%A Jie WANG, Jianfei WANG %T GENERALIZED ROPER-SUFFRIDGE OPERATOR FOR $\epsilon$ STARLIKE AND BOUNDARY STARLIKE MAPPINGS %0 Journal Article %D 2020 %J Acta mathematica scientia,Series B %R 10.1007/s10473-020-0610-y %P 1753-1764 %V 40 %N 6 %U {http://121.43.60.238/sxwlxbB/CN/abstract/article_16286.shtml} %8 2020-12-25 %X This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties. First, we prove that the Roper-Suffridge extension operator preserves $\epsilon$ starlikeness on the open unit ball of a complex Banach space $\mathbb{C}\times X$, where $X$ is a complex Banach space. This result includes many known results. Secondly, by introducing a new class of almost boundary starlike mappings of order $\alpha$ on the unit ball $B^n$ of ${\mathbb{C}}^{n}$, we prove that the Roper-Suffridge extension operator preserves almost boundary starlikeness of order $\alpha$ on $B^n$. Finally, we propose some problems.