Abstract
We consider support points of the class $S^0(\mathbb{D}^n)$ of normalized univalent mappings on the polydisc $\mathbb{D}^n$ with parametric representation and we prove sharp estimates for coefficients of degree 2.
Results & Lemmas (8)
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Theorem 2.2
Theorem 2.2 ([Suf70]). Let be locally biholomorphic, i.e. is invertible for every, with and. Then if and only if the function belongs to.
Theorem 2.2 ([Suf70]). Let $f : D^{n} \to C^{n}$ be locally biholomorphic, i.e. $Df(z)$ is invertible for every $z \in D^{n}$, with $f(0) = 0$ and $Df(0) = I_n$. Then $f \in S^{*}(D^{n})$ if and only if the function $z \mapsto -(Df(z))^{-1} \cdot f(z)$ belongs to $M(D^{n})$.
Theorem 2.3
Theorem 2.3 ([Por87a]). Let be a Herglotz vector field. For every and, let be the solution of the initial value problem (2.2). Then the…
Theorem 2.3 ([Por87a]). Let $G(z, t)$ be a Herglotz vector field. For every $s \ge 0$ and $z \in D^{n}$, let $\phi_{s,t}(z)$ be the solution of the initial value problem (2.2). Then the limit $\lim_{t\to\infty} e^{t} \phi_{s,t}(z) =: f_s(z)$ (2.4) exists for all $s \ge 0$ locally uniformly on $D^{n}$ and $f_s \in S(D^{n})$. Furthermore, the functions $\{f_t\}_{t\ge 0}$ satisfy $f_s(z) = f_t(\phi_{s,t}(z))$ for all $z \in D^{n}$ and $0 \le s \le t$, and $\{f_t\}_{t\ge 0}$ is a normalized Loewner chain having the property that $\{e^{-t} f_t\}_{t\ge 0}$ is a normal family on $D^{n}$. Finally, $f_t$ satisfies the Loewner PDE $\frac{\partial f_t(z)}{\partial t} = -Df_t(z) G(z, t)$ for all $z \in D^{n}$ and for almost all $t \ge 0$.
Theorem 2.6
Theorem 2.6 (Theorem 1 and Theorem 2 in [Por87a]). If, then for all. In particular,. (Koebe quarter theorem for the class.)
Theorem 2.6 (Theorem 1 and Theorem 2 in [Por87a]). If $f \in S_0(D^{n})$, then $\frac{\|z\|_{\infty}}{(1 + \|z\|_{\infty})^{2}} \le \|f(z)\|_{\infty} \le \frac{\|z\|_{\infty}}{(1 - \|z\|_{\infty})^{2}}$ for all $z \in D^{n}$. In particular, $\frac{1}{4} D^{n} \subseteq f(D^{n})$. (Koebe quarter theorem for the class $S_0(D^{n})$.)
Theorem 2.7
Theorem 2.7 (Theorem 2.9 in [GKK03]). The class is compact.
Theorem 2.7 (Theorem 2.9 in [GKK03]). The class $S_0(D^{n})$ is compact.
Theorem 2.9
Theorem 2.9. Let and let be a normalized Loewner chain with such that is a normal family. Then a). b) is a Runge domain. c) For, is dense…
Theorem 2.9. Let $f \in S_0(D^{n})$ and let $\{f_t\}_{t\ge 0}$ be a normalized Loewner chain with $f = f_0$ such that $\{e^{-t} f_t\}_{t\ge 0}$ is a normal family. Then a) $\bigcup_{t\ge 0} f_t(D^{n}) = C^{n}$. b) $f(D^{n})$ is a Runge domain. c) For $n \ge 2$, $S^{*}(D^{n}) \cap \operatorname{Aut}(C^{n})$ is dense in $S^{*}(D^{n})$ and $S_0(D^{n}) \cap \operatorname{Aut}(C^{n})$ is dense in $S_0(D^{n})$.
Theorem 3.3
Theorem 3.3. Let and let be a normalized Loewner chain with such that is a normal family on, then for all.
Theorem 3.3. Let $f \in \operatorname{supp} S_0(D^{n})$ and let $\{f_t\}_{t\ge 0}$ be a normalized Loewner chain with $f_0 = f$ such that $\{e^{-t} f_t\}_{t\ge 0}$ is a normal family on $D^{n}$, then $e^{-t} f_t \in \operatorname{supp} S_0(D^{n})$ for all $t \ge 0$.
Theorem 3.4
Theorem 3.4. Let and let be a normalized Loewner chain with such that is a normal family on, then for all.
Theorem 3.4. Let $f \in \operatorname{ex} S_0(D^{n})$ and let $\{f_t\}_{t\ge 0}$ be a normalized Loewner chain with $f_0 = f$ such that $\{e^{-t} f_t\}_{t\ge 0}$ is a normal family on $D^{n}$, then $e^{-t} f_t \in \operatorname{ex} S_0(D^{n})$ for all $t \ge 0$.
Theorem 4.3 · coeff
Theorem 4.3. Let and,. Then the following statements hold: a) for all with and. This estimate is sharp for all such due to the mappings…
Theorem 4.3. Let $n \ge 2$ and $(f_1, \ldots, f_n) \in S_0(D^{n})$, $f_1(z) = z_1 + \sum_{|\alpha|\ge 2} A_{\alpha} z^{\alpha}$. Then the following statements hold: a) $|A_{\alpha}| \le 2$ for all $\alpha$ with $|\alpha| = 2$ and $\alpha_1 \ne 0$. This estimate is sharp for all such $\alpha$ due to the mappings $F_1(z) = (\frac{z_1}{(1 - z_1)^{2}}, z_2, \ldots, z_n)$ for $\alpha = (2, 0, \ldots, 0)$, $F_2(z) = (z_1(1 + z_2)^{2}, z_2, \ldots, z_n)$, $F_3(z) = (\frac{z_1(1 + z_2)}{1 - z_2}, \frac{z_2}{1 - z_2}, z_3, \ldots, z_n)$ for $\alpha = (1, 1, 0, \ldots, 0)$. b) $|A_{\alpha}| \le 1$ for all $\alpha$ with $|\alpha| = 2$ and $\alpha_1 = 0$. This estimate is sharp for all such $\alpha$ due to the mappings $F_4(z) = (z_1 + z_2^{2}, z_2, \ldots, z_n)$, $F_5(z) = (\frac{z_1 - z_1 z_2 + z_2^{2}}{1 - z_2}, \frac{z_2}{1 - z_2}, z_3, \ldots, z_n)$ for $\alpha = (0, 2, 0, \ldots, 0)$, $F_6(z) = (z_1 + z_2 z_3, z_2, \ldots, z_n)$, $F_7(z) = (z_1 + z_2 z_3 \frac{\log(1 + z_2) - \log(1 + z_3)}{z_2 - z_3}, \frac{z_2}{1 + z_2}, \frac{z_3}{1 + z_3}, z_4, \ldots, z_n)$ for $\alpha = (0, 1, 1, 0, \ldots, 0)$.
Definitions (1)
Def 2.4
Definition 2.4..
Definition 2.4. $S_0(D^{n}) := \{f \in S(D^{n}) \mid f \text{ has parametric representation}\}$.