Let $A, B, X \\in \\mathbb{B}(\\mathcal{H})$ for some complex Hilbert space $\\mathcal{H}$ and $\\tilde{A}$ denote the Aluthge transformation of $A$. Then $AX=XB$ implies $\\tilde{A}X=X\\tilde{B}$, whenever it implies that $A^* X=XB^*$. We show that if $A ,B^*$ are log-hyponormal and $\\tilde{A}X=X\\tilde{B}$ then $AX=XB$. If $A=U|A|$ be the polar decomposition of $A$, $U|A|^{1\\over2}\\geq a\\geq0$ and $X$ is an operator such that $U^* X=XU$, then we prove that $\\|\\tilde{A}^* X-X\\tilde{A}\\|_p\\geq 2a\\||A|^{1\\over2}X-X|A|^{1\\over2}\\|_p$.

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