An operator extension of Bohr's inequality

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

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