Coefficient Estimates for a Class of Star-Like Functions

THEOREM 2.1. Let oo /(*) = 2 + Z akz* belong to S*(a) (0 < a ^ 1). 77z£w |a2| ^ 2a, with equality if and only if /(*) Ll - ezj U€i X)- If 0 < a < -|, ^ew [«a] ^ a ÎW% equality if and only if if I < a ^ 1, //^n |a3| ^ 3a2 wi/& equality if and only if (1*1 = i); and if a = ^, then |a3| ^ |, w£& equality if and only if <"> 51? - MHH)

THEOREM 2.2. Let f(z) = z + 2*=2 akzk belong to 5*(a), 0 < a ^ 1, awd let n > 1 be a fixed integer. There exists a number /3n (0 < fin < 1) swc& / t o if j3w < a ^ 1, |an| = An(a) if and only if f(z) = efa(ez)y where fu{z) is defined by (2.8) and |e| = 1.

THEOREM 2.3 (Rogosinski [6, p. 70]). Let f(z) = a + ]£fc=i akzk be sub- ordinate to F{z) = a + Ylk=iAjcZk in U. If F(z) is univalent in U and F{U) is convex, then \an\ S 1-41|. If F(U) is not a half plane, then equality can hold for a given n only if f(z) = F(ezn) (|e| = 1). If P(z) e SPa (0 < a < 1), then P(z) is subordinate to ((1 + z)/(l - z))a. It follows from Theorem 2.3 that if CO P(z) = 1 + D ptz\ k=l then \pn\ ^ 2a. Moreover, \pn\ = 2a if and only if '« = fe)° «•!-')• We shall also need

THEOREM 2.4. For each integer n > 1, £Âer# aw/5 a number ynj 0 < yn < 1, 5^c/ï / t o if 0 < a < yn, then An(a) = 2a/ (n — 1). Moreover, if oo /(*) = 2 + £ %2* A;=2 is a function in S*(a) for which \an\ = 2a/(n — 1), then zf'jz) m where lei = 1. Zf'jz) _ f l + e/- 1!" Ll - 6 2 ^ ] '

THEOREM 3.1. Let 1 °° 2 £=0 belong to 2* (a) (0 < a ^ 1). Then for n ^ 1, (3.1) H.| ^ ^ wiift equality if and only if zF'jz) _ _ (l + tzn+1Y F(2) ~ \1 - 6S"+7 * where [el = 1. https://doi.org/10.4153/CJM-1970-055-8 Published online by Cambridge University Press