  We prove a Berry-Esseen type inequality for approximating expectations of
sufficiently smooth functions $f$, like $f=|\cdot|^3$, with respect to
standardized convolutions of laws $P_1,\ldots, P_n$ on the real line by
corresponding expectations based on symmetric two-point laws $Q_1,\ldots,Q_n$
isoscedastic to the $P_i$. Equality is attained for every possible
constellation of the Lipschitz constant $\|f"\|^{}_{\mathrm{L}}$ and the
variances and the third centred absolute moments of the $P_i$. The error bound
is strictly smaller than $\frac 16$ times the Lyapunov ratio times
$\|f"\|^{}_{\mathrm{L}}$, and tends to zero also if $n$ is fixed and the third
standardized absolute moments of the $P_i$ tend to one.
  In the homoscedastic case of equal variances of the $P_i$, and hence in
particular in the i.i.d. case, the approximating law is a standardized
symmetric binomial one.
  The inequality is strong enough to yield for some constellations, in
particular in the i.i.d. case with $n$ large enough given the standardized
third absolute moment of $P_1$, an improvement of a more classical and already
optimal Berry-Esseen type inequality of Tyurin (2009).
  Auxiliary results presented include some inequalities either purely
analytical or concerning Zolotarev's $\zeta$-metrics, and some binomial moment
calculations.
