The growth and 1/4-theorems for starlike mappings in Cn

THEOREM 1.1 (Suffήdge [5]). Suppose that f(z) is a normalized holomorphic mapping on the unit ball B and that J/(z) is nonzero in B. Then f is a one-to-one mapping that takes the ball onto a domain that is starlike with respect to the origin if and only if the normalized mapping (1.1) w(z) = Jf(z)- ιf(z) has the property that (1.2) Re~z'w(z) > 0 for every point z in the ball B. An easy corollary will be of use later in our work: If / is starlike with respect to the origin, then, when / is restri

LEMMA 1.2 (Pfaltzgraff [4]). Let w(z) be a normalized mapping such that (1.3) Re Σ

THEOREM 2.1. Let f = (f\, fι, ... , fn)' be a starlike mapping of the unit ball B in C n . For any point z e B, Furthermore the estimates are sharp.

Lemma 1.2. The rest of the argument follows as before. The inequalities in Theorem 2.1 are best possible. To demonstrate this fact we define If the variable z is restricted to be of the form (z\, 0)', then equality can be obtained in both inequalities of Theorem 2.1. That / satisfies the hypothesis that it be normalized at the origin is clear from power series expansions. That it is one-to-one in the ball is obvious since each coordinate function of / is a univalent function of one variable: thu

COROLLARY 3.1. If f{z) is a holomorphic mapping on the unit ball in Cn and is starlike with respect to the origin, then the image of f contains a ball of radius 1/4 centered at the origin. The value 1/4 is best possible.

COROLLARY 3.2. If f is holomorphic in the unit ball and is starlike with a k-fold symmetric image for k > 1, then ~ I / ( Z ) I " ( l - | z Then the image of the ball under f contains a ball of radius 2~ 2l k and these estimates are best possible.

COROLLARY 3.3. The only balanced domain which is the image of the unit ball under a normalized biholomorphic mapping is the unit ball

COROLLARY 3.4 (Poincare 1907, Cartan 1932). For n > 2, the unit ball in C n is not biholomorphically equivalent to a polydisc in C n.