Close-to-starlike holomorphic functions of several variables

THEOREM 1. If f(x) = x + e £ίf(B) is locally biholomorphic in B and close-to-starlike relative to the starlike function g(x) = x + • then (2.3) F(x, t) = f{x) + (e* - is a univalent subordination chain. Hence f(x) is univalent in B. We shall give the proof of Theorem 1 in §3 below. The subor- dination chain characterization (2.3) yields the linear accessibility of the images of the balls Br = {x e 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(Br) is the union of nonintersecting rays.

THEOREM 2. Suppose f(x) = x + is holomorphic in B and that g(x) = x + e ^f{B) is starlike. If (2.3) F(x, t) = f{x) + (e ι 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 + e £ϊf{B) is starlike univalent if and only if it is close-to-starlike relative to itself, i.e., (2.1) holds with g = / . Thus from Theorems 1 and 2 we have immediately the

COROLLARY 2. Let f(x) — x + be locally biholomorphic in B. Then f is univalent and starlike in B if and only if F(x, t) = e*f(x) is a univalent subordination chain. This extends to higher dimensional spaces Pommerenke's one dimen- sional 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 fr(x) = f(rx)/rf gr{x) = g(rx)/r and Fr(x, t) - fr(x) + (e* - ί)gr(x) for 0 ^ r ^ 1, t ^ 0. Let R = {r: 0 ^ r ^ 1 and Fp(x, t) is a un

LEMMA 3.1. If r e R then Fr(x, t) is a univalent subordination chain.

LEMMA 3.2. // reR,r<l then there exists ε0 > 0 such that Fr+ε(x, t) is a univalent function of xeB for each t ^ 0 and 0 ^ e < e0.

LEMMA 3.3. Let ε0 be as determined in Lemma 3.2. Then for 0 :£ ε < ε0 Fr+ε (x, t) is a nnivalent subordination chain.

Lemma 3.1. 4*