Subordination theorems for some classes of starlike fumctions

Theorem 1. Let $\alpha$ , d, and M be fixed nonnegative numbers satisfying $0 < \alpha < \infty$ , $0 \le d < 1$ , and $1 < M \le \infty$ . Then there exists a function $F = F(\cdot, \alpha, d, M) \in S(\alpha, d, M)$ with the following properties: - (A) The function $g(z) = \log F(z)/z$ , $z \in K$ , (g(0) = 0) is univalent and convex in the direction of the imaginary axis, - (B) If $f \in S(\alpha, d, M)$ , then $\log f(z)/z, z \in K$ , is subordinate to g. In order to describe F we first make the following definition.

Corollary 1. Let $\alpha$ , d, M, and F be as in Theorem 1. Let $\Phi$ be a given nonconstant entire function. If $f \in S(\alpha, d, M)$ , then (A) For given $z \in K - \{0\}$ $$\operatorname{Re}\left\{\Phi\left[\log\frac{f(z)}{z}\right]\right\} \leq \max_{0 < \theta \leq 2\pi} \operatorname{Re}\left\{\Phi\left[\log\frac{F(e^{i\theta}z)}{e^{i\theta}z}\right]\right\},$$ (B) For given r, 0 < r < 1, and $\lambda > 0$ , $$\int_0^{2\pi} |f(re^{i\theta})|^{\lambda} d\theta \leq \int_0^{2\pi} |F(re^{i\theta})|^{\lambda} d\theta,$$ (C) For a given positive integer $N \ge 2$ , $$\sum_{k=2}^{N} |a_k|^2 \leq \sum_{k=2}^{N} |A_k|^2,$$ where $f(z) = z + \sum_{k=2}^{\infty} a_k z^k$ and $F(z) = z + \sum_{k=2}^{\infty} A_k z^k$ , $z \in K$ . Equality holds in any one of (A), (B), or (C) only if for some real $\theta$ , $f(z) = e^{-i\theta}F(e^{i\theta}z), z \in K$ . We note that with the appropriate choice of $\Phi$ in Corollary 1, some of the classical extremal problems follow for $S(\alpha, d, M)$ . For example, the quantities |f(z)/z|, |Arg f(z)/z|, $Re\{[f(z)/z]^p\}$ , where |z|=r, 0 < r < 1, $f \in S(\alpha, d, M)$ , and p > 0, are all maximized or minimized on $\partial K_r$ by F. We remark that Krzyz [10] proved (A) of Corollary 1 for S(1, 0, M), and Barnard [1] proved (A) of Corollary 1 for S(1, d, M). However, they did not show the F in their respective classes was the unique function with property (A). Also Miller [11] proved (A) of Corollary 1 for $S(\alpha, 0, \infty)$ , $0 < \alpha < \infty$ , and $\Phi(w) = \pm w$ . Next for fixed d, $0 \le d < 1$ , and M, $1 < M \le \infty$ , let $S^(d, M)$ denote the class of normalized starlike univalent functions f in K which satisfy (1.3). We observe for given r, 0 < r < 1, and $f \in S^(d, M)$ that f(rz)/r, $z \in K$ , is in $S(\alpha, d, M)$ for $\alpha > 0$ small enough. Moreover if f, $g \in S^(d, M)$ and 0 < r < 1, then the function $z[f(z)/z]^r[g(z)/z]^{1-r}$ , $z \in K$ , is in $S^(d, M)$ . Using these observations and Theorem 1, we easily obtain in §9, the following theorem. - (A) The function $g(z) = \log F^*(z)/z$ , $z \in K$ , (g(0) = 0) is convex univalent, - (B) If $f \in S^*(d, M)$ , then $\log f(z)/z$ , $z \in K$ , is subordinate to g. Theorem 2 implies, as in the discussion after Theorem 1, the following corollary.

Corollary 2. Let d and M be as in Corollary 1. Replace F by $F$ and $S(\alpha, d, M)$ by $S^(d, M)$ in Corollary 1. Then Corollary 1 is valid for $F^*$ . We remark that Theorem 2 and Corollary 2 are well known in the simple case $d = \frac{1}{4}$ , $M = \infty$ (see Goluzin [4, Thm. 1, p. 531]). Moreover, Suffridge [16] proved (C) of Corollary 2 for $S^(d, \infty)$ and N = 2. Barnard [2] proved (A) of Corollary 2 for $S^(d, M)$ and $\Phi(w) = \pm w$ . With these exceptions, Theorem 2 and Corollary 2 are new results for starlike functions. For given M, $\alpha$ , d, as in Theorem 1, let f be in $S(\alpha, d, M)$ and put D = f(K). Then the proof of Theorem 1 is based upon a geometric description of $\partial D$ and a use of the Julia variational formula similar to Krzyz [10] and Barnard [1]. This geometric description of $\partial D$ is obtained in Lemmas 1-3 of §2. In §3 we use Lemmas 1-3 to determine $\delta(M, \alpha)$ and show $F \in S(\alpha, d, M)$ is uniquely defined by (i)-(iv). In §4 we define our variations of D when $\partial D$ contains an arc of an $\alpha$ curve. In §5 we show that the Hadamard variational formula holds for the Greens functions of our varied domains. In §6 we deduce the Julia variational formula from the Hadamard variational formula, and show how it can be used to solve an extremal problem. In §7 we prove Lemmas 4–7. We use these lemmas in §8 to prove Theorem 1. In §9 we deduce Theorem 2 from Theorem 1 and describe $\partial F^*(K, d, M)$ . As motivation for the proof of Theorem 1, we first remark that it turns out (A) of Corollary 1 implies Theorem 1. Second, we remark that our geometric description implies $\partial D$ is made up of a finite number of arcs with the following property: each arc is the image, under $\zeta^{\alpha}$ , of an arc contained in the boundary of a convex domain. Since the Julia variation is a local boundary variation, it follows that the solution to (A) of Corollary 1 in $S(\alpha, d, M)$ should be obtainable from a local use of conformal mapping and arguments similar to those of Krzyz [10] and Barnard [1]. Furthermore, a general description of $\partial F(K, \alpha, d, M)$ from considering local $\alpha$ should follow powers of Z $\partial F(K, 1, d, M)$ . This is indeed the case, as we see from (i)-(iv). emphasize, though, that the extremal functions in Theorem 1, corresponding to different values of $\alpha$ , do not bear such a simple relationship. Even though the bounds on $F(K, \alpha, d, M)$ make it quite difficult to obtain an explicit representation formula for F, this function is completely described by its geometric properties. Since the Julia variational method allows to preserve both bounds and the class, it seems the most natural way to prove Theorem 1. Finally the authors would like to thank Professor Frank Keogh for some helpful comments concerning the geometric description of $S(\alpha)$ . 2. A geometric description of the image domains of $\alpha$ starlike functions. Given $w \neq 0$ , let $\operatorname{Arg} w$ , $-\pi \leq \operatorname{Arg} w \leq \pi$ , denote the principal argument of w. Let $\gamma$ be an $\alpha$ curve as in Definition 1. Since $\gamma$ is the image of a line, not containing $\zeta = 0$ , under a continuous $\alpha$ power of $\zeta$ , it follows for $0 < \alpha \leq 2$ , that $\gamma$ divides the w plane into two disjoint domains. Moreover the domain containing w = 0 is starlike. However for $\alpha > 2$ , $\gamma$ intersects itself, and consequently there exist rays through w = 0 which intersect $\gamma$ more than once. Since we shall be studying starlike domains in which part of the boundary is an arc of $\gamma$ , it is necessary to make the following definition for fixed $\alpha$ , $0 < \alpha < \infty$ .

Lemma 1. Let D be a bounded domain containing w=0 with the property that each ray through w=0 intersects $\partial D$ in exactly one point. Let $\alpha$ be a fixed positive number and suppose there exists a sufficiently small $\eta>0$ such that whenever $A,B\in\partial D$ and $0<|\operatorname{Arg}(\bar{A}B)|<\eta<\pi\alpha$ , then either $\beta\subset D\cup\{A,B\}$ or $\beta\subset\partial D$ . Then there is a function $g\in S(\alpha)$ and a number t>0 such that tg(K)=D. Proof. Let A, B, be any two points of $\partial D$ with $0 < |Arg(\bar{A}B)| < \eta \le \pi \alpha$ . Define $T_1$ and $\beta \in T_1 \cup \{A, B\}$ relative to A and B as in (2.1a). Let $h_1$ be an analytic $1/\alpha$ power of w on $T_1$ which is continuous on $\partial T_1$ . Put $D_1 = D \cap T_1$ , $\lambda = \partial D \cap T_1$ , and suppose that $E, F, E \ne F$ , are any two points of $\lambda$ . Then from the hypotheses of Lemma 1 (with E, F, F, F, F, F, F, F, F, F, F, F, F, F, Taking the preimage of $\tau$ under $h_1$ , we find that $\lambda$ may be approximated by an arc $\sigma_1 \subset D_1 \cup \lambda \cup \{A, B\}$ , made up of $\alpha$ arcs, with endpoints A, B. Moreover each point of $\sigma_1$ is within C/n of a point of $\lambda$ , where C is a positive constant which depends only on $\alpha$ and D. Also the tangent to $\sigma_1$ rotates counterclockwise as we pass from one $\alpha$ arc to another in the counterclockwise direction. Since $\partial D$ may be written as a finite union of sets of the form $\lambda$ , we see that $\partial D$ may be approximated by a Jordan curve $\sigma$ with the same properties as $\sigma_1$ . bounded domain D(n), with $\partial D(n) = \sigma$ , is clearly starlike with respect to w = 0. Let $g_n$ denote the Riemann mapping function satisfying $g_n(0) = 0$ , $g'_n(0) > 0$ , and $g_n(K) = D(n)$ . Then $g_n$ is continuous in $K \cup \partial K$ and a continuous $1/\alpha$ power of $g_n$ maps $\partial K - \{1\}$ onto a polygonal arc. Moreover, as $\partial K - \{1\}$ is described in the counterclockwise direction, a piecewise continuous argument of the tangent to this polygonal arc does not decrease. Using this fact and a Schwarz-Christophel type argument we deduce that $$\alpha \operatorname{Re} \left\{ 1 + z g_n''(z) / g_n'(z) \right\} + (1 - \alpha) \operatorname{Re} \left\{ z g_n'(z) / g_n(z) \right\}$$ $$= \sum_{k=1}^m b_k \operatorname{Re} \left( \frac{1 + e^{-i\theta_k} z}{1 - e^{-i\theta_k} z} \right), \ z \in K,$$ where $b_k$ and $\theta_k$ are positive, and m is a positive integer. Hence $$(2.2) g_n/g'_n(0) \in S(\alpha).$$ The sequence $(g_n)_1^{\infty}$ is a uniformly bounded sequence of univalent functions in K. Moreover from the construction of D(n), we see that $g_n(K) = D(n) \rightarrow D$ in the sense of kernel convergence. Using these facts and applying a theorem of Carathéodory (see Goluzin [4, Thm. 1, p. 55]), we deduce that $\lim_{n\to\infty} g_n = \hat{g}$ , $\hat{g}'(0) > 0$ , and $\hat{g}(K) = D$ . Using the compactness of $S(\alpha)$ and (2.2), we further deduce that $\hat{g}/\hat{g}'(0) = g \in S(\alpha)$ . Hence Lemma 1 is true. To continue our geometric description of the image domains of $\alpha$ starlike functions we prove

Lemma 2. Let $f \in S(\alpha, 0, M)$ for some $M < \infty$ and put D = f(K). Then each ray through w = 0 intersects $\partial D$ in exactly one point. If A, B, $A \ne B$ , are in $\partial D$ and if $\beta$ is an $\alpha$ arc with endpoints A and B, then - (a) either $\beta \subset \partial D$ or $\beta \subset D \cup \{A, B\}$ , - (b) if $\Omega$ denotes the component of $D \beta$ containing w = 0, then there exists $a \in S(\alpha)$ and t > 0 such that $tg(K) = \Omega$ .

Lemma 4. If $\Omega \in \mathcal{A}$ , then there exists a sequence of domains $\{\Omega_n\}$ with $\Omega_n \in \mathcal{A}_n$ such that $\Omega_n \to \Omega$ in the sense of kernel convergence. Proof. It obviously suffices to show that for each $\eta > 0$ there exists an integer n and $\Omega_n \in \mathcal{A}_n$ such that $\partial \Omega_n$ is contained in an $\eta$ neighborhood of $\partial \Omega$ . Let $f \in S(\alpha, d, M)$ be such that $f(K) = \Omega$ . For given r, 0 < r < 1, we consider the function $f_r(z) = f(rz)/r$ , $z \in K$ . From (1.1) we see that $f_r$ is an $\alpha$ starlike function. Moreover the maximum and minimum modulus principle guarantee the existence of a $d_1$ and $d_2$ such that $$(7.1) d < d_1 < |f_r(z)/z| < M_1 < M, z \in K.$$ We put $\Omega^ = f_r(K)$ . Then since f is continuous on $K \cup \partial K$ , we may choose r near enough 1, such that each point of $\partial \Omega$ is contained in an $\eta/2$ neighborhood of $\partial \Omega$ . From Lemma 2 we see that $\Omega^*$ may be approximated by an $\alpha$ starlike domain G with the following properties: - (i) $G \subset \Omega^*$ , - (ii) $\partial G \subset L(d_1, M_1) = \{z : d_1 \leq |z| \leq M_1\},$ - (iii) $\partial G$ is the union of a finite number of $\alpha$ arcs, - (iv) $\partial G$ is contained in an $\eta/4$ neighborhood of $\partial \Omega^*$ , (v) if $$\rho = \text{m.r. } G < 1$$ , then $\frac{1}{\rho} \le \min \left\{ M/M_1, 1 + \frac{\eta}{4M} \right\}$ . From (ii), (iii), and (v) we see that $1/\rho G \in \mathcal{A}_n$ for some n. Also, (iv) and (v) imply that $\partial(1/\rho G)$ is contained in an $\eta/2$ neighborhood of $\Omega^*$ . Hence if $\Omega_n = 1/\rho G$ , then $\partial \Omega_n$ is contained in an $\eta$ neighborhood of $\Omega$ . This completes the proof of Lemma 4. Lemma 4 and A Theorem of Carathéodory imply that if $f(K) = \Omega$ , $f_n(K) = \Omega_n$ , where $f, f_n \in S(\alpha, d, M)$ , then $$\lim_{n \to \infty} f_n = f$$ uniformly on compact subsets of K. For a given positive integer n, let $C_n$ denote the class of functions $f \in S(\alpha, d, M)$ with $f(K) \in \mathcal{A}_n$ . As in the proof of Lemma 1, we see that if $f \in C_n$ , then f may be written in the form $$\alpha \operatorname{Re}\left\{1 + \frac{zf''(z)}{f'(z)}\right\} + (1 - \alpha) \operatorname{Re}\left\{\frac{zf'(z)}{f(z)}\right\} = \sum_{k=1}^{m} a_k \operatorname{Re}\left\{\frac{1 + e^{i\theta_k}z}{1 - e^{i\theta_k}z}\right\},\,$$ where $m \le n$ , $a_k > 0$ $(1 \le k \le m)$ , and $\sum_{k=1}^m a_k = 1$ . From this formula it is easily seen that $C_n$ is compact. Hence if $C_n \ne \{\phi\}$ , then there exists an extremal function $F_n$ for Problem 1 with $C = C_n$ . Choose a subsequence $(n_j)_1^{\infty}$ of $(n)_1^{\infty}$ such that if $H_j = F_{n_j}$ , $j = 1, 2, \cdots$ , then $\lim_{j \to \infty} H_j = H \in S(\alpha, d, M)$ uniformly on compact subsets of K. Then from (7.2) we see that H is an extremal function for Problem 1 with $C = S(\alpha, d, M)$ . We note for given $f \in S(\alpha, d, M)$ and $t \in K$ that the function $f(tz)/t, z \in K$ , is also in $S(\alpha, d, M)$ . It follows from this fact and a result of Kirwan [9] that (7.3) $$\Phi' \left[ \log \frac{H(z)}{z} \right] \neq 0.$$ Here $\Phi$ and z are as in Problem 1, and $H = \lim_{j \to \infty} H_j$ is as above. Hence we may choose $n_0$ large enough such that (7.4) $$\Phi'[\log(H_j(z)/z)] \neq 0, j \geq n_0.$$ We use (7.4) to obtain a partial description of $\Omega_j = H_j(K), j \ge n_0$ . Indeed, we have

Lemma 5. Let $H_j$ and $\Omega_i = H_j(K)$ be as above for $j \ge n_0$ . Then all but at most two of the $\alpha$ sides of $\partial \Omega_j$ are either $\alpha$ chords of $\partial K_M$ or are tangent to $\partial K_d$ .

Lemma 6. Let $d_1$ and $M_1$ be fixed positive numbers satisfying $d < d_1 < M_1 < M$ . Let $\Omega_j$ be as in Lemma 5 for $j \ge n_0$ . Then there exists, independently of j, a maximum number N of $\alpha$ sides of $\partial \Omega_j$ that intersect the closed annulus $L(d_1, M_1)$ . Proof. We first consider those $\alpha$ sides of $\partial \Omega_i$ (j fixed) that have their endpoints on $\partial K_M$ . If an $\alpha$ chord of $\partial K_M$ intersects L(d, M), this chord subtends a minor arc of $\partial K_M$ of arc length at least $t_1$ . $t_1$ may be taken to be the arc length of the minor arc subtended by an $\alpha$ chord of $\partial K_M$ which is tangent to $\partial K_M$ . Again this statement is proved using (2.3b) and properties of convex domains. Choose an integer $l_1$ such that $l_1t_1 > 2\pi M$ . Then, independently of j, no more that $l_1$ sides of $\partial \Omega_j$ which are $\alpha$ chords of $\partial K_M$ , can intersect $L(d_1, M_1)$ . Suppose next that $\beta$ is an $\alpha$ side tangent to $\partial K_d$ which intersects $L(d_1, M_1)$ . Let P be a point of $\beta \cap L(d_1, M_1)$ and let $P_1$ be the radial projection of P on $\partial K_d$ . Let $P_2$ be the point where $\beta$ is tangent to $\partial K_d$ . Then the length of the minor arc of $\partial K_d$ with endpoints $P_1, P_2$ , has length at least $t_2$ , where $t_2$ depends only on d, $d_1$ , and $\alpha$ . Since $\Omega_i$ is starlike, two arcs of $\partial K_d$ , obtained from two different $\alpha$ sides tangent to $\partial K_d$ in the above way, cannot overlap. Hence if $l_2$ is a positive integer satisfying $l_2t_2 \ge 2\pi d$ , then $\partial \Omega_i$ has at most $l_2$ , $\alpha$ sides intersecting $L(d_1, M_1)$ which are tangent to $\partial K_d$ . Using Lemma 5 we conclude that $\partial \Omega_i$ has at most $N = l_1 + l_2 + 2$ sides which intersect $L(d_1, M_1)$ . Next we use Lemma 6 to characterize $\Omega = H(K)$ . We shall need some notation. Given $\theta$ , $0 < \theta \le 2\pi$ , and $j \ge n_0$ , let $w_j(\theta)$ denote the unique point of intersection of $\partial \Omega_j$ with the ray from w = 0 which has direction $e^{i\theta}$ . The uniqueness of $w_j(\theta)$ is guaranteed by Lemma 2. $w(\theta)$ is defined relative to $\partial \Omega$ in a similar way. For given $\epsilon > 0$ and $\theta$ , $0 < \theta \le 2\pi$ , we claim there exists a positive integer $n_1 = n_1(\epsilon, \theta) \ge n_0$ such that $$(7.5) |w_j(\theta) - w(\theta)| < \epsilon \text{for } j \ge n_1.$$ This claim is a direct consequence of the fact that $\Omega_j \to \Omega$ as $j \to \infty$ in the sense of kernel convergence. We use (7.5) and Lemma 6 to prove.

Lemma 7. Let $d_1$ and $M_1$ be as in Lemma 6. Let $\Omega = H(K)$ . Then $\partial \Omega \cap \{w : d_1 < |w| < M_1\}$ consists of a finite number of $\alpha$ arcs. Proof. Suppose $x_i$ , $1 \le i \le N+2$ , are N+2 points of $\partial \Omega$ with $0 \le \operatorname{Arg}(x_i \bar{x}_1) < \operatorname{Arg}(x_{i+1} \bar{x}_1) < \pi \alpha$ , $1 \le i \le N+1$ , and $d_1 < |x_i| < M_1$ , $1 \le i \le N+2$ . Let V be a sector, whose boundary consists of two rays drawn from w=0, which contains each $x_i$ , $1 \le i \le N+2$ , in its interior. We may assume V has angle opening less than $\pi \alpha$ . Draw the $\alpha$ arcs $\beta_i$ , $1 \le i \le N+1$ , which are contained in V, and have endpoints $x_i$ , $x_{i+1}$ . Let $\phi_i$ denote the smallest angle between the tangents of $\beta_i$ and $\beta_{i+1}$ at $x_{i+1}$ for $1 \le i \le N$ . Then if Lemma 7 is false we clearly can choose $x_i$ , $1 \le i \le N+2$ , as above, and such that $$(7.6) 0 < \phi_i < \pi, \quad 1 \le i \le N.$$ Furthermore, from (7.5) we can choose, for arbitrarily small $\epsilon > 0$ and j large enough, N+2 points of $\partial \Omega_j$ , say $y_1, y_2, \dots, y_{N+2}$ , so that $$|x_i - y_i| < \epsilon$$ , $1 \le i \le N + 2$ . However if $\epsilon$ is small enough this inequality and (7.6) imply that $\partial \Omega_i$ has N+1, $\alpha$ sides which intersect $L(d_1, M_1)$ . We have reached a contradiction to Lemma 6. Hence Lemma 7 is true. 8. Proof of Theorem 1. Finally we prove.

Lemma 8. For some real $\theta$ , $\Omega = e^{i\theta}F(K)$ , where $F = F(\cdot, \alpha, d, M)$ is as in (i)-(iv).

Definition 1. Let $\alpha$ be given, $0 < \alpha < \infty$ . Then $\gamma$ is said to be an $\alpha$ curve in the w plane, if there exists a line in the $\zeta$ plane, not containing $\zeta = 0$ , which is mapped onto $\gamma$ by a continuous $\alpha$ power of $\zeta$ . Second, we let $\partial E$ denote the boundary of a set E, and $K_r = \{z: |z| < r\}$ , $0 < r < \infty$ , $r \ne 1$ . Third, we let $\delta(M, \alpha)$ denote the radius of the largest disk with center at the origin contained in f(K) for all $f \in S(\alpha, 0, M)$ . Here $\alpha$ and M are fixed numbers satisfying $0 < \alpha < \infty$ and $1 < M \le \infty$ . It is easily seen that $S(\alpha, d, M) = S(\alpha, \delta(M, \alpha), M)$ for $0 \le d \le \delta(M, \alpha)$ . Hence in describing F we assume for given $\alpha$ and M as above that $\delta(M, \alpha) \le d < 1$ . For such values of $\alpha$ , d, and d we now describe $\partial F(K)$ . If $d = \delta(M, \alpha)$ , then $\partial F(K)$ contains - (i) An arc with endpoints $C, \bar{C}$ , of the $\alpha$ curve tangent to $\partial K_d$ at -d. - If $\delta(M, \alpha) < d < 1$ , then $\partial F(K)$ contains - (ii) An arc of $\partial K_d$ through -d with endpoints $A, \bar{A}$ , - (iii) Two arcs with endpoints A, C, and $\bar{A}, \bar{C}$ , of the two $\alpha$ curves tangent to $\partial K_d$ at A and $\bar{A}$ respectively. - Either (a) $0 < C = \overline{C} \le M$ or (b) $C \ne \overline{C}$ , $M < \infty$ , and |C| = M. If (a) occurs, then $\partial F(K)$ is the arc in (i) for $d = \delta(M, \alpha)$ , and the union of the arcs in (ii) and (iii) for $\delta(M, \alpha) < d < 1$ . If (b) occurs, then $\partial F(K)$ contains - (iv) The arc of $\partial K_M$ through M with endpoints $C, \bar{C}$ . $\partial F(K)$ is now the union of the arcs in (i) and (iv) for $d = \delta(M, \alpha)$ , and the union of the arcs in (ii)-(iv) for $\delta(M, \alpha) < d < 1$ . This completes the description of $\partial F(K)$ . The function F is uniquely determined by the above description of $\partial F(K)$ and the requirement that $F \in S(\alpha, d, M)$ , as we show in §3. We remark that Theorem 1 is well known in the simple case $\alpha = 1$ , $M = \infty$ , $d = \frac{1}{2}$ . In this case it is a simple consequence of the fact that a normalized convex function is starlike of order $\frac{1}{2}$ (see Suffridge [15] for a proof of this fact). However, in all other cases Theorem 1 is new. The subordination result in (B) implies the following corollary (see for example Golusin [4, Ch.8, §8]).

Definition 2. Let $\beta$ denote a closed arc of an $\alpha$ curve $\gamma$ . Then we shall call $\beta$ an $\alpha$ arc of $\gamma$ , if each ray through w=0 intersects $\beta$ in at most one point. We shall determine the number of $\alpha$ arcs with endpoints $A, B(A \neq B)$ in the w plane. Clearly the number is zero if either $Arg(\bar{A}B) = 0$ , or one of A and B is zero. Hence we assume $A \neq 0$ , $B \neq 0$ , and $Arg(\bar{A}B) \neq 0$ . Next we draw the rays from w = 0 through A and B. These rays divide the w plane into two sectors, $T_1$ and $T_2$ , with angular openings $\theta_1$ and $\theta_2$ respectively. We may suppose that $0 < \theta_1 \leq \theta_2 < 2\pi$ , since otherwise we renumber. We observe that if $\beta$ is an $\alpha$ arc with endpoints A and B, then either $\beta \subset T_1 \cup \{A, B\}$ , or $\beta \subset T_2 \cup \{A, B\}$ , as follows from Definition 2. We claim for fixed $\alpha, 0 < \alpha < \infty$ , that (2.1a) Let i be fixed, i = 1 or 2. Then if $0 < \theta_i < \pi \alpha$ , there exists exactly one $\alpha$ arc $\beta$ with endpoints A and B for which $\beta \subset T_i \cup \{A, B\}$ . If $\pi \alpha \leq \theta_i$ , there does not exist an $\alpha$ arc $\beta$ with endpoints A and B for which $\beta \subset T_i \cup \{A, B\}$ . To prove (2.1a), let $h_i$ denote an analytic $1/\alpha$ power of w in $T_i$ (i = 1 or 2) which is continuous on $\partial T_i$ . Then the line segment with endpoints $h_i(A)$ and $h_i(B)$ is contained in $h_i(T_i) \cup \{h_i(A), h_i(B)\}$ , if and only if $0 < \theta_i < \pi \alpha$ . Using this fact and considering the inverse mapping to $h_i$ , we get (2.1a). From (2.1a) we see for $0 < \alpha \le 1$ that if $0 < |\operatorname{Arg}(\bar{A}B)| < \pi\alpha$ , then there exists exactly one $\alpha$ arc with endpoints A, B. For $1 < \alpha < \infty$ , it follows from (2.1a) that there is at least one $\alpha$ arc with endpoints $A, B(\operatorname{Arg}(\bar{A}B) \ne 0)$ . Also for $2 < \alpha < \infty$ , there are exactly two $\alpha$ arcs with endpoints $A, B(\operatorname{Arg}(\bar{A}B) \ne 0)$ . Next we determine a geometric criterion for a bounded domain to be a magnification of the image domain of an $\alpha$ starlike function. This criterion is given by Lemma 1. In Lemma 1, $\beta$ denotes the $\alpha$ arc with endpoints A, B, satisfying $\beta \subset T_1 \cup \{A, B\}$ .