In this paper, we investigate the perturbation for the Moore-Penrose inverse of closed operators on Hilbert spaces. By virtue of a new inner product defined on $H$, we give the expression of the Moore-Penrose inverse $bar{T}^dag$ and the upper bounds
of $|bar{T}^dag|$ and $|bar{T}^dag -T^dag|$. These results obtained in this paper extend and improve many related results in this area.
