A sharper Lyapunov-Katz central limit error bound for i.i.d. summands Zolotarev-close to normal

We prove a central limit error bound for convolution powers of laws with finite moments of order $r \in \mathopen]2, 3\mathclose]$, taking a closeness of the laws to normality into account. Up to a universal constant, this generalises the case of $r=3$ of the sharpening of the Berry (1941) - Esseen (1663) theorem obtained by Mattner (2024), namely by sharpening here the Katz (1963) error bound for the i.i.d. case of Lyapunov's (1901) theorem. For this purpose, we introduce Zolotarev's (1976) $\zeta$ distances for probability measures and a certain variant thereof due to Senatov (1980). Our proof uses a convolution inequality obtained by Mattner (2024) and a partial generalisation of a theorem of Senatov and Zolotarev (1986).