncommutative generalization of Bohr's inequality.