This paper is concerned with the problem of estimating $|a_4-a_2a_3|$, where $a_k$ are the coefficients of a given close-to-convex function. The bounds of this expression for various classes of analytic functions have been applied to estimate the third Hankel determinant $H_3(1)$. The results for two subclasses of the class $\mathcal{C}$ of all close-to-convex functions are sharp. This bound is equal to 2. It is conjectured that this number is also the exact bound of $|a_4-a_2a_3|$ for the whole class $\mathcal{C}$.