Showing $1-\Re \varphi(2t)\leq 4(1-\Re \varphi(t)) $

Question

I need to show that $1-\Re \varphi(2t)\leq 4(1-\Re \varphi(t)) $ where $t \in \mathbb{R}$ for evry characteristic function $\varphi$.

I know that if the random variable is symmetrical then the characteristic function will be real. So how should I start to prove given inequality and what properties to apply?

Answer 1

Note that $\Re \varphi(t) = \Bbb E[\cos tX]$. For all $u\in\Bbb R$, we have $$\begin{eqnarray} 1-\cos 2u =2\sin^2 u&=&2-2\cos^2 u\le 4-4\cos u \end{eqnarray}$$ since $2(\cos u-1)^2=2\cos^2 u +2-4\cos u\ge 0$ holds for all $u$. Now we have $$\begin{eqnarray} 1-\Re \varphi(2t)=\Bbb E[1-\cos 2tX]&\leq &\Bbb E[4-4\cos tX]=4-4\Re \varphi(t) \end{eqnarray}$$ which follows from $1-\cos 2tX\le 4-4\cos tX$.