Let B be the unit ball in Cn with respect to an arbitrary norm on Cn. In this paper, we give a necessary and su.cient condition that a Loewner chain f(z, t), such that {e-tf(z, t)}t is more than or equal to 0 is a normal family on B, is k-fold symmetrical. As a corollary, we give a necessary and su.cient condition that a normalized locally biholomorphic mapping on B is spirallike of type and k-fold symmetrical. When alpha = 0, this result solves a natural problem that is similar to an open problem posed by Liczberski. We also give two examples of k-fold symmetrical Loewner chains.
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In this paper, we will give coefficient conditions for mappings of the form f(z) = z/(1+sum for k=1 bkzk1 to infinity) to be starlike or convex on the Euclidean unit ball B in Cn. Our results give concrete examples of strongly starlike mappings of order a, starlike mappings of order a and convex mappings on B.
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In this paper, we will consider some sufficient conditions for locally biholomorphic mappings defined in the unit ball in complex Banach spaces to be biholomorphic and to have (phi)-like images. As a corollary, we obtain some sufficient condition for locally biholomorphic mappings to be starlike mappings.
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In this paper we will give some conditions of spirallikeness for holomorphic mappings defined on bounded balanced pseudoconvex domains in Cn with C1 plurisubharmonic defining functions.
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