A Remark on Esseen's Paper “A Moment Inequality with an Application to the Central Limit Theorem”

It is proved that
$$\lim_{n\to\infty}\inf_{\substack{-\infty<a<\infty\\<0<\sigma<\infty}}\sup_x\sqrt n\left|F_n(x)-\Phi\left(\frac{x-a}\sigma\right)\right|\leq\frac1{\sqrt{2\pi}}\rho_3,$$
where $\Phi (x)$ is a normal distribution function and $F_n (x)$ is a distribution function of a normed sum of independent identically distributed random variables. The constant $(2\pi)^{-1/2}$ cannot be improved.