We present new additive results for the group inverse in a Banach algebra under certain perturbations. The upper bound of $|(a+b)^{#}-a^d|$ is thereby given. These extend the main results in [X. Liu, Y. Qin and H. Wei, Perturbation bound of the group inverse and the generalized Schur complement in Banach algebra, Abstr. Appl. Anal., 2012, 22 pages. DOI:10.1155/2012/629178].
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