THE SUBORDINATION PRINCIPLE AND ITS APPLICATION TO THE GENERALIZED ROPER-SUFFRIDGE EXTENSION OPERATOR

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doi:10.1007/s10473-022-0213-x

Abstract: This note is devoted to applying the principle of subordination in order to explore the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator with special analytic properties. First, we prove that both the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator preserve subordination. As applications, we obtain that if $\beta\in[0,1],\gamma\in[0,\frac{1}{r}]$ and $\beta+\gamma\leq1$ , then the Roper-Suffridge extension operator

Φ_{β, γ} (f) (z) = (f (z_{1}), {(\frac{f (z_{1})}{z_{1}})}^{β} (f^{'} (z_{1}))^{γ} w), z \in Ω_{p, r}

$\Phi_{\beta,\,\gamma}(f)(z)=\left(f(z_{1}), \left(\frac{f(z_1)}{z_1}\right)^{\beta}(f'(z_{1}))^{\gamma}w\right),\,\,z\in \Omega_{p,r}$ preserves an almost starlike mapping of complex order

λ

$\lambda$ on

Ω_{p, r} = {z = (z_{1}, w) \in C \times X : | z_{1} |^{p} + ‖ w ‖_{X}^{r} < 1}

$\Omega_{p,r}=\{z=(z_1,w)\in \mathbb C\times X :|z_1|^{p}+\|w\|_X^{r}<1\}$ , where

1 \leq p \leq 2

$1\leq p\leq 2$ ,

r \geq 1

$r\geq 1$ and

X

$X$ is a complex Banach space. Second, by applying the principle of subordination, we will prove that the Pfaltzgraff-Suffridge extension operator preserves an almost starlike mapping of complex order

λ

$\lambda$ . Finally, we will obtain the lower bound of distortion theorems associated with the Roper-Suffridge extension operator. This subordination principle seems to be a new idea for dealing with the Loewner chain associated with the Roper-Suffridge extension operator, and enables us to generalize many known results from

p = 2

$p=2$ to

1 \leq p \leq 2

$1\leq p\leq 2$ .

Key words: Biholomorphic mappings, starlike mappings, subordination, Loewner chain

CLC Number:

32H02

Jianfei WANG, Xiaofei ZHANG. THE SUBORDINATION PRINCIPLE AND ITS APPLICATION TO THE GENERALIZED ROPER-SUFFRIDGE EXTENSION OPERATOR[J].Acta mathematica scientia,Series B, 2022, 42(2): 611-622.

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