The reverse order laws for $\{1,2,3M\}$- and $\{1,2,4N\}$- inverse of three matrix products

The reverse order laws for weighted generalized inverses often appear in linear algebra problems of several applied fields, having attracted considerable attention. In this paper, by using the maximal and minimal ranks of the generalized Schur complement, we obtained some necessary and sufficient conditions for the reverse order laws

$$A_3\{1,2,3M_3\}{A}_2\{1,2,3M_2\}{A}_1\{1,2,3M_1\}\subseteq (A_1{A}_2{A}_3)\{1,2,3M_1\}$$

and

$$A_3\{1,2,4N_4\}{A}_2\{1,2,4N_3\}{A}_1\{1,2,4N_2\}\subseteq (A_1{A}_2{A}_3)\{1,2,4N_4\}.$$