UNIVALENT «-SPIRAL FUNCTIONS

Theorem 1. If $f \in \mathfrak{T}_{\alpha,\rho}$ and (3.1) $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n, \quad z \in \Delta,$$ then (3.2) $$|a_n| \leqslant \prod_{k=0}^{n-2} \frac{|2(1-\rho)\cos\alpha \cdot e^{-i\alpha} + k|}{k+1}, \qquad n = 2, 3, \dots,$$ and these bounds are sharp for all admissible $\alpha$ and $\rho$ and for each n.

Corollary 1. If $f \in \mathfrak{T}_{\alpha}$ and has the representation (3.1), then (3.14) $$|a_n| \leqslant \prod_{j=0}^{n-2} \frac{|2e^{-i\alpha}\cos\alpha + j|}{j+1}, \quad n = 2, 3, \dots.$$ Taking $\alpha = 0$ , we obtain a theorem of Robertson (11) which was also obtained recently by Schild (14).

Corollary 2. If f is starlike of order $\rho$ and has (3.1) as its Maclaurin series, then (3.15) $$|a_n| \leqslant \frac{\prod_{k=2}^n (k-2\rho)}{(n-1)!}, \qquad n=2,3,\dots.$$ The technique of using (3.3) and the integration in (3.7) is due to Clunie (1). The same method applied to (3.3) yields coefficient estimates for functions in $\mathfrak{P}_{\rho}$ .

Theorem 2. If $$P(z) = 1 + \sum_{k=1}^{\infty} p_k z^k \in \mathfrak{P}_{\rho},$$ then $$|p_k| \leqslant 2(1-\rho), \qquad k = 1, 2, \ldots;$$ $P_{\rho}$ , (2.9), renders these bounds sharp. For $\rho = 0$ we obtain the classical theorem of Carathéodory (12), and a proof of it by Clunie's method is given elsewhere by the author (6, Lemma 3.2). It is easily adapted to give a proof of Theorem 2. It is interesting to note that the method of equating coefficients in (2.6) and using Theorem 2, as can be done for $\mathfrak{S}$ , (9), or for $\mathfrak{S}_{\rho}$ (11), does not yield sharp estimates for the functions of Theorem 1 or Corollary 1. 4. The $\mathfrak{g}$ -spiral radius. If $f \in \mathfrak{T}_{\alpha,\rho}$ , then by making use of Definitions 3 and 5 and (2.6) we see that the $\beta$ -spiral radius of f is the largest number r such that for some P in $\mathfrak{P}_{\rho}$ and |z| < r (4.1) $$\operatorname{Re}\left\{e^{i(\beta-\alpha)}\left[\cos\alpha \cdot P(z) + i\sin\alpha\right]\right\} \geqslant 0.$$ It is well known and indeed follows from (2.8) and (3.3) that every member of $\mathfrak{P}$ is subordinate to $P_0$ (9); therefore every function in $\mathfrak{P}_{\rho}$ is subordinate to $P_{\rho}$ . Consequently, if (4.1) holds for $P_{\rho}$ , then it holds, a fortiori, for all P in $\mathfrak{P}_{\rho}$ . Let (4.2) $$B(z) = e^{i(\beta - \alpha)} [\cos \alpha \cdot P_{\rho}(z) + i \sin \alpha]$$ $$= e^{i(\beta - \alpha)} \left\{ \cos \alpha \cdot \left[ \frac{1 + (1 - 2\rho)z}{1 - z} \right] + i \sin \alpha \right\}.$$ B(z) maps the disk $|z| \le r$ onto a disk whose centre is (4.3) $$e^{i(\beta-\alpha)} \left\{ \cos \alpha \cdot \left[ \frac{1 + (1 - 2\rho)r^2}{1 - r^2} \right] + i \sin \alpha \right\}$$ and whose radius is (4.4) $$\left| e^{i(\beta-\alpha)} \cos \alpha \cdot \frac{2(1-\rho)r}{1-r^2} \right|.$$ Hence, $Re\{B(z)\} > 0$ if and only if $$(4.5) \quad \operatorname{Re}\left\{e^{i(\beta-\alpha)}\left[\cos\alpha \cdot \frac{1+(1-2\rho)r^2}{1-r^2} + \mathrm{i}\sin\alpha\right]\right\} \geqslant \cos\alpha \cdot \frac{2(1-\rho)r}{1-r^2},$$ or $$(4.6) \qquad \cos\beta \cdot (1-r^2) + 2(1-\rho)r^2\cos(\beta-\alpha) \cdot \cos\alpha \geqslant \cos\alpha \cdot 2(1-\rho)r.$$ These results are summarized below:

Theorem 3. $\beta$ -s.r. $\mathfrak{T}_{\alpha,\rho}$ is the smallest positive root r of the equation (4.7) $$[2(1-\rho)\cos(\beta-\alpha)\cdot\cos\alpha-\cos\beta]r^2 - 2(1-\rho)\cos\alpha\cdot r + \cos\beta = 0.$$ $F_{\alpha,\rho}$ , defined in (3.13), shows this result is sharp. By fixing the parameters $\alpha$ , $\beta$ , and $\rho$ in the theorem we obtain some interesting special cases.

Corollary 3. The radius of starlikeness of $\mathfrak{T}_{\alpha}$ is (4.8) $$0-s.r. \mathfrak{T}_{\alpha} = 1/(\cos \alpha + |\sin \alpha|).$$ The last result was obtained recently by M. S. Robertson (13).

Corollary 4. $\beta$ -s.r. $\mathfrak{S}_{\rho}^*$ is the smallest positive root r of (4.11) $$\cos \beta \cdot (1 - 2\rho)r^2 - 2(1 - \rho)r + \cos \beta = 0.$$ When $\rho = \frac{1}{2}$ , (4.11) is linear; since $\Re$ , the class of convex functions in $\mathfrak{S}$ , is contained in $\mathfrak{S}_{\frac{1}{2}}^*$ , (7), we may state the following interesting conclusion.

Corollary 5. $\beta$ -s.r. $\Re = \cos \beta$ .

Definition 1. f is regular in $\Delta$ and satisfies (2.1); f is an $\alpha$ -spiral function, $f \in \mathfrak{T}_{\alpha}$ , if and only if there is a real number $\alpha$ such that (2.2) $$\operatorname{Re}\left[e^{i\alpha}\left\{zf'(z)/f(z)\right\}\right] > 0, \quad z \in \Delta.$$ Špaček's results, described above, show that $\mathfrak{T}_{\alpha} \subset \mathfrak{S}$ . It is clear that $-\pi/2 \leqslant \alpha \leqslant \pi/2$ and that $|\alpha| = \pi/2$ only for the identity function f(z) = z; hence we exclude that case from further consideration. Received October 26, 1965. This work was supported by the National Science Foundation Grant NSF-GP-4314. A geometric interpretation of (2.2) may be given as follows: Let f satisfy (2.2) on the circle $z = re^{i\theta}$ , 0 < r < 1, $\theta$ real, and let C(r) be the image of this circle under f. Denote by $T(r, \theta)$ the tangent vector to C(r) at the point $f(re^{i\theta})$ with direction determined by that of increasing $\theta$ . Then, with proper choice of arguments, (2.2) is equivalent to $$(2.3) 0 < \arg[ie^{i\alpha} \{re^{i\theta}f'(re^{i\theta})/f(re^{i\theta})\}] < \pi$$ and, since $$arg\{ire^{i\theta} f'(re^{i\theta})\} = arg\{T(r, \theta)\}$$ (9, p. 223), we may rewrite (2.3) as (2.4) $$0 < \arg\{T(r,\theta)\} - [\arg\{f(re^{i\theta})\} - \alpha] < \pi.$$ Consequently, the vector $T(r, \theta)$ is directed into the open half-plane whose boundary has the equation (2.5) $$w = f(re^{i\theta}) + t \exp i[\arg\{f(re^{i\theta})\} - \alpha], \quad t \text{ real.}$$ This half-plane does not contain the origin in its interior when $\alpha > 0$ , but does contain the origin when $\alpha < 0$ . It is not difficult to show that the line given by (2.5) is parallel to the line tangent to the spiral $\rho = ke^{\phi \cot(-\alpha)}$ at the point $(\rho, \phi)$ . (The spiral and the point are given in polar coordinates; the direction of the radius vector is identified with the direction $\arg\{f(re^{i\theta})\}$ .) S. Ozaki (10) has discussed some of these ideas for the case when f is multivalent. Let $\mathfrak{P}$ represent the class of normalized holomorphic functions of positive real part, i.e., $P \in \mathfrak{P}$ if and only if P is holomorphic in $\Delta$ , $\text{Re}\{P(z)\} > 0$ for z in $\Delta$ and P(0) = 1. If $f \in \mathfrak{T}_{\alpha}$ , the introduction of appropriate normalizing factors enables us to write $$\sec \alpha [e^{i\alpha} \{zf'(z)/f(z)\} - i \sin \alpha]_{z=0} = 1.$$ This leads to useful representation formulas for members of $\mathfrak{T}_{\alpha}$ in terms of functions in $\mathfrak{P}$ . f is an $\alpha$ -spiral function if and only if there exists a function P in $\mathfrak{P}$ such that (2.6) $$\frac{zf'(z)}{f(z)} = \frac{\cos \alpha \cdot P(z) + i \sin \alpha}{\cos \alpha + i \sin \alpha}, \quad z \in \Delta,$$ or, equivalently, (2.7) $$f(z) = z \exp \left\{ \cos \alpha \cdot e^{-i\alpha} \int_0^z \frac{P(t) - 1}{t} dt \right\}, \quad z \in \Delta.$$ Many geometric and analytic properties of $\alpha$ -spiral functions can be obtained from (2.6), (2.7), and known properties of the class $\mathfrak{P}$ which has been studied extensively (12). These results can be sharpened considerably by a suitable refinement of $\mathfrak{P}$ into subclasses.

Definition 2. P is a function with positive real part of order $\rho$ , $P \in \mathfrak{P}_{\rho}$ , $0 \leq \rho \leq 1$ if and only if $P \in \mathfrak{P}$ and $Re\{P(z)\} \geqslant \rho$ for z in $\Delta$ . Evidently, $\mathfrak{P}_{\rho} \subset \mathfrak{P}_{\sigma}$ whenever $\rho \leqslant \sigma$ ; therefore $\mathfrak{P}_{\rho} \subset \mathfrak{P}_{0} = \mathfrak{P}$ for all $\rho$ . Furthermore, Q is in $\mathfrak{P}_{\rho}$ if $Q(z) = (1 - \rho)P(z) + \rho$ for some P in $\mathfrak{P}$ , and conversely. A particularly distinguished (6, 12) member of $\mathfrak{P}$ is the function $P_{0}$ defined by (2.8) $$P_0(z) = (1+z)/(1-z), \quad z \in \Delta.$$ For convenience we let $$(2.9) P_{\rho}(z) = (1-\rho)P_{0}(z) + \rho = \{1 + (1-2\rho)z\}/(1-z).$$

Definition 3. $f \in \mathfrak{T}_{\alpha,\rho}$ , or f is an $\alpha$ -spiral function of order $\rho$ , if and only if f is defined by (2.7) with $P \in \mathfrak{P}_{\rho}$ . For $\rho=0$ , we obtain the sets of functions of Definition 1 and we write $\mathfrak{T}_{\alpha,0}=\mathfrak{T}_{\alpha}$ . On the other hand when $\alpha=0$ we get the functions which are starlike of order $\rho$ with respect to the origin (6, 11, 14) and which are frequently denoted by $\mathfrak{S}_{\rho}$ , i.e., $\mathfrak{S}_{\rho}^=\mathfrak{T}_{0,\rho}$ . The class of all starlike functions is usually symbolized by $\mathfrak{S}^*$ ; in keeping with this we let $\mathfrak{T}$ represent all $\alpha$ -spiral functions, that is, $\mathfrak{T}=\bigcup_{\alpha,\rho}\mathfrak{T}_{\alpha,\rho}$ , $0\leqslant\rho\leqslant1$ , $-\frac{1}{2}\pi\leqslant\alpha\leqslant\frac{1}{2}\pi$ . The concepts "radius of convexity" and "radius of starlikeness" for classes of univalent functions are useful and have attracted investigators (6, 9, 13). More recently Krzyź (4) obtained the radius of the largest disk centred at the origin whose map by every member of $\mathfrak{S}$ is close-to-convex (3). A similar question can be posed for the notion given in Definition 1.

Definition 4. If f is in $\mathfrak{S}$ and $-\pi/2 < \beta < \pi/2$ , then the $\beta$ -spiral radius of f is $$\beta\text{-s.r.}\{f\} = \sup \left[r : \operatorname{Re} \left\{ e^{i\beta} \frac{zf'(z)}{f(z)} \right\} > 0, \qquad |z| < r \right].$$

Definition 5. If $\mathfrak{A} \subset \mathfrak{S}$ and $-\pi/2 < \beta < \pi/2$ , then the $\beta$ -spiral radius of $\mathfrak{A}$ is $$\beta$$ -s.r. $\mathfrak{A} = \inf_{f \in \mathfrak{A}} [\beta$ -s.r. $\{f\}$ ]. The geometric interpretation of the $\beta$ -spiral radius is clear from the preceding discussion. It is evident that $\alpha$ -s.r. $\mathfrak{T}_{\alpha}=1$ and that the 0-s.r. $\mathfrak{A}$ is the radius of starlikeness of $\mathfrak{A}$ . By using the principle of subordination (6, 9, 11) we shall find $\beta$ -s.r. $\mathfrak{T}_{\alpha,\rho}$ for arbitrary $\alpha$ , $\beta$ , and $\rho$ , $-\pi/2 < \alpha$ , $\beta < \pi/2$ , and $0 \le \rho \le 1$ .