By a modification of the method that was applied in (Korolev and Shevtsova,
2009), here the inequalities
$$\rho(F_n,\Phi)\le\frac{0.335789(\beta^3+0.425)}{\sqrt{n}}$$ and
$$\rho(F_n,\Phi)\le \frac{0.3051(\beta^3+1)}{\sqrt{n}} $$ are proved for the
uniform distance $\rho(F_n,\Phi)$ between the standard normal distribution
function $\Phi$ and the distribution function $F_n$ of the normalized sum of an
arbitrary number $n\ge1$ of independent identically distributed random
variables with zero mean, unit variance and finite third absolute moment
$\beta^3$. The first of these inequalities sharpens the best known version of
the classical Berry--Esseen inequality since
$0.335789(\beta^3+0.425)\le0.335789(1+0.425)\beta^3<0.4785\beta^3$ by virtue of
the condition $\beta^3\ge1$, and 0.4785 is the best known upper estimate of the
absolute constant in the classical Berry--Esseen inequality. The second
inequality is applied to lowering the upper estimate of the absolute constant
in the analog of the Berry--Esseen inequality for Poisson random sums to 0.3051
which is strictly less than the least possible value of the absolute constant
in the classical Berry--Esseen inequality. As a corollary, the estimates of the
rate of convergence in limit theorems for compound mixed Poisson distributions
are refined.

通过改进 Korolev 和 Shevtsova（2009）的方法，针对满足零均值、单位方差和有限三阶绝对矩的正态分布函数和归一化随机变量和的分布函数，证明了两个不等式。第一个不等式是关于经典 Berry-Esseen 不等式最优版本的加强版本，第二个不等式应用于降低泊松随机和 Berry-Esseen 不等式的绝对常数上限。作为推论，改进了混合泊松分布极限定理中的收敛速度估计。