  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.
