In probability theory, if we have $\xi$ and $\eta$ as random variables, the Cauchy Inequality can be designated as:$$E(\xi \eta) \leqslant \sqrt{E(\xi^2)E(\eta^2)}$$Then I have an idea to make an promotion of this inequality:$$E\left(\prod_{k=1}^n \xi_k\right) \leqslant \sqrt[n]{\prod_{k=1}^n E(\xi_k^n)}\tag{1}$$I tried to prove this inequality by using Carlson Inequality:$$\left(\int_{-\infty}^{+\infty} \prod_{k=1}^n x_k \mathrm{d}F(x_1,x_2,\cdots,x_n)\right)^n \leqslant \prod_{k=1}^n \int_{-\infty}^{+\infty} x_k^n \mathrm{d}F_k(x) $$Is inequality (1) true? If so, is this a proper way to prove (1) ?
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