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.
Jie WANG
,
Jianfei WANG
. GENERALIZED ROPER-SUFFRIDGE OPERATOR FOR $\epsilon$ STARLIKE AND BOUNDARY STARLIKE MAPPINGS[J]. Acta mathematica scientia, Series B, 2020
, 40(6)
: 1753
-1764
.
DOI: 10.1007/s10473-020-0610-y
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