On the absolute constants in the Berry-Esseen-type inequalities

By a modification of the method that was applied in (Korolev and Shevtsova, 2010), here the inequalities $\Delta_n\leq0.3328(\beta_3+0.429)/\sqrt{n}$ and $\Delta_n\leq0.33554(\beta_3+0.415)/\sqrt{n}$ are proved for the uniform distance $\Delta_n$ between the standard normal distribution function and the distribution function of the normalized sum of an arbitrary number $n\geq1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $\beta_3$. The first of these two inequalities improves one that was proved in (Korolev and Shevtsova, 2010), and as well sharpens the best known upper estimate for the absolute constant $C_0$ in the classical Berry--Esseen inequality to be $C_0<0.4756$, since $0.3328(\beta_3+0.429)\leq0.3328\cdot1.429\beta_3<0.4756\beta_3$ by virtue of the condition $\beta_3\geq1$. The second of these inequalities is also a structural improvement of the classical Berry--Esseen inequality, and as well sharpens the upper estimate for $C_0$ still more to be $C_0<0.4748$.