Starlike and convex mappings in several complex variables

Lemma 1.1. If D is a bounded domain, the Bergman kernel function $K_D(z, \overline{z})$ is strictly plurisubharmonic and (1.2) $$1/\omega(D) \leq K_D(z, \bar{z}) \leq 1/\pi^n (l(z))^{2n},$$ where $l(z) = \min_{\tau \in \partial D} \rho(\tau, z)$ , $\rho(\tau, z) = \max_{j} \{ |\tau_j - z_j|, j = 1, \dots, n \}$ and $\omega(D)$ signifies the euclidean volume of D.

Lemma 2.1. For a fixed value t, a holomorphic univalent function w=f(z) of D have convex image $\Delta_t$ of $D_t$ defined by (1.1) if and only if at every point z on the boundary $\partial D_t$ $$(2.12) \quad \mathscr{R} \left[ \alpha^ \frac{\partial^2 K_{\scriptscriptstyle D}(z,\,\overline{z})}{\partial z^ \partial z} \alpha + \left\{ \frac{\partial^2 K_{\scriptscriptstyle D}(z,\,\overline{z})}{\partial z^2} - \frac{\partial K_{\scriptscriptstyle D}(z,\,\overline{z})}{\partial z} \left( \frac{df}{dz} \right)^{-1} \frac{d^2 f}{dz^2} \right\} \alpha^2 \right] > 0$$ for all unit vectors $\alpha$ satisfying $$\mathscr{R}\Big[\frac{\partial K_{\scriptscriptstyle D}(z,\,\overline{z})}{\partial z}\,\alpha\,\Big]=0$$ . DEFINITION. We define the class $\mathscr{D}$ of bounded schlicht domains D for which the kernel function $K_D(z, \overline{z})$ becomes infinite everywhere on the boundary $\partial D$ , $K_D(0, 0) = \min_{z \in D} K_D(z, \overline{z})$ only at z = 0, $\partial K_D(z, \overline{z})/\partial z \rightleftharpoons 0$ , $z \rightleftharpoons 0$ , in D, and there is the holomorphic mapping g(z) of D into D satisfying g(0) = 0, for some one $z^{(1)}$ of two arbitrary points $z^{(1)}$ , $z^{(2)}(\rightleftharpoons 0)$ in D $g(z^{(1)}) = z^{(2)}$ , and $K_D(z, \overline{z}) \geqq K_D(g(z), \overline{g(z)})$ . For example, let D be a minimal domain or representative domain with center at the origin which is the image domain of $E=\{\zeta\colon |\zeta|=(\sum_{j=1}^n|\zeta_j|^2)^{1/2}<1\}$ under the biholomorphic mapping $z=\varphi(\zeta)$ satisfying $0=\varphi(0)$ . Then $\det(d\varphi(\zeta)/d\zeta)\equiv \mathrm{const.}$ when D is a minimal, domain and $d\varphi(\zeta)/d\zeta\equiv \mathrm{const.}$ when D is a representative domain (see [4], Theorem 3.1). Hence, for any holomorphic mapping g(z) of D into D satisfying g(0)=0, we have $K_D(z,\overline{z})\geq K_D(g(z),\overline{g(z)})$ because $K_E(\zeta,\overline{\zeta})\geq K_E(\Phi(\zeta),\overline{\Phi(\zeta)})$ under the holomorphic mapping $\Phi(\zeta)\equiv \varphi^{-1}[g(\varphi(\zeta))],\Phi(0)=0$ , of E into E. Also we have $K_D(0,0)=\min_{z\in D}K_D(z,\overline{z})$ at only the origin. Moreover, for arbitrary points $z^{(1)},z^{(2)}\in D$ , if $|\varphi^{-1}(z^{(2)})|\leq |\varphi^{-1}(z^{(1)})|$ , then $$g(z) \, \equiv \, \varphi \Big( \frac{ \big| \varphi^{-1} (z^{(2)}) \big|}{\big| \varphi^{-1} (z^{(1)}) \big|} \, U_2 \, U_1^* \varphi^{-1} (z) \Big)$$ is a holomorphic mapping of D into D satisfying g(0) = 0 and $g(z^{(1)}) = z^{(2)}$ where $$\varphi^{-1}(z^{(1)}) \,=\, U_1 \begin{pmatrix} \varphi^{-1}(z^{(1)}) \,| \ 0 \ \vdraightarrow \ 0 \end{pmatrix}, \,\,\, \varphi^{-1}(z^{(2)}) \,=\,\, U_2 \begin{pmatrix} \varphi^{-1}(z^{(2)}) \,| \ 0 \ \vdraightarrow \ 0 \end{pmatrix}$$ and $U_1$ , $U_2$ are unitary matrices. And we observe $$\partial K_D(z,\bar{z})/\partial z = \partial K_E(\zeta,\bar{\zeta})/\partial \zeta \cdot (d\varphi(\zeta)/d\zeta)^{-1} \Longrightarrow 0, z \Longrightarrow 0$$ because $$\partial K_{\scriptscriptstyle E}(\zeta,\,\overline{\zeta})/\partial\zeta=(n\,+\,1)\zeta^*K_{\scriptscriptstyle E}(\zeta,\,\overline{\zeta})/(1\,-\,|\zeta\,|^2) \rightleftharpoons 0,\,\zeta \rightleftharpoons 0$$ .

Theorem 2.1. Let D be a bounded schlicht domain of the class $\mathscr{D}$ . Suppose $f: D \to C^n$ is holomorphic, f(0) = 0, and $\det(df/dz) \rightleftharpoons 0$ for all $z \in D$ . Then f is a univalent map of D onto a convex domain if and only if $$(2.13) \quad \mathscr{R} \bigg[ \alpha^ \frac{\partial^2 K_{\scriptscriptstyle D}(z,\,\overline{z})}{\partial z^ \partial z} \alpha \, + \, \Big\{ \frac{\partial^2 K_{\scriptscriptstyle D}(z,\,\overline{z})}{\partial z^2} \, - \, \frac{\partial K_{\scriptscriptstyle D}(z,\,\overline{z})}{\partial z} \Big( \frac{df}{dz} \Big)^{\!-1} \frac{d^2 f}{dz^2} \Big\} \alpha^2 \, \bigg] > 0$$ for all unit vectors $\alpha$ satisfying $$\mathscr{R}\Big[\frac{\partial K_{\scriptscriptstyle D}(z,\, \overline{z})}{\partial z}\alpha\Big]=0$$ .

Theorem 2.2. Let D be the unit hypersphere and let $f: D \rightarrow C^n$ be holomorphic, f(0) = 0 and $\det(df/dz) \neq 0$ for all $z \in D$ . Then f(D) is convex if and only if $$\mathscr{R}\left[|Az|^2+z^*\left(\frac{df}{dz}\right)^{-1}\frac{d^2f}{dz^2}(Az\times Az)\right]\geqq 0,$$ where $$A=\begin{pmatrix} A_1 & 0 \ \ddots \ 0 & A_n \end{pmatrix}$$ , $A_j\geqq 0,\, j=1,\, \cdots$ , $n$ , and the equality holds only if Az = 0.

Theorem 2.3. Let D be the polydisk and let $f: D \to C^n$ be holomorphic, f(0) = 0 and $\det(df/dz) \rightleftharpoons 0$ for all $z \in D$ . Then f is a univalent map of D onto a convex domain if and only if the condition (2.26) is fulfilled.

Theorem 3.1. Let D be a bounded schlicht domain for which the kernel function $K_D(z, \overline{z})$ becomes infinite everywhere on the boundary, $K_D(0, 0) = \min_{z \in D} K_D(z, \overline{z})$ at only the origin, and $K_D(z, \overline{z}) \geq K_D(g(z), \overline{g(z)})$ for any holomorphic mapping g(z) of D into D satisfying g(0) = 0. Suppose $f: D \to C^n$ is holomorphic, f(0) = 0 and $\det(df/dz) \approx 0$ for all $z \in D$ . Then f is starlike if and only if (3.1) $$\mathscr{R}\left[\frac{\partial K_{D}(z, \overline{z})}{\partial z} \left(\frac{df}{dz}\right)^{-1} f\right] > 0$$ for all $z \in D$ , $z \approx 0$ . REMARK 2. Domains which belong to the above mentioned class $\mathscr D$ satisfy the conditions of this Theorem.

Corollary 3.1. Let D be the unit hypersphere, and let $f: D \rightarrow C^n$ be holomorphic, f(0) = 0 and $\det(df/dz) \approx 0$ for all $z \in D$ . Then f(z) is starlike if and only if $$\mathscr{R}\left[z^*\left(\frac{df}{dz}\right)^{-1}f\right] > 0$$ for all $z \in D$ , $z \rightleftharpoons 0$ .