Close-to-starlike holomorphic functions of several variables

Theorem 1. If $f(x) = x + \cdots \in \mathcal{H}(B)$ is locally biholomorphic in B and close-to-starlike relative to the starlike function $g(x) = x + \cdots$ then (2.3) $$F(x, t) = f(x) + (e^{t} - 1)g(x), 0 \le t, x \in B,$$ is a univalent subordination chain. Hence f(x) is univalent in B. We shall give the proof of Theorem 1 in §3 below. The subordination chain characterization (2.3) yields the linear accessibility of the images of the balls $B_r = \{x \in X : ||x|| < r\}$ (0 < r < 1) (compare [1] and [3]).

Corollary 1. If f and g satisfy the hypotheses of Theorem 1 then for each r, 0 < r < 1, the complement (in X) of $f(B_r)$ is the union of nonintersecting rays.

Theorem 2. Suppose $f(x) = x + \cdots$ is holomorphic in B and that $g(x) = x + \cdots \in \mathcal{H}(B)$ is starlike. If (2.3) $$F(x, t) = f(x) + (e^{t} - 1)g(x), 0 \le t, x \in B$$ is a univalent subordination chain then f is close-to-starlike relative to g. We shall prove this theorem in §4 below. By the results in [8] a mapping $f(x) = x + \cdots \in \mathcal{H}(B)$ is starlike univalent if and only if it is close-to-starlike relative to itself, i.e., (2.1) holds with g = f. Thus from Theorems 1 and 2 we have immediately the

Corollary 2. Let $f(x) = x + \cdots$ be locally biholomorphic in B. Then f is univalent and starlike in B if and only if $F(x, t) = e^t f(x)$ is a univalent subordination chain. This extends to higher dimensional spaces Pommerenke's one dimensional result in Folgerung 2 of [6]. 3. Proof of Theorem 1. We shall give the proof in a sequence of three lemmas. We use the notation $f_r(x) = f(rx)/r$ , $g_r(x) = g(rx)/r$ and $F_r(x, t) = f_r(x) + (e^t - 1)g_r(x)$ for $0 \le r \le 1$ , $t \ge 0$ . Let $R = \{r: 0 \le r \le 1 \text{ and } F_\rho(x, t) \text{ is a univalent subordination chain for } \rho < r\}$ . Then $0 \in R$ and clearly R is closed. We wish to show that R is open so R = [0, 1].

Lemma 3.1. If $r \in R$ then $F_r(x, t)$ is a univalent subordination chain.

Lemma 3.2. If $r \in R$ , r < 1 then there exists $\varepsilon_0 > 0$ such that $F_{r+\varepsilon}(x,\,t)$ is a univalent function of $x \in B$ for each $t \ge 0$ and $0 \le \varepsilon < \varepsilon_0$ .

Lemma 3.3. Let $\varepsilon_0$ be as determined in Lemma 3.2. Then for $0 \le \varepsilon < \varepsilon_0$ $F_{r+\varepsilon}$ (x, t) is a univalent subordination chain.

Definition 1. A mapping $f(x) = x + \cdots \in \mathcal{H}(B)$ is said to be close-to-starlike if there exist $h(x) \in \mathcal{M}$ and a starlike map $g \in \mathcal{H}(B)$ such that (2.1) $$Df(x)(h(x)) = g(x), x \in B$$ . REMARK. By this definition, close-to-starlike maps in X are a generalization of Kaplan's close-to-convex functions in C. Indeed when X = C a function $h \in \mathcal{M}$ has the form h(z) = zP(z) where $$P(0) = 1$$ , Re $P(z) > 0(|z| < 1)$ (see [8, p. 576]) and (2.1) is equivalent to the condition (1.1) for the convex function $\phi(z) = \int_0^z g(x)/x \, dx$ . By using the criteria for star-likeness and convexity of vector valued maps established in [7] one can easily construct examples showing that Alexander's theorem ( $\phi(z)$ is convex if and only if $g(z) = z\phi'(z)$ is starlike) fails in spaces of dimension greater than one. Hence the name close-to-starlike seems most natural in our work. A mapping $v(x) \in \mathcal{H}(B)$ is called a Schwarz function if $||v(x)|| \le ||x||$ for all $x \in B$ . A subordination chain ([5], [6]) is a function f(x, t) from $B \times [0, \infty)$ into X such that for each $t \ge 0$ , $f_t(x) = f(x, t) = e^t x + \cdots$ is in $\mathcal{H}(B)$ and there exist Schwarz functions v(x, s, t) such that $$(2.2) f(x, s) = f(v(x, s, t), t), 0 \le s \le t, x \in B,$$ for all $0 \le s \le t < \infty$ . A univalent subordination chain is a sub-ordination chain f(x, t) such that for each $t \ge 0$ , $f_t(x)$ is univalent in B.