A class of univalent functions

A sharp coefficient estimate is obtained for a class D(cx) of functions univalent in the open unit disc. The radius of convexity and an arclength result are also determined for the class. Let D(c) denote the class of functions f (z) =z+a2z2+ * analytic in the open unit disc E and satisfying (1) f(f'(z) 1)/(f'(z) + 1)1 < a, z cE, for some a, O<o? < 1. The valuesf'(z) lie inside the circle in the right half plane with center (1 +O2)/(l -oC2) and radius 2c/(1 -oC2). The class D(oa) is a subclass of the class of functions whose derivative has positive real part and hence a function in D(oc) is univalent in E. Iff eD(oc) it follows from Schwarz lemma thatf'(z)=(1-ocz0(z))/(l+ocz0(z)), where 0(z) is analytic and I 0(z)f ? 1 in E. A class of starlike functions has been studied by Padmanabhan [5] in whichf'(z) is replaced by zf'(z)/f (z) in inequality (1). A sharp coefficient estimate for the class D(z) is proved in Theorem 1 using a technique of Clunie and Keogh [2]. In Theorem 2 the radius of convexity of the class is obtained and in Theorem 3 an arclength result is given. THEOREM 1. Iff(z)=z+2n'=2 aaz?l is in D(zc) for some cc, O<c<1, then Ia,,I 2c/n, n=2, 3, * . The inequality is sharp. PROOF. Since f(z) is in D(z), then f'(z)=(1+ccz0(z))/(1-ccz0(z)), where 0(z)= En-l trZn is analytic and I 0(z)f < 1 for z E E. Then f'(z) 1 = cz0(z){f '(z) + 1}, or 00 00 00 (2) Znanzn1 = cc ( tnZ 2z + Enanzn n=2 n= O n=2 Equating corresponding coefficients in (2) gives nan = cc{(n 1)toan-1 + (n 2)tlan-2 + * * * + 2tn-3a2 + 2tn-2}. Presented to the Society, January 26, 1973; received by the editors September 5, 1972 and, in revised form, September 25, 1972. AMS (MOS) subject classifications (1970). Primary 30A32, 30A34.