Inequality in characteristic function

Question

Let $\phi$ be a characteristic function of random variable $X$. Prove that $1-|\phi(2u)|^2\leq 4(1-|\phi(u)|^2)$. I don't even have a clue how to start this.

Answer 1

Observe that: $$\phi_{-X}(u)=\mathbb Ee^{-iuX}=\mathbb E\overline{e^{iuX}}=\overline{\mathbb Ee^{iuX}}=\overline{\phi_X(u)}$$

Showing that - if $\phi(u)$ is a characteristic function - then so is $\overline{\phi(u)}$.

Also it is well known that a product of characteristic functions is again a characteristic function.

This together tells us that $|\phi(u)|^2=\phi(u)\overline{\phi(u)}$ is a characteristic function so this answer on this question (the link provided by bubububub) proves the statement.

Also you can take a look at the first part of this answer.