The principle of subordination applied to functions of several variables

Lemma 1. Let v(z;t): $E \times I \rightarrow C^*$ be holomorphic for each $t \in I = [0,1]$ , v(z;0) = z, v(0,t) = 0 and |v(z;t)| < 1 when $z \in E$ . If (2) $$\lim_{t \to 0^+} [(z - v(z; t))/t^{\rho}] = w(z)$$ exists and is holomorphic in E for some $\rho > 0$ , then $w \in \mathscr{P}$ .

Lemma 2. Let $f: E \to C^n$ be holomorphic and univalent and satisfy f(0) = 0. Let $F(z;t): E \times I \to C^n$ be a holomorphic function of z for each $t \in I = [0,1]$ , F(z;0) = f(z), F(0,t) = 0 and suppose F(z;t) < f for each $t \in I$ (i.e., $F(E;t) \subset f(E)$ for each $t \in I$ ). Let $\rho > 0$ be such that $\lim_{t\to 0^+} F(z;0) - F(z;t)/t^\rho = F(z)$ exists and is holomorphic. Then F(z) = Jw where $w \in \mathscr{P}$ . Here F and w are written as column vectors and J is the complex Jacobian matrix for the mapping f.

Theorem 1. Suppose $f: E \to C^n$ is starlike and that J is the complex Jacobian matrix of f. There exists $w \in \mathscr{S}$ such that f = Jw where f and w are written as column vectors.

Theorem 2. Suppose $f: E \to C^*$ is holomorphic, f(0) = 0, J is nonsingular and that $$f(z) = Jw, w \in \mathscr{P}.$$ Then f is starlike.

Theorem 3. Suppose $f: E \to C^n$ is holomorphic, f(0) = 0 and that J is nonsingular for all $z \in E$ . Then f is a univalent map of E onto a convex domain if and only if there exist univalent mappings $f_j$ $(1 \le j \le n)$ from the unit disk in the plane onto convex domains in the plane such that $f(z) = T(f_1(z_1), f_2(z_2), \dots, f_n(z_n))$ where T is a nonsingular linear transformation.

Lemma 3. Let v(z;t): $D_p \times I \longrightarrow C^n$ be holomorphic for each $t \in I$ , v(z,0)=z, v(0,t)=0 and $|v(z;t)|_p < 1$ when $z \in D_p$ . If $$\lim_{t\to 0^+} \left[ (z - v(z; t))/t^{\rho} \right] = w(z)$$ exists and is holomorphic in $D_p$ for some $\rho > 0$ , then $w \in \mathscr{P}_p$ .

Lemma 4. Let $f\colon D_{\mathfrak{p}}\to C^{\mathfrak{n}}$ be holomorphic and univalent and satisfy f(0)=0. Let $F(z;t)\colon D_{\mathfrak{p}}\times I\to C^{\mathfrak{n}}$ be a holomorphic function of z for each $t\in I$ , F(z,0)=f(z), F(0;t)=0 and suppose F(z;t)< f for each $t\in I$ . Let $\rho>0$ be such that $\lim_{t\to 0^+}(F(z;0)-F(z;t))/t^\rho=F(z)$ exists and is holomorphic. Then F(z)=Jw where $w\in \mathscr{F}_{\mathfrak{p}}$ .

Theorem 4. If $f: D_p \to C^n$ is starlike then there exists $w \in \mathscr{S}_p$ such that f = Jw. Conversely, if $f: D_p \to C^n$ , f(0) = 0, J is nonsingular and f = Jw, $w \in \mathscr{S}_p$ then f is starlike.

Theorem 5. Let $f: D_p \to C^n$ , f(0) = 0 and suppose J is non-singular. Then $f(D_p)$ is convex if and only if F = Jw where $w \in \mathscr{S}_p$ for each choice of $A = (A_1, A_2, \dots, A_n)$ , $A_j \geq 0$ $(1 \leq j \leq n)$ and F is given by (7) with $z \in D_p$ . Now set p=2. It is easy to see that Theorem 4 above is equivalent to Matsuno's Theorem 1 [4, p. 91]. Consider $f: D_2 \to C^2$ given by $f(z)=(z_1+az_2^2,z_2)$ . Theorem 5 shows that $f(D_2)$ is convex if and only if $|a| \le 1/2$ while Matsuno's Lemma 3 [4, p. 94] implies f is convex-like if and only if $|a| \le 3\sqrt{3}/4$ . This shows that convex-like is not equivalent to geometrically convex.