Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Suppose $p \geq 2$. Show that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } \leq \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p }$$
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Suppose $p \geq 2$. Show that
$$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } \leq \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p }$$