A generalized Kahane-Khinchin inequality

The inequality ${\displaystyle \int \log |\sum a_n{e}^{2\pi i{\phi }_n}|d{\phi }_1\dots d{\phi }_n\ge C{\log (\sum {\left|a_n\right|}^2)}^{1/2}}$ with an absolute constant C, and similar ones, are extended to the case of ${\displaystyle a_n}$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by ${\displaystyle e^{2\pi i\phi }}$.