Modified Roper-Suffridge operator for some holomorphic mappings

Abstract

In this paper, we will study the operator given by F(z)={(f(z_{\mathrm{1}})+f^{\prime }(z_{\mathrm{1}})P(z_{\mathrm{0}}),{(f^{\prime }(z_{\mathrm{1}}))}^{\mathrm{1}/k}{z}_{\mathrm{0}}^{\mathrm{T}})}^{\mathrm{T}}, where z={(z_{\mathrm{1}},z_{\mathrm{T}}^{\mathrm{0}})}^{\mathrm{T}} belongs to the unit ball Bⁿ in {\mathrm{ℂ}}^n, z_{\mathrm{1}}\in U=B^{\mathrm{1}}, z_{\mathrm{0}}={(z_{\mathrm{2}},\cdots ,z_n)}^{\mathrm{T}}\in {\mathrm{ℂ}}^{n-\mathrm{1}}, and P:{\mathrm{ℂ}}^{n-\mathrm{1}}\to \mathrm{ℂ} is a homogeneous polynomial of degree k (k≥2), the holomorphic branch is chosen such that {(f^{\prime }(\mathrm{0}))}^{\mathrm{1}/k}=\mathrm{1}. We will give different conditions for P such that the modified operator preserves the properties of almost spirallikeness of type β and order α, spirallikeness of type β and order α, and strongly spirallikeness of type β and order α, respectively.

Key words： Roper-Suffridge operator; almost spirallike mappings of type β and order α; spirallike mappings of type β and order α; strongly spirallikeness of type β and orderα

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