UNIVALENT «-SPIRAL FUNCTIONS

THEOREM 1. If f <E Xa,P and oo (3.1) /(*) =z + ^anzn, s € A , w=2 https://doi.org/10.4153/CJM-1967-038-0 Published online by Cambridge University Press

COROLLARY 1. If f Ç Ta and has the representation (3.1), then (3.i4) K i < n | 2 g " t e . c ; ? + i l , « = 2 , 3 , . . . . Taking a = 0, we obtain a theorem of Robertson (11) which was also obtained recently by Schild (14).

COROLLARY 2. If f is starlike of order p and has (3.1) as its Maclaurin series, then n c* - 2P) (3.15) k | < k ^ _ , * = 2,3, . . . . 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 func- tions in ^p.

THEOREM 2. If P{z) = 1+£&«*£ ^ then \pk\<2(l-p), * = 1 , 2 , . . . ; Pp, (2.9), renders these bounds sharp. For p = 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 ©*, (9), or lor ©p* (11), does not yield sharp estimates for the functions

THEOREM 3. 0-s.r. Xa,P is the smallest positive root r of the equation (4.7) [2(1 - p) cos(0 - a) -cos a - cos 0]r2 — 2(1 — p) cos a • r + cos 0 = 0. Fa,p, defined in (3.13), shows this result is sharp. By fixing the parameters a, 0, and p in the theorem we obtain some inter- esting special cases.

COROLLARY 3. The radius of starlikeness of Ta is (4.8) 0-s.r. Za = l/(cos a + |sin a\). The last result was obtained recently by M. S. Robertson (13).

COROLLARY 4. /3-s.r. ©p* is the smallest positive root r of (4.11) cos 0- (1 - 2p)r2 - 2(1 - p)r + cos p = 0. When p = | , (4.11) is linear; since $, the class of convex functions in ©, is contained in @i*, (7), we may state the following interesting conclusion.$

COROLLARY 5. 0-s.r. $ = cos 0. https://doi.org/10.4153/CJM-1967-038-0 Published online by Cambridge University Press$