Abstract. 1. Introduction

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1 Chin. Ann. o Math. 19B: 4(1998), THE GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS** Liu Taishun* Ren Guangbin* Abstract 1 4 The authors obtain the growth and covering theorem or the class o normalized biholomorphic starlike mappings on bounded starlike circular domains. This type o domain discussed is rather general, since the domain must be starlike i there exists a normalized biholomorphic starlike mapping on it. In the unit disc, it is just the amous growth and covering theorem or univalent unctions. This theorem successully realizes the initial idea o H. Cartan about how to extend geometric unction theory rom one variable to several complex variables. Keywords Starlike mappings, Growth theorem, Starlike circular domains 1991 MR Subject Classiication 32A30, 30C25 Chinese Library Classiication O Introduction On the geometric unction theory o one complex variable, the ollowing growth and -covering theorem is well known (see [2]). Theorem A. For each normalized univalent unction on the unit disc D C, z (1 + z ) 2 (z) z (1 z ) 2, z D. Especially, the let-hand side o the above inequality implies (D) 1 4D. For each z D, z 0, equality occurs in the above inequality i and only i is Koebe unction K(z) = z (1 z) 2 or its rotation e iθ K(e iθ z). It is natural to extend this and other results on the geometric unction theory o one variables to several variables. But as early as ity years ago, H. Cartan pointed out where the diiculty lies. For example, the ollowing is a counter-example: ( z ) 2 (z) = z 1, (1 z 1 ) k, k N. Obviously this is a normalized biholomorphic mapping on the unit ball B 2 in C 2, which indicates that the corresponding result in several complex variables ails, Then the question arises: to extend under some suitable restrict on biholomorphic mappings. Hence H. Cartan suggested the study o biholomorphic starlike mappings, convex Manuscript received October 24, Department o Mathematics, University o Science and Technology o China, Heei , China. Project supported by the National Natural Science Foundation o China and the National Education Committee Doctoral Foundation.

2 402 CHIN. ANN. OF MATH. Vol.19 Ser.B mappings and some other important classes o mappings in several complex variables. Along this direction, in 1988, the irst airmative result about the growth and 1 4-covering theorem in several complex variables was obtained by R. W. Barnard, C. H. Fitzgerald, S. Gong [3]. They gave the growth and 1 4-covering theorem or the class o normalized biholomorphic starlike mappings in the unit ball B n C n. Ater that, or the class o normalized biholomorphic starlike mappings, several authors, Liu [9], Gong, Yu, Wang [5,17], Paltzgra [12], Zhang, Dong [18], Chen [1] obtained respectively the generalization on other domains such as the bounded symmetric domains, egg domains, and the unit ball in inite dimensional Banach spaces. See [4] or more details. In this paper, we obtain the growth and 1 4-covering theorem on bounded starlike circular domains or the class o normalized biholomorphic starlike mappings, which extend all the corresponding results mentioned above. This paper was once reported by the irst author at the Conerence o Several Complex Variables o China held in Beijing in Main Theorem Our main result is the ollowing theorem. Main Theorem. Suppose Ω to be a bounded starlike circular domain in C n, its deining unction ρ(z) is a C 1 unction except or a lower dimensional set. I (z) is a normalized biholomorphic starlike mapping on Ω, then ρ(z) (1 + ρ(z)) 2 ρ((z)) ρ(z) (1 ρ(z)) 2, (2.1) or equivalently z (1 + ρ(z)) 2 (z) z (1 ρ(z)) 2. (2.2) Especially (Ω) 1 Ω. (2.3) 4 Note 2.1. (i) This kind o domain on which we discuss is rather general, in the sense that the domain must be starlike, i there exists a normalized biholomorphic starlike mapping on it, such that the contractive mappings 1 r (rz) are also starlike. This is simply because the identity mapping then as the limit o 1 r (rz) is also starlike. On the other hand it includes the ollowing domains as examples. (a) any bounded symmetric domain in C n with its standard Harish-Chandra realization (See Lemma 3.4 in the next section or the proo). (b) Egg domains B p = {z = (z 1,, z n ) C n : z 1 p z n p n < 1}, p 1 > 0,, p n > 0, or more general domains { (z,, w) C n 1 + +n k : A 1 z p A k w p k < 1 }, p 1 > 0,, p k > 0, z C n1,, w C n k, where A 1,, A k are the non-singular linear transormations in C n 1,, C n k respectively. (ii) Taking Ω = B n ; R I, R II, R III, R IV (the classical domains [6] ); Bp; the unit ball in inite dimensional Banach spaces; and bounded strictly balanced domains respectively in

3 No.4 Liu, T. S. & Ren, G. B. GROWTH THEOREM FOR STARLIKE MAPPINGS 403 the Main Theorem gives the corresponding growth and 1 4-covering theorems in [1, 3, 9, 17, 18]. 3. Preliminaries Deinition 3.1. A domain Ω C n is said to be circular i e iθ z Ω, whenever z Ω, θ R. A domain Ω C n is said to be starlike (with respect to the origin) i the line segment joining 0 Ω and every other point in Ω lies entirely in Ω. Deinition 3.2. Let Ω be a domain containing the origin. We call a holomorphic mapping (z) = ( 1 (z),, n (z)) rom Ω into C n normalized i (0) = 0, J (0) = I, where J is the Jacobian o, I is the identity matrix. I (Ω), the graph o, is a starlike set with respect to the origin in C n, we call a starlike mapping. The ollowing lemmas are needed in the proo o main theorem and its note. Lemma 3.1. Ω C n is a bounded starlike and circular domain i and only i there exists a unique real continuous unction ρ : C n R, called the deining unction o Ω, such that (i) ρ(z) 0, z C n ; ρ(z) = 0 z = 0, (ii) ρ(tz) = t ρ(z), or any t C, z C n, (iii) Ω = {z C n : ρ(z) < 1}. Note 3.1. The unction ρ(z) in the lemma is just the distance unction in the book o [8], which plays an important role in the characterization o pseudoconvex domains. Proo. I the continuous unction ρ(z) satisies (i), (ii), and (iii), then clearly Ω is a starlike circular domain. Its boundedness is also easy to prove, since i there exists a ray coming rom the origin which completely alls in the starlike domain Ω, then or any ixed point z 0 in this ray we have rom (iii) ρ(tz 0 ) < 1, t [0, + ). But we then obtain ρ(z 0 ) = 0 by (ii). This contradicts (i). Conversely, i Ω is a bounded starlike and circular domain in C n, then we deine ρ(z) = in{c > 0 : c 1 z Ω}. Obviously ρ(z) satisies (ii), (iii) and ρ(z) 0, z C n. I there exists a point z 0 0 such that ρ(z 0 ) = 0, then by (ii) and (iii), Ω includes the whole ray which comes rom the origin and through the point z 0, hence Ω is unbounded. Thus (i) holds. Finally we prove that ρ(z) is continuous. Clearly {z C n : r < ρ(z) < R} = RΩ \ rω is an open set in C n, which implies the continuity o ρ. This completes the proo o Lemma 3.1. Lemma 3.2. I Ω is a Harish-Chandra realization o some bounded symmetric domain, then its deining unction ρ(z) is holomorphic about z, z (except or a lower dimensional set). Proo. From [15, Proposition 4.6], Ω = {z C n : ad(rez) < 1}. Here C n is regarded as a subset p + o complexiication o Lie algebra o Aut(Ω), is the operator norm determined by the inner product o Killing orm, Rez = 1 2 (z + z) or z p + C n. Note that p + = p, p + p = 0. Denote ρ(z) = ad(rez). From [15, Lemma 4.5], it is easy to check that ρ(z) satisies Lemma 3.1 (i), (ii) and (iii). Note that the unction which satisies the condition o Lemma 3.1 (ii) and (iii) is unique, hence ρ(z) = ρ(z) = ad(rez). Since the computation o the operator norm ad(rez) only involves the computation o Lie algebra and o solv-

4 404 CHIN. ANN. OF MATH. Vol.19 Ser.B ing characteristic polynomial to get the maximum characteristic value, we see that ρ(z) is holomorphic about z, z except or a lower dimensional set. Lemma 3.3. I Ω C n is a bounded symmetric domains, and is convex and circular, then its deining unction ρ(z) is holomorphic about z, z (except or a lower dimensional set). Proo. Firstly suppose Ω is a irreducible bounded symmetric domain o rank 2, and is convex and circular domain, then Ω and its Harish-Chandra imbedding are the same up to an aine linear transormation by Main Theorem in [10]. Thus the result ollows rom Lemma 3.2. Secondly, i Ω is a irreducible bounded symmetric domain o rank 1, and is a convex and circular domain, then Ω is exactly the common ball. The result also holds. Note that i ρ 1 (z), ρ 2 (z) are the deining unctions o Ω 1, Ω 2 respectively, then the deining unction o Ω 1 Ω 2 is ρ(z, w) = max (ρ 1 (z), ρ 2 (w)). Thus Lemma 3.3 holds or the reducible situation. Finally, let us recall the well-known result in the theory o one complex variable. Lemma 3.4. Let g be a holomorphic unction on the unit disc in C. I Re g(0) = 1, Re g() 0, D, then the ollowing inequality holds: Re g(), D Preparatory Theorem Now, we provide a necessary condition o starlikeness or normalized biholomorphic mapping, which has its own interest. Preparatory Theorem. Suppose Ω to be a bounded starlike circular domain in C n, its deining unction ρ(z) is a C 1 unction except or a lower dimensional set. I (z) is a normalized biholomorphic starlike mapping, then we have Re ρ 1 (z)j (z)(z) 0, z Ω. (4.1) Moreover ρ(z) 1 + ρ(z) 1 ρ(z) 2Re ρ 1 ρ(z) (z)j (z)(z) ρ(z)1, z Ω, (4.2) 1 + ρ(z) ( where (z) is denoted by a column vector and ρ (z) = ρ 1 (z),, ρ n (z). Note 4.1. In the special cases, that is, in the bounded strictly balanced domains, this result was also obtained by Chen [1, Lemma 3.2]. Proo. Now, we list some simple properties o deining unction derived directly rom Lemma 3.2(ii), which will be used very oten below. 2Re ρ (z)z = ρ(z), z Cn, (4.3) 2Re ρ (z 0)z 0 = 1, z 0 Ω, (4.4) ρ ρ (λz) = (z), λ [0, ), (4.5) ρ (eiθ iθ ρ z) = e (z), θ R. (4.6) )

5 No.4 Liu, T. S. & Ren, G. B. GROWTH THEOREM FOR STARLIKE MAPPINGS 405 The above equalities hold except or a lower dimensional set. In the ollowing, we ix 0 z Ω and denote z 0 = z ρ(z). Note that z 0 Ω. First we begin to prove (4.1), which will be treated in two cases. Case 1. z 0 be a non-essential boundary point [8, p.125] o Ω. This means (ξ), J 1 (ξ) can be holomorphically continued to some neighbourhood o z 0, so is the mapping 1 ((1 r)(ξ)) or any ixed r (0, 1). Note that under this homomorphic extension, (z 0 ) is deined. Clearly 1 ((1 r)(z 0 )) Ω, that is, ρ ( 1 ((1 r)(z 0 )) ) 1, 0 < r < 1. By expanding in Taylor series o r, the above inequality changes into Furthermore { } ρ z 0 J 1 (z 0)(z 0 )r + O(r 2 ) 1. ρ(z 0 ) 2Re ρ (z 0)J 1 (z 0)(z 0 )r + O(r 2 ) 1. Recall that ρ(z 0 ) = 1, thus Re ρ (z 0)J 1 (z 0)(z 0 ) 0. (4.7) Since Ω is a circular domain, or each θ R, e iθ z 0 is also a non-essential boundary point o Ω just as z 0, the same argument as (4.7) gives Denote Re ρ (eiθ z 0 )J 1 (eiθ z 0 )(e iθ z 0 ) 0.. (4.8) h() = Re ρ (z 0) J 1 (z 0)(z 0 ), D, where D is the unit disc in C. Recall that all z 0 ( D) are nonessential boundary points o Ω, thus h() is the real part o a homomorphic unction, or a harmonic unction on D. By (4.6) and (4.8), h() 0 on D. Using the minimal value principle o harmonic unction, we have h() 0, D. Particularly taking = ρ(z), by (4.5) and (4.6), we obtain Re ρ 1 (z)j (z)(z) 0. Case 2. z 0 be a essential boundary point o Ω. From the Theorem in [8], there exists a neighbourhood o z 0 in Ω which consists o essential boundary points o Ω. Now Choosing some suicient small neighborhood o z 0 on Ω and joining every point in this neighbourhood with the origin, we obtain a cone-type domain Ω in C n, which is a domain o holomorphy. Consider the closed analytic disc in C n : φ r,t () = 1 ((1 r)(tz 0 )), D, or any ixed t (0, 1), r [0, 1]. It is easy to observe that φ r,t ( D) Ω. Moreover, note that or the suicient small r, φ r,t () = 1 ((1 r)(tz 0 )) = tz 0 + O(r) is very closed to the ray tz 0, the center ray o the cone-type domain Ω. Hence φ r,t ( D) Ω, t (0, 1), r [0, δ], (4.9)

6 406 CHIN. ANN. OF MATH. Vol.19 Ser.B where δ is a suicient small positive number. Now we claim that φ r,t (D) Ω, t (0, 1), r [0, δ]. (4.10) I not, we can assume that there exists a positive constant s 0 < 1 such that sφ r,t (D) Ω or any 1 2 s 0 < s < s 0, while or s 0 < s < 1, sφ r,t (D) Ω. This means φ s,r,t (D) is not relative compact in Ω, where φ s,r,t (z) = sφ r,t (z). 1 2 s0<s<s0 At the same time recalling that Ω is a cone-type domain, by (4.9) we have φ s,r,t ( D) = sφ r,t ( D) Ω or 1 2 s 0 < s < 1. Furthermore φ s,r,t ( D) Ω. 1 2 s0<s<1 But on the other hand, rom the property o analytic disc o holomorphic domain (see [8, Theorem ( )]), we immediately obtain φ s,r,t (D) Ω, which is a 1 2 s0<s<s0 contradiction. This proves the desired claim. By Lemma 3.1, we know that (4.10) implies ( 1 ) ((1 r)(tz 0 )) ρ 1, < 1. Letting t 1, we have ρ ( 1 ((1 r)(z 0 )) ), acquire ρ ( 1 ((1 r)(z)) ) ρ(z). Namely, ρ(z) 2Re ρ Thereore Re ρ 1 (z)ρ(z)j Secondly, we prove the urthermore inequality (4.2). Denote < 1. Then take ζ = ρ(z) to 1 (z)j (z)(z)r + o(r2 ) ρ(z). (z)(z) 0. This establishes the inequality (4.1). g() = 2 ρ (z 0) J 1 (z 0)(z 0 ), D. From (4.5), (4.6), we know that i let = e iθ, then ρ (z 0) = ρ (z 0)e iθ. Thereore g() = 2 ρ (z 0)J 1 (z 0 )(z 0 ). Clearly rom the normalized condition o, (4.4), and (4.1), we know that g() satisies the conditions o Lemma 3.4. This implies Re ρ (z 0)J 1 (z 0)(z 0 ) 1, D. 1 + Thus, set = ρ(z) in the above equality to obtain the desired equality (4.2). This completes the proo o the preparatory theorem. 5. The proo o Main Theorem Now we can give the proo o the Main Theorem. Proo o Main Theorem. Denote z(t) = 1 (t(z)), t (0, 1). Geometrically this represents a curve in the domain Ω, obtained by pulling back the straight line segment linking the point (z) and the origin in (Ω) through the mapping 1. Then it is easy to

7 No.4 Liu, T. S. & Ren, G. B. GROWTH THEOREM FOR STARLIKE MAPPINGS 407 see that Obviously, (z(t)) = t(z), (5.1) J 1(t(z)) = J 1 (z(t)), (5.2) dz(t) = 1 dt t J 1 (z(t))(z(t)), (5.3) z(t) = t(z) + O(t 2 ). (5.4) dρ(z(t)) dt Thus, by (4.2), (5.3) and (5.5), we obtain at once ( ) ρ = 2Re (z(t))dz(t). (5.5) dt 1 + ρ(z(t)) ρ(z(t))1 t 1 ρ(z(t)) dρ(z(t)) 1 ρ(z(t)) ρ(z(t))1 dt t 1 + ρ(z(t)). (5.6) Integrate the both sides o the right inequality above to yield An easy computation gives that is, log 1 ϵ (1 + ρ(z(t))) dρ(z(t)) ρ(z(t)) (1 ρ(z(t))) 1 ϵ dt t. ρ(z) (1 ρ(z)) 2 log ρ(z(ϵ)) (1 ρ(z(ϵ))) 2 log 1 ϵ, ρ(z) (1 ρ(z)) 2 ρ(z(ϵ)) (1 ρ(z(ϵ))) 2 ϵ. I we set ϵ 0 in the above equality, then by Lemma 3.1(ii) and (5.4) we obtain the ollowing ρ((z)). Similarly, integrating the both sides o the let inequality in ρ(z) (1 ρ(z)) 2 (5.6) gives ρ((z)) ρ(z) (1+ρ(z)) 2. Now, we turn to the equivalence o the inequalities (2.1) and (2.2). The method is the same as [5]. Denote r = z. Obviously dr = 1 n (z i dz i + z i dz i ), dρ = 2r i=1 In the standard inner product by (4.3), we have n i=1 ( ρ dz i + ρ ) dz i. i i dz i, dz i = dz i ; dz i = 2, dz i, dz j = 0; dz i, dz j = dz i, dz j = 0 (i j), dρ, dr = 1 r 2Re ρ z = ρ r. (5.7) On the other hand, it is easy to check that dρ, dr = ρ ρ dr, dr = r r. (5.8) Combining (5.7) and (5.8) gives ρ r = ρ r. (5.9)

8 408 CHIN. ANN. OF MATH. Vol.19 Ser.B Then integrating the both sides o the equality rom (z) to (ϵz) or any small positive number ϵ, we obtain ρ((z)) ρ((ϵz)) = (z) (ϵz). (5.10) We will soon see that this equality gives the equivalence o (2.1) and (2.2). In act, assume (2.1), then combining it with (5.10) gives (1 ϵρ(z))2 + ϵρ(z))2 (ϵz) (z) (ϵz) (1 ϵ(1 + ρ(z)) 2 ϵ(1 ρ(z)) 2. Then, letting ϵ 0, we obtain (2.2) since lim. (ϵz) ϵ 0 ϵ = z by the normalized condition o The same argument shows that (2.2) implies (2.1). This completes the proo o Main Theorem. Acknowledgement. The authors would like to thank Proessor Gong Sheng or his help and encouragement. Reerences [ 1 ] Chen Zheng, On the characterization o biholomorphic mappings and starlike mappings in a class o bounded strictly balanced domains in C n, Chin. Ann. o Math., 16A:2 (1995), [ 2 ] Duren, P. L., Univalent Functions, Grundlehren der mathematischen Wissenschaten 259, Springer Verlag, New York, [ 3 ] Barnard, R. W., Fitzgerald, C. H. & Gong, S, The growth and 1 4 -theorems or starlike mappings in Cn, Chin. Sci. Bull., 34 (1989), , Paciic Jour. o Math., 150(1991), [ 4 ] Gong, S., Biholomorphic mappings in several complex variables, Contemp. Math., 142 (1993). [ 5 ] Gong Sheng, Wang Shikun & Yu Qihuang, The growth theorem or biholomorphic mappings in several complex variables, Chin. Ann. o Math., 14B:1 (1993), [ 6 ] Hua, L. K., Harmonic Analysis o Functions o Several Complex Variables in the Classical Domains, Amer. Math. Soc. Transl. Ser. 6, Providence, RI, [ 7 ] Kikuchi, K., Starlike and convex mappings in several complex variables, Paciic J. o Math., 44(1973), [ 8 ] Krantz, S. G., Function theory o several complex variables, John Wiley and Sons, New York, [ 9 ] Liu, T. S., The growth theorems, covering theorems and distortion theorems or biholomorphic mappings on classical domains., University o Science and Technology o China, Dissertation (1989). [10] Mok Ngaiming & Tsai I-Hsun, Rigidity o convex realizations o irreducible bounded symmetric domains o rank 2, J. Rein Angew. Math., 431 (1992), [11] Paltzgra, J. A., Subordination chains and univalence o holomorphic mappings on C n, Math. Ann., 210 (1974), [12] Paltzgra, J. A., Loeimer theory in C n, Abstracts o Papers Presented to the Amer. Math. Soc. J., 11(1990), [13] Range, R. M., Holomorphic unctions and integral representations in several complex variables, GTM 108, Springer-Verlag New York Inc., [14] Rudin, W., Function theory in the unit ball o C n, Grundlehren der Mathematischen Wissenschaten in Einzeldarstellungen, Springer, Berlin, [15] Satake, I., Algebraic structures o symmetric domains, Publications o the Mathematical Society o Japan, 14, Kano Memorial Lectures, 4, Iwanami Shoten, Tokyo, Princeton University Press, Princeton, N. J, [16] Suridge, T. J., The principle o subordinition applied to unctions o several variables, Paciic J. Math., 1 (1970), [17] Yu QiHuang, Wang Shikun & Gong Sheng, The growth and 1 4 -theorems or starlike mappings in Bp, Chin. Ann. o Math., 11B:1 (1990), [18] Zhang, W. J. & Dong, D. Z., The growth and 1 -theorem or starlike mappings in a Banach space, Chin. 4 Sci. Ann., 18 (1991),