Hamada, Hidetaka

Kohr, Gabriela

Kyoto University Research Information Repository

Title The growth theorem of spirallike mappings in several complexvariablesAuthor(s)Hamada, Hidetaka; Kohr, GabrielaCitation数理解析研究所講究録 (1999), 1112: 26-35Issue Date1999-09URL http://hdl.handle.net/2433/63365RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityThe,$\mathrm{g}$rowth theorem of spirallike mappings inseveral complex variables(Hidetaka Hamada)Faculty of Mathematics, Babe\S -Bolyai Univ., Gabriela KohrAbstractLet $\mathrm{B}$ be the unit ball in an arbitrary complex Banach space $X$ . Let $\alpha\in$$\mathrm{R},$ $|\alpha|<\pi/2$ . First, we give the growth theorem for normalized spirallike map-pings of type $\alpha$ on B. We also show that the growth theorem does not hold fornormalized $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}\iota \mathrm{A}\mathrm{i}\mathrm{k}\mathrm{e}$ mappings defined by Suffridge. Next, we give an alternatecharacterization of normalized spirallike mappings of type $\alpha$ on $\mathrm{B}$ in terms ofsubordination chains, when the dimension of $X$ is finite.1 IntroductionLet $f$ be a univalent mapping in the unit disc $\triangle$ with $f(0)=0$ and $f’(0)=1$ .Then the classical growth theorem is as follows:$\frac{|z|}{(1+|_{Z}|)^{2}}\leq|f(z)|\leq\frac{|z|}{(1-|_{Z}|)^{2}}$.It is well known that the above growth theorem cannot be generalized to nor-malized biholomorphic mappings on the Euclidean unit ball $\mathrm{B}^{n}$ in $\mathrm{C}^{n}(n\geq 2)$ .Barnard, FitzGerald and Gong [1] and Chuaqui [2] extended the above growththeorem to normalized starlike mappings on $\mathrm{B}^{n}$ . Dong and Zhang [3] generalizedthe above result to normalized starlike mappings on the unit ball in complexBanach spaces.In this paper, we will generalize the above growth theorem to spirallike map-pings of type $\alpha$ on the unit ball $\mathrm{B}$ in an arbitrary complex Banach space. Onemight consider that the same growth theorem holds for all normalized spirallikemappings defined by Suffridge [12]. However, we can give an example of a normal-ized spirallike mapping such that the same growth theorem does not hold. ThisMathematics Subject Classification: Primary $32\mathrm{A}30$ ; Secondary $30\mathrm{C}45$Key words and phrases: Growth theorem, spirallike, subordination chain.1112 1999 26-35 26example also shows that the growth of normalized spirallike mappings cannot beestimated from above.We also give an alternate characterization of normalized spirallike mappingsof type $\alpha$ on the unit ball $\mathrm{B}$ with respect to an arbitrary norm on $\mathrm{C}^{n}$ in termsof subordination chains.2 IntroductionFor complex Banach spaces $X,$ $Y$ , let $\mathcal{L}(X, Y)$ be the space of all continuouslinear operators from $X$ into $Y$ with the standard operator norm. By $I$ we denotethe identity in $\mathcal{L}(X, X)$ . Let $G$ be a domain in $X$ and let $f$ : $Garrow Y$ . $f$ is saidto be holomorphic on $G$ , if for any $z\in G$ , there exists a $Df(z)\in \mathcal{L}(X, Y)$ suchthat$\lim_{harrow 0}\frac{||f(_{Z+}h)-f(Z)-Df(Z)h||}{||h||}=0$ .Let $\mathcal{H}(G)$ be the set of holomorphic mappings from a domain $G\subset X$ into $X$ .A mapping $f\in \mathcal{H}(G)$ is said to be locally biholomorphic on $G$ if its R\’echetderivative $Df(z)$ as an element of $\mathcal{L}(X, X)$ is nonsingular at each $z\in G$ . Let $\mathrm{B}$denote the unit ball with respect to the norm $||\cdot||$ on $X$ . A mapping $f\in \mathcal{H}(\mathrm{B})$is said to be normalized if $f(0)=0$ and $Df(\mathrm{O})=I$ . For each $z\in X\backslash \{0\}$ , wedefine$T(z)=\{z*\in \mathcal{L}(x, \mathrm{c}).\cdot. ||Z^{*}||=1, z^{*}(Z)=||_{Z}||\}$ .By the Hahn-Banach theorem, $T(z)$ is nonempty. Let$\Lambda^{r}=\{g\in \mathcal{H}(\mathrm{B})$ : $g(\mathrm{O})=0,$ $\Re z^{*}(\mathit{9}(z))>0$ for all $z\in \mathrm{B}\backslash \{0\},$ $z^{*}\in T(z)\}$ ,and also, let$\mathcal{M}=\{g\in N : Dg(0)=I\}$ .The following definition generalizes the notion of spirallike functions of type $\alpha$on the unit disc to B.Definition 2.1 Let $f$ : $\mathrm{B}arrow X$ be a normalized biholomorphic mapping on B.Let $\alpha\in \mathrm{R},$ $|\alpha|<\pi/2$ . We say that $f$ is a spirallike mapping of type $\alpha$ if thespiral $\exp(-e^{-i\alpha}t)f(z)(t\geq 0)$ is contained in $f(\mathrm{B})$ for any $z\in \mathrm{B}$ .We obtain the following theorem from Corollary 2 and Theorem 6 of Gurganus[4] (cf. [6], see also [7]).27Theorem 2.1 Let $f$ be a normalized locally biholomorphic mapping on B. If$f$ is a spirallike mapping of type $\alpha_{f}$ then $e^{-i\alpha}[Df(z)]-1(f(z))\in N$ . $Moreover_{J}$when $X$ is a finite dimensional complex Banach space, $f$ is a spirallike mappingof type $\alpha$ if and only if $e^{-i\alpha}[Df(z)]^{-}1(f(z))\in N$ .Remark 2.1 In Lemma 5 of Gurganus [4], he claimed that for each $h\in N$ andfor each $x\in \mathrm{B}$ , the initial value problem$\frac{\partial v}{\partial t}=-h(v)$ , $v(0)=x$ ,has a unique solution $v(t)$ for all $t\geq 0$ . For the proof, he uses Theorem 2.1of Pfaltzgraff [9]. One of the conditions on $h$ in the theorem is that for each$r\in(0,1)$ , there exists a constant $K(r)$ such that $||h(z)||\leq K(r)$ for all $z$ with$||z||\leq r$ . However, in general, holomorphic mappings on the unit ball is notnecessarily bounded on $||z||\leq r$ . We do not know whether the above condition issatisfied for all $h\in N$ or not. So, we restrict ourselves to the finite dimensionalcase in Theorem 2.1.The following definition is due to Suffridge [12].Definition 2.2 Let $f$ : $\mathrm{B}arrow X$ be a normalized biholomorphic mapping. Let$A\in \mathcal{L}(X, X)$ such that$\inf\{\Re z^{*}(A(Z)) : ||z||=1, z^{*}\in T(z)\}>0$ .We say that $f$ is spirallike relative to $A$ if $e^{-tA}f(\mathrm{B})\subset f(\mathrm{B})$ for all $\mathrm{t}\geq 0$ , where$e^{-\iota A}= \sum_{=k0}\infty\frac{(-1)^{k}}{k!}tAkk$ .Let $\mathrm{B}$ denote the unit ball with respect to an arbitrary norm $||\cdot||$ on $\mathrm{C}^{n}$ . Amapping $v\in \mathcal{H}(\mathrm{B})$ is called a Schwarz mapping if $||v(z)||\leq||z||$ for all $z\in$ B.This condition is equivalent to the condition that $v(\mathrm{O})=0$ and $||v(Z)||\leq 1$ for$z\in \mathrm{B}$ .If $f,$ $g\in \mathcal{H}(\mathrm{B})$ , we say that $f$ is subordinate to $g(f\prec g)$ if there exists aSchwarz mapping $v\in \mathcal{H}(\mathrm{B})$ such that $f(z)=g(v(z))$ for $z\in \mathrm{B}$ .Let $\{f(z, \mathrm{t})\}_{t\geq}0$ be a family of mappings such that $f_{t}(z)=f(z, t)\in \mathcal{H}(\mathrm{B})$ and$f_{t}(0)=0$ for each $t\geq 0$ . We call $\{f(z,t)\}$ a subordination chain if $f(z, s)\prec$$f(z, t)$ for all $z\in \mathrm{B}$ and $0\leq s\leq t$ . Moreover, $f(z, t)$ is called univalent if $f(\cdot, t)$is univalent on $\mathrm{B}$ for each $t\geq 0$ .283 The growth theoremIn this section, we will prove the following theorem.Theorem 3.1 Let $f$ be a normalized spirallike mapping of type $\alpha$ from $\mathrm{B}$ to $X$ .Then$\frac{||z||}{(1+||Z||)^{2}}\leq||f(z)||\leq\frac{||z||}{(1-||z||)2}$.Proof. Let$h(z)=e^{-i\alpha}[Df(Z)]^{-}1f(z)$ .Since $h\in N$ , we obtain the following inequalities from Lemma 4 of Gurganus[4].$\cos\alpha||Z||\frac{1-||_{Z}||}{1+||_{Z}||}\leq\Re z^{*}(h(z))\leq||z||\frac{1+||z||}{1-||_{Z}||}\cos\alpha$ (3.1)for $z\in \mathrm{B}\backslash \{0\},$ $z^{*}\in T(z)$ . Let $0<r_{1}<r_{2}<1$ . Let $z_{2}$ be a point such that$||z_{2}||=r2$ . The curve $c(t)=\exp(-e^{-}ti\alpha)f(z_{2})$ is contained in $f(\mathrm{B})$ for all $t\geq 0$ .Also $c(t)arrow \mathrm{O}$ as $tarrow\infty$ . Since $f$ is biholomorphic, the curve $v(t)=f^{-1}(c(t))$is well-defined and intersects the sphere $||z||=r_{1}$ at some point $z_{1}=f^{-1}(c(t_{1}))$ .Since$\frac{\partial v}{\partial t}=-h(v)$ ,we can show that $||v(t)||$ is absolutely continuous. Therefore, $||v(t)||$ is differen-tiable $\mathrm{a}.\mathrm{e}$ . on $[0, \infty)$ and$\frac{\partial||v||}{\partial t}=\Re v^{*}(\frac{\partial v}{\partial t})-$for $v^{*}\in T(v(t))\mathrm{a}.\mathrm{e}$ . on $[0, \infty)$ by Lemma 1.3 of Kato [8]. Then$\frac{\partial||v||}{\partial t}=-\Re v^{*}(h(v))<0$ (3.2)for $v^{*}\in T(v(t))$ . Let $F(t)=||f(v(t))||=e^{-t\cos\alpha}||f(z_{2})||$ . Then we have$\frac{1+||v(t)||}{||v(t)||(1-||v(t)||)}\frac{\partial||v||}{\partial t}$ $\leq$ $\frac{1}{F}\frac{dF}{dt}=-\cos\alpha$$\leq$ $\frac{1-||v(t)||}{||v(t)||(1+||v(t)||)}\frac{\partial||v||}{\partial t}$29from (3.1) and (3.2), Since $||v(t)||$ is strictly decreasing on $[0,t_{1}]$ by (3.2), wehave$\log F(t_{1})-\log F(0)$ $\geq$ $\int_{0}^{t_{1}}\frac{1+||v(t)||}{||v(t)||(1-||v(t)||)}\frac{\partial||v||}{\partial t}dt$$=$ $\int_{||v()||}^{1}|v(t_{1})||\frac{1+x}{x(1-X)}0d_{X}$$=$ $\log||v(t_{1})||-2\log(1-||v(t_{1})||)$$-\{\log||v(\mathrm{o})||-2\log(1-||v(0)||)\}$and$\log F(t_{1})-\log F(\mathrm{o})$ $\leq$ $\log||v(t_{1})||-2\log(1+||v(t_{1})||)$$-\{\log||v(\mathrm{o})||-2\log(1+||v(\mathrm{o})||)\}$ .Then$\frac{(1-||v(\mathrm{o})||)^{2}}{||v(0)||(1-||v(t_{1})||)^{2}}F(0)\leq\frac{F(t_{1})}{||v(t_{1})||}\leq\frac{(1+||v(\mathrm{o})||)^{2}}{||v(0)||(1+||v(t_{1})||)^{2}}F(0)$ .Namely,$\frac{(1-||z_{2}||)^{2}}{||z_{2}||(1-||v(t_{1})||)^{2}}||f(Z_{2})||\leq\frac{||f(v(t1))||}{||v(t_{1})||}\leq\frac{(1+||z_{2}||)^{2}}{||z_{2}||(1+||v(t_{1})||)^{2}}||f(_{Z_{2}})||$. (3.3)Letting $r_{1}arrow 0$ , we obtain that$\frac{(1-||z_{2}||)^{2}}{||z_{2}||}||f(z_{2})||\leq 1\leq\frac{(1+||z_{2}||)^{2}}{||z_{2}||}||f(z_{2})||$ ,since$\lim_{zarrow 0}\frac{||f(z)||}{||z||}=1zarrow 0\mathrm{i}\ln\frac{||Df(0)z||}{||z||}=1$ .This completes the proof.When $\alpha=0$ , we obtain the growth theorem of normalized starlike mappingson the unit ball in complex Banach spaces [3] (cf. $[1],[2]$ ).Let$M_{\infty}(r, f)=||| \sup_{z|=r}||f(_{Z})||$ .Then we obtain the following corollary (cf. Tsurumi [13]) from (3.3) and Theorem3.1.30Corollary 3.1 Let $f$ be a normalized $\mathit{8}pirallike$ mapping of type $\alpha$ from $\mathrm{B}$ to $X$ .Then the limit$\beta=\lim_{rarrow 1}(1-r)2M\infty(r, f)$exists. Moreover, we have $0\leq\beta\leq 1$ .Example 3.1 Consider the holomorphic mapping $f(z_{12}, z)=(z_{1}+az_{2}^{2}, Z_{2})$ onthe Euclidean unit ball in $\mathrm{C}^{2}$ . Let $A$ be a linear mapping such that $\mathrm{A}(z_{1,2}z)=$$(2z_{1}, z_{2})$ . Then $[Df(z)]^{-}1Af(Z_{1,2}z)=(2z_{1}, z_{2})$ . Therefore, $f$ is a normalizedspirallike mapping relative to A for any $a\in$ C. Let $a\in \mathrm{R}$ and $a>2\sqrt{15}$ . Let$z^{0}=(0,1/2)$ . Then $f(z^{0})=(a/4,1/2)$ and $||f(z^{0})||>2$ . On the other hand,$\frac{||z^{0}||}{(1-||Z^{0}||)2}=2$ .Therefore,$||f(z0)||> \frac{||Z^{0}||}{(1-||Z^{0}||)2}$ .Also, $||f(z^{0})||arrow\infty$ as $aarrow\infty$ . Therefore, the growth of normalized spirallikemappings cannot be estimated from above.4 Subordination chainsIn this section, we will give an alternate characterization of normalized spiral-like mappings of type $\alpha$ on $\mathrm{B}$ in terms of subordination chains, where $\mathrm{B}$ is theunit ball in $\mathrm{C}^{n}$ with respect to an arbitrary norm on $\mathrm{C}^{n}$ .Let $f$ : $\mathrm{B}arrow \mathrm{C}^{n}$ be a holomorphic mapping on $\mathrm{B}$ and let $\alpha\in \mathrm{R},$ $|\alpha|<\pi/2$ .Let$f(z, t)=e(1-ia)tf(e^{i}Z)at,$ $z\in \mathrm{B},$ $t\geq 0$ ,where $a=\tan\alpha$ . Then, we have the following theorem($\mathrm{c}\mathrm{f}$. [ $11$ , Theorem 6.6], [5,Theorem 2.4]).Theorem 4.1 Let $f$ : $\mathrm{B}arrow \mathrm{C}^{n}$ be a normalized locally biholomorphic mappingon $\mathrm{B}$ and let $\alpha\in \mathrm{R}$ , $|\alpha|<\pi/2$ . Then $\{f(z, t)\}$ is a univalent subordinationchain if and only if $f$ is a spirallike mapping of type $\alpha$ .Proof. First, assume that $f$ is a spirallike mapping of type $\alpha$ , Then $f_{t}$ isunivalent. It is easy to check that $f_{t}(z)\in \mathcal{H}(\mathrm{B}),$ $Df_{t}(0)=e^{t}I,$ $t\geq 0$ and also,31$f(z, t)$ satisfies the absolute continuity hypothesis of Theorem 2.2 of Pfaltzgraff[9].On the other hand, we have$\frac{\partial_{\mathit{1}}f}{\partial t}(z, t)=Df(z, t)g(z, t),$ $z\in \mathrm{B},$ $t\geq 0$ , (4.1)where$g(z, t)=iaz+(1-ia)e-iat[Df(eiat_{Z})]^{-}1f(e^{it}aZ)$ .Clearly $g(z, \mathrm{t})$ is a measurable function for each $z\in \mathrm{B},$ $g(\mathrm{O}, t)=0$ and$Dg(\mathrm{O}, t)=I$ . For any $z\in \mathrm{B}\backslash \{0\},$ $z^{*}\in T(z)$ , we have$\Re z^{*}(g(z, t))$ $=$ $\Re(ia||Z||)+\frac{1}{\cos\alpha}\Re(e^{-i}e-iabz^{*}[Df(eZ)iat]^{-}1f(e)\alpha iat_{Z})$$>$ $0$ ,since $f$ is a spirallike mapping of type $\alpha$ and $e^{-iat}z^{*}\in T(e^{iat}Z)$ . Therefore$g_{t}(z)\in \mathcal{M}$ . It is easy to show that, for each $r\in(0,1)$ , there exists a constant$M=M(r)>0$ such that$||g(z, t)||\leq M(r)$ ,for all $z\in\overline{\mathrm{B}}_{r}$ and $t\geq 0$ .Let $t_{m}=m$ if $a=0$ and $\mathrm{t}_{m}=2\pi m/a$ if $a\neq 0$ . Then $e^{-t_{m}}f(z, t_{m})=f(z)$ holdsfor any $m\in \mathrm{Z}$ .Hence, from Theorem 2.2 of Pfaltzgraff [9], it follows that $\{f(z, t)\}$ is a subor-dination chain.Conversely, assume that $\{f(z, t)\}$ is a univalent subordination chain. Thenthere exist Schwarz mappings $v(z, s, t)$ such that$f(z, s)=f(v(z, S, t), t),$ $z\in \mathrm{B},$ $t\geq s\geq 0$ .Then $v(z, s, \mathrm{t})=e^{-iat}f^{-1}(e^{(1-ia})(_{S-t})f(ez)iaS)$ and therefore $v(z, s, t)$ is differen-tiable with respect to $t$ for each $z\in \mathrm{B},$ $s\geq 0$ and $t\geq s$ . Hence, differentiatingthis equality with respect to $t$ , we obtain that$Df(v(_{Z}, s, t), t) \frac{\partial v}{\partial t}(Z, s, t)+\frac{\partial f}{\partial t}(v(_{Z,s}, t),$ $t)=0,$ $z\in \mathrm{B},$ $t\geq s\geq 0$ . (4.2)If we compare the relations (4.1) and (4.2) and use the univalence of $f(\cdot, t)$ for$t\geq 0_{7}$ we obtain that$g(v(z, s, t), \mathrm{t})=-\frac{\partial v}{\partial t}(z, s, t),$ $z\in \mathrm{B},$ $t\geq s\geq 0$ . (4.3)32Since $v(z, s, t)$ is a Schwarz mapping, we obtain that$\Re z^{*}(v(_{Z}, S, t))\leq||v(_{Z,s}, t)||\leq||z||$for all $z\in \mathrm{B}\backslash \{0\},$ $t\geq s$ and $z^{*}\in T(z\rangle$ .Therefore,$\Re z^{*}(\frac{\partial v}{\partial t}(z, 0, \mathrm{o}))$ $=$ $t arrow\lim_{0+}\Re_{z^{*}}(\frac{v(_{Z},\mathrm{o},t)-z}{t})$$\leq$ $t arrow\lim_{0+}\frac{||z||-||Z||}{t}$$=$ $0$ .Now, from (4.3) and the above inequality, we obtain that$\Re[z^{*}(e^{-i\alpha}[Df(z)]-1f(_{Z}))]\geq 0,$ $z\in \mathrm{B}\backslash \{0\},$ $z^{*}\in T(Z)$ .Moreover, for fixed $z\in\partial \mathrm{B},$ $0<r<1,$ $z^{*}\in T(rz)$ , let$\phi(\zeta)=\Re[Z*(e^{-i)]}\alpha_{\frac{[Df(\zeta z)]^{-1}f(\zeta z)}{\zeta}}$for ( $\in\triangle$ , where $\triangle$ denotes the unit disc in C. Then $\phi$ is harmonic on $\triangle$ . Since$(|\zeta|/\zeta)z^{*}\in T(\zeta_{Z}),$ $|\zeta|\phi(\zeta)\geq 0$ for $\zeta\in\triangle$ . Since$\phi(0)=\Re[z^{*}(e-i\alpha z)]=\Re[e^{-i\alpha}||z||]>0$ ,we have $\phi(\zeta)>0$ for all $\zeta\in\triangle$ by the minimum principle for harmonic functions.Considering $\phi(r)$ , we obtain that $f$ is a spirallike mapping of type $\alpha$ from Theorem2.1. This completes the proof.If we consider the case $\alpha=0$ in Theorem 4.1, we obtain the following resultdue to Pfaltzgraff and Suffridge [10, Corollary 2].Corollary 4.1 Let $f$ : $\mathrm{B}arrow \mathrm{C}^{n}$ be a normalized locally biholomorphic mapping$on$ B. Then $f$ is starlike if and only if $\{e^{t}f(Z)\}$ is a univalent subordination chain.Remark 4.1 Let $D$ be a bounded balanced pseudoconvex domain with $C^{1}$ pluri-subharmonic defining functions in $\mathrm{C}^{n}$ . Namely, for any $\zeta\in\partial D$ , there exist aneighborhood $U$ of $\zeta$ in $\mathrm{C}^{n}$ and a $C^{1}$ plurisubharmonic function $r$ on $U$ such that$D\cap U=\{z\in U : r(z)<0\}$ . In [5], the authors obtained similar results as inthis section for normalized locally biholomorphic mappings on $D$ .33References[1] R.W. Barnard, C.H. FitZGerald and S. Gong, The growth and 1/4-theorem8for starlike mappings in $\mathrm{C}^{n}$ , Pacific J. Math. 150 (1991), 13-22.[2] M. Chuaqui, Application of subordination chain8 to starlike mappings in $\mathrm{C}^{n}$ ,Pacific J. Math. 168 (1995), 33-48.[3] D. Dong and W. Zhang, Growth and 1/4-theorem for starlike maps in theBanach space, Chin. Ann. Math. Ser.A 13, No.4 (1992), 417-423.[4] K.R. Gurganus, $\Phi$ -like holomorphic functions in $\mathrm{C}^{n}$ and Banach spaces,Trans. Amer. Math. Soc. 205 (1975), 389-406.[5] H. Hamada and G. Kohr, Spirallike mappings on bounded balanced pseudo-convex domain8 in $\mathrm{C}^{n}$ , Zesz. Nauk. Politechniki Rzeszowskiej, Matematyka22 (1998), 23-27.[6] H. Hamada and G. Kohr, Spirallike non-holomorphic mapping8 on balancedpseudoconvex domains, to appear in Complex Variables.[7] H. Hamada, G. Kohr and P. Liczberski, $\Phi$ -like holomorphic mappings onbalanced pseudoconvex domain8, to appear in Complex Variables.[8] T. Kato, Nonlinear semigroups and evoluation equations, J. Math. Soc.Japan 19 (1967), 508-520.[9] J.A. Pfaltzgraff, Subordination chains and univalence of holomorphic map-pings in $\mathrm{C}^{n}$ , Math. Ann. 210 (1974), 55-68.[10] J.A. Pfaltzgraff and T.J. Suffridge, Close-to-8tarlike holomorphic functionsof several complex variable8, Pacif. J. Math. 57 (1975), 271-279.[11] Ch. Pommerenke, Univalent Functions, Vandenhoeck, Gottingen, 1975.[12] T.J. Suffridge, Starlikenessf convexity and other geometric propertie8 of holo-morphic maps in higher $dimen\mathit{8}i_{\mathit{0}}ns$, Lecture Notes in Mathematics, vol. 599,$\mathrm{S}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{r}_{\rangle}$ Berlin New York Heidelberg, 1976, pp.146-159.[13] K. Tsurumi, Radial growth of starlike holomorphic mappings in the unit ballin $\mathrm{C}^{n}$ , Spaces of analytic and harmonic functions and operator theory, RIMSKokyuroku 946(In Japanese), April 1996, 61-69.34H. HamadaFaculty of EngineeringKyushu Kyoritsu University1-8 Jiyugaoka, Yahatanishi-kuKitakyshu 807-8585, Japanemail: hamada@kyukyo-u.ac.jpG. KohrFaculty of MathematicsBabe\S -Bolyai University1 M. $\mathrm{K}\mathrm{o}\mathrm{g}\dot{\mathrm{a}}\mathrm{l}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{u}$ Str.3400 Cluj-Napoca, Romaniaemail: gkohr@math.ubbcluj.ro35

The growth theorem of spirallike mappings in several complex variables

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The growth theorem of spirallike mappings in several complex variables

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