The principle of subordination applied to functions of several variables

LEMMA 1. Let v(z; t): E x I—+C n be holomorphic for each te I = [0,1], v(z; 0) = z, v(0, t) = 0 and | v(z; ί) | < 1 ^/^βπ 2 e £7. 7/ ( 2 ) lim \{z - v(z; t))/t p] = w(z) ΐ->0+ exists and is holomorphic in E for some p > 0, then we^.

LEMMA 2. Let f:E—>C n be holomorphic and univalent and satisfy /(0) = 0. Let F(z; t): E x I-+C n be a holomorphic function of z for each tel = [0,1], F(z; 0) = f(z), F(0, t) = 0 and suppose F(z) t) <f for each t e I (i.e., F(E; t) af(E)

THEOREM 1. Suppose f:E-+C n is starlike and that J is the complex Jacobian matrix of f. There exists w e & such that f = Jw where f and w are written as column vectors.

Theorem 1 therefore says that if / is starlike then Re [det (Js)/z3- det J] ^ 0 when | z | = | zd \ > 0. Also, { 3 ) fj = &LW, + Mlw2 + + y±wn , 1 ^ j ^ n dz, dz2 dzn

THEOREM 2. Suppose f:E—>C n is holornorphίc, /(0) = 0, J is nonsingular and that {5 ) f(z) - Jw, w e &> . Then f is starlike.

THEOREM 3. Suppose f:E—>C n is holomorphic, f(0) = 0 and that J is nonsingular for all zeE. Then f is a univalent map of E onto a convex domain if and only if there exist univalent mappings fj (1 fg j ^ n) from the unit disk in the plane onto convex domains in the plane such that f(z) = T(f(zι)Jf2(z2) - ,fn(zn)) where T is a nonsingular linear transformation.

LEMMA 3. Let v(z; t): Dp x I—+C n be holomorphic for each tel> v(z, 0) = z, v(0, t) = 0 and \ v(z; t) \p < 1 when z e Dp. If lim [(z - v(z; t))/t p] = w(z) exists and is holomorphic in Dp for some p > 0, then w e &v.

LEMMA 4. Let f:Dp—+C n be holomorphic and univalent and satisfy /(0) = 0. Let F(z; t): Dp x I—+C n be a holomorphic function of z for each tel, F(z, 0) = f(z), F(0; t) = 0 and suppose F(z; t) <J for each tel. Let ρ>0 be such that \imt^0+(F(z; 0) - F(z; t))/t p = F(z) exists and is holomorphic. Then F(z) = Jw where w e &\.

THEOREM 4. If f: Dp—+C n is starlike then there exists we^p such thatf— Jw. Conversely, iff: Dp—>C n,f(0) = 0, J is nonsingular and f — Jw, w e &p then f is starlike.

THEOREM 5. Let f: Dp—*C n,f(0) = 0 and suppose J is non- singular. Then f(Dp) is convex if and only if F = Jw where w e &P for each choice of A — (Aly A2, , An), Aά ^ 0 (1 ^ j ^ n) and F is given by (7) with z e Dp. Now set p = 2. It is easy to see that Theorem 4 above is equiva- lent to Matsuno's Theorem 1 [4, p. 91]. Consider f:D2—>C 2 given by f( z) — ( zi + azli zz)

Theorem 5 shows that f(D2) is convex if and only if | a \ ^ 1/2 while Matsuno's Lemma 3 [4, p. 94] implies / is convex- like if and only if | a \ ^ 3α/"3"/4. This shows that convex-like is not equivalent to geometrically convex. REFERENCES 1. S. Bergman, The kernel function and conformal mapping, Mathematical Surveys, Vol. V., Amer. Math. Soc, New York, 1950. 2. S. Bochner and W. T. Martin, Several complex variables, Princeton Univ. Press, 1948.