The generalized Roper-Suffridge extension operator in Banach spaces (III)

Abstract

Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)$ on the domain $\Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} }$ defined by

$$ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)(x) = \begin{array}{*{20}c} {\sum\limits_{j = 1}^n {\left( {\frac{{f(x_1^* (x))}} {{x_1^* (x)}}} \right)} ^{\beta _j } (f'(x_1^* (x)))^{\gamma _j } x_1^* (x)x_j } \\ { + \left( {\frac{{f(x_1^* (x))}} {{x_1^* (x)}}} \right)^{\beta _{n + 1} } (f'(x_1^* (x)))^{\gamma _{n + 1} } \left( {x - \sum\limits_{j = 1}^n {x_1^* (x)x_j } } \right)} \\ \end{array} $$

for $x \in \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} }$, where β ₁ = 1, γ ₁ = 0 and

$$ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } = \left\{ {x \in X:\sum\limits_{j = 1}^n {|x_j^* (x)|^{p_j } } + \left\| {x - \sum\limits_{j = 1}^n {x_j^* (x)x_j } } \right\|^{p_{n + 1} } < 1} \right\} $$

with p _j > 1 (j = 1, 2,..., n + 1), the linearly independent family {x ₁, x ₂,..., x _n } ⊂ X and {x ^*₁ , x ^*₂ ,..., x ^* _n } ⊂ X* satisfy x ^* _j (x _j ) = ‖x _j ‖ = 1 (j = 1, 2,..., n) and x ^* _j (x _k ) = 0 (j ≠ k), and we choose the branch of the power functions such that $(\frac{{f(\xi )}} {\xi })^{\beta _j } |_{\xi = 0} = 1$ and $(f'(\xi ))^{\gamma _j } |_{\xi = 0} = 1$, j = 2,..., n + 1. We prove that the operator $\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)$ preserves almost spirallike mapping of type β and order α or spirallike mapping of type β and order α on $\Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} }$ for some suitable constants β _j , γ _j .