pr.probability – $L^p$ inequality for “positively correlated” random variables Answer

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pr.probability – $L^p$ inequality for “positively correlated” random variables

Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following “positive correlation” property: for all non-negative integers $\alpha_1,\ldots,\alpha_m,\beta_1,\ldots,\beta_m$ we have
$$ \mathbb E(\xi_1^{\alpha_1}\overline{\xi_1^{\beta_1}}\ldots \xi_m^{\alpha_m}\overline{\xi_m^{\beta_m}})\geq 0. $ $
Now take any complex numbers $a_1,\ldots,a_m$ with $\max_i|a_i|\leq 1$ and set
$$ M_p(a)=\mathbb E\left|a_1\xi_1+\ldots+a_m\xi_m\right|^p. $$
When $p=2k$ other $k$ is a positive integer, $M_p(a)$ is a polynomial in $a_i$ other $\overline{a_i}$ with non-negative coefficients, therefore
$$ M_{p}(a)\leq M_{p}(1,1,\ldots,1). $$
My question is: will this inequality hold for all real $p\geq 2$? And if not, what if we take $\xi_j=\exp(i\lambda_j N)$where $\lambda_j$ are real and $N$ is (the same for different $j$) standard normally distributed variable?

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