Let us consider a matrix with positive elements: ${\bf X}^{k\times2}=[X_1:\ldots:X_k]'$ with $X_i=(1,X_{1i})',\;i=1,\ldots,k$. Also consider ${\bf X}^{(-1)}$ as the Moore-Penrose inverse of ${\bf X}$ and the projection matrix $P_X={\bf X}{\bf X}^{(-1)}$.

Let us consider three diagonal matrices with dimension $k\times k$ as $D_1,D_2$ and $D_3$.

Then is it true that: $$[{\bf X}'D_1P_XD_2P'_XD_3{\bf X}]_{[1,1]}\geq[{\bf X}'D_1D_2D_3{\bf X}]_{[1,1]},$$ and $$[{\bf X}'D_1P_XD_2P'_XD_3{\bf X}]_{[2,2]}\geq[{\bf X}'D_1D_2D_3{\bf X}]_{[2,2]},$$ where $M_{[1,1]}$ is the $[1,1]$ elements of $M$.

Let us consider a matrix with positive elements: ${\bf X}^{k\times2}=[X_1:\ldots:X_k]'$ with $X_i=(1,X_{1i})',\;i=1,\ldots,k$. Also consider ${\bf X}^{(-1)}$ as the Moore-Penrose inverse of ${\bf X}$ and the projection matrix $P_X={\bf X}{\bf X}^{(-1)}$.

Let us consider three $k\times k$ diagonal matrices with positive elements and denoted as $D_1,D_2$ and $D_3$.

Then is it always true that: $$[{\bf X}'D_1P_XD_2P'_XD_3{\bf X}]_{[1,1]}\geq[{\bf X}'D_1D_2D_3{\bf X}]_{[1,1]},$$ and $$[{\bf X}'D_1P_XD_2P'_XD_3{\bf X}]_{[2,2]}\geq[{\bf X}'D_1D_2D_3{\bf X}]_{[2,2]},$$ where $M_{[1,1]}$ is the $[1,1]$ elements of $M$.