On Asymptotically Best Constants in More Exact Forms of Mean Limit Theorems

Let $\mathfrak{B}$ be a class of distribution functions $F$ with a finite third absolute moment $\beta$ and first moment equal to zero. Also, let $F_n (x)=F^{* n}(x\sigma\sqrt n)$, where $\sigma^2 $ is the second moment $F$, $\Phi _{a,b}$ is the normal distribution with parameters $(a,b)$ and $\rho _3 $ is the mean metric, i.e., \[ \rho _3 (G,H)=\int|G - H|dx.\] C. Esseen [1] has calculated the value of
$$ A_3(F)=\lim_{n\to\infty}\sqrt n\rho _3(F_n,\Phi _{0,1}). $$
In this paper we calculate the value of \[ \bar A_3 (F)=\mathop{\lim }\limits_{n \to\infty}\sqrt n\mathop{\inf }\limits_{a,b}\rho _3 (F_n,\Phi _{0,1}),\] the asymptotic of parameters $a_n$, $b_n$, which realize $\inf.\rho _3$, and we prove that \[ B_3=\mathop{\sup }\limits_\mathfrak{B}\frac{{\sigma ^3 }}{\beta }A_3 (F)=\bar B_3=\mathop{\sup}\limits_\mathfrak{B} \frac{{\sigma ^3}}{\beta }\bar A_3 (F)=\frac{1}{2}.\]