We prove a central limit error bound for convolution powers of laws with finite moments of order r ]2, 3], 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 (1942) 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. Our proof uses a partial generalisation of the theorem of Senatov and Zolotarev (1986) used for the earlier special case. A result more general than our main one could be obtained by using instead another theorem of Senatov (1980), but unfortunately an auxiliary inequality used in the latter's proof is wrong.
Jonas et al. (Sun,) studied this question.