On the Berry–Esseen type inequalities for poisson random sums

For the uniform distance between the distribution function $\Phi(x)$ of the standard normal random variable and the distribution function $F_\lambda(x)$ of the Poisson random sum of independent identically distributed random variables $X_1, X_2,\dots$ with finite third absolute moment, $\lambda>0$ being the parameter of the Poisson index, it is proved the inequality
$$ \sup_{x}|F_\lambda(x)-\Phi(x)|\le 0.4532\frac{\mathsf E|X_1-\mathsf E X_1|^3}{(\mathsf D X_1)^{3/2}\sqrt{\lambda}}\,,\quad \lambda>0, $$
which is similar to the Berry–Esseen estimate and uses the central moments, unlike the known analogous inequalities based on the noncentral moments.