We prove that whenever $S$ is a weighted sum of n independent Rademacher random variables, then $||S||_p / ||S||_4 \leq ||G||_p / ||G|| _4$, where $G$ is a standard Gaussian random variable and $p \geq 4$. Moreover, we prove that for fixed $n$ and $p \geq 5$, the maximum is attained in a case where all, except at most one, coefficients of Rademacher sum are equal. As a corollary of the main result, we show that $||S||_p \leq (1 - \Omega(1/n))||G||_p$ and $1/n$ is the optimal order in such an estimate.