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Results & Lemmas (4)
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Theorem 2.1
Theorem 2.1. Let be a starlike mapping of the unit ball B in. For any point, Furthermore the estimates are sharp.
Theorem 2.1. Let $f = (f_1, f_2, ..., f_n)'$ be a starlike mapping of the unit ball B in $\mathbb{C}^n$ . For any point $z \in B$ ,
$$\frac{|z|}{(1+|z|)^2} \le |f(z)| \le \frac{|z|}{(1-|z|)^2}.$$
Furthermore the estimates are sharp.
Corollary 3.1 · radius
Corollary 3.1. If f(z) is a holomorphic mapping on the unit ball in and is starlike with respect to the origin, then the image of f…
Corollary 3.1. If f(z) is a holomorphic mapping on the unit ball in $\mathbb{C}^n$ and is starlike with respect to the origin, then the image of f contains a ball of radius 1/4 centered at the origin. The value 1/4 is best possible.
Corollary 3.2 · radius
Corollary 3.2. If f is holomorphic in the unit ball and is starlike with a k-fold symmetric image for, then Then the image of the ball…
Corollary 3.2. If f is holomorphic in the unit ball and is starlike with a k-fold symmetric image for $k \ge 1$ , then
$$\frac{|z|}{(1+|z|^k)^{2/k}} \le |f(z)| \le \frac{|z|}{(1-|z|^k)^{2/k}}.$$
Then the image of the ball under f contains a ball of radius $2^{-2/k}$ and these estimates are best possible.
Corollary 3.3
Corollary 3.3. The only balanced domain which is the image of the unit ball under a normalized biholomorphic mapping is the unit ball.
Corollary 3.3. The only balanced domain which is the image of the unit ball under a normalized biholomorphic mapping is the unit ball.