A theorem on stability of the argument of characteristic function

Let $f(x)$ be the characteristic function of a probability distribution on the line. If $1-|f(t)|\le\varepsilon$ for $|t|\le a$ and, moreover, $\varepsilon\le C_1$, then
$$ \min_{\beta\in R} \max_{|t|\leq a}|\arg f(t)-\beta t|\leq C_2\varepsilon^{3/4}, $$
where $C_1$, $C_2$ are suitable absolute constants.