| We investigate relationships between the Moore-Penrose inverse $(ABA^{*})^{\dag}$ and the product $[(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ through some rank and inertia formulas for the difference of $(ABA^{*})^{\dag} - [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$, where $B$ is Hermitian matrix and $(AB)^{(1,2,3)}$ is a $\{1,\, 2,\,3\}$-inverse of $AB$. We show that there always exists an $(AB)^{(1,2,3)}$ such that $(ABA^{*})^{\dag}=[(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ holds. In addition, we also establish necessary and sufficient conditions for the two inequalities $(ABA^{*})^{\dag} \succ [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ and $(ABA^{*})^{\dag} \prec [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ to hold in the L\"owner partial ordering. Some variations of the equalities and inequalities are also presented. In particular, some equalities and inequalities for the Moore-Penrose inverse of the sum $A + B$ of two Hermitian matrices $A$ and $B$ are established. |
