Geodesic
A Remark on Esseen's Paper ``A Moment Inequality with an Application to the Central Limit Theorem''
Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 1, pp. 125-128

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It is proved that $$\lim_{n\to\infty}\inf_{\substack{-\infty\infty\\0\sigma\infty}}\sup_x\sqrt n\left|F_n(x)-\Phi\left(\frac{x-a}\sigma\right)\right|\leq\frac1{\sqrt{2\pi}}\rho_3,$$ 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. 
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     author = {B. A. Rogozin},
     title = {A {Remark} on {Esseen's} {Paper} {``A} {Moment} {Inequality} with an {Application} to the {Central} {Limit} {Theorem''}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {125--128},
     publisher = {mathdoc},
     volume = {5},
     number = {1},
     year = {1960},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1960_5_1_a9/}
}
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B. A. Rogozin. A Remark on Esseen's Paper ``A Moment Inequality with an Application to the Central Limit Theorem''. Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 1, pp. 125-128. http://geodesic.mathdoc.fr/item/TVP_1960_5_1_a9/