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

An optimal Berry-Esseen type theorem for integrals of smooth functions