An operator extension of Bohr's inequality

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}: $$\mathrm{\mid }\sum _{i=1}^n{z}_i{\mathrm{\mid }}^r\le (\sum _{i=1}^n{\alpha }_i^{1\mathrm{/}(1-r)}{)}^{r-1}\sum _{i=1}^n{\alpha }_i\mathrm{\mid }{z}_i{\mathrm{\mid }}^r(r>1,z_i\in \mathbb{C},{\alpha }_i>0,1\le i\le n)\mathrm{.}$$ We also present some norm inequalities related to our noncommutative generalization of Bohr’s inequality.