On the generalized Roper‐Suffridge extension operator in Banach spaces

Abstract

The generalized Roper-Suffridge extension operator in Banach spaces is introduced. We prove that this operator preserves the starlikeness on some domains in Banach spaces and does not preserve convexity in some cases. Furthermore, the growth theorem and covering theorem of the corresponding mappings are given. Some results of Roper and Suffridge and Graham et al. in C n are extended to Banach spaces.

On ε quasi-convex mappings in the unit ball of a complex Banach space

“…They generalized this extension operator in C n and discussed their properties (see [3][4][5][6][7][8][9][10][11], etc.). Recently, we generalized the Roper-Suffridge operator to Banach spaces in [14,23]. We have proved that the generalized Roper-Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε ∈ [0, 1].…”

Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

“…Recently, we generalized Roper-Suffridge operator to Banach spaces in [13,21]. Now we introduce a linear operator in purpose to construct some other concrete examples about the biholomorphic convex mappings on B in a Hilbert space X.…”

The Roper-Suffridge extension operator and classes of biholomorphic mappings

“…In fact several types of generalizations have been considered (see refs. [10,12,15,[18][19][20][36][37][38]). …”

“…M. S. Liu and Y. Zhu [18,20] gave a generalization to complex Banach spaces. In particular, they considered operators and domains analogous to (4.2) and (4.3), and showed that ε-starlikeness is preserved.…”