Ukrainian Mathematical Journal Vol. 48, No. 7, 1996 ON THE RADII OF CONVEXITY AND STARLIKENESS FOR SOME SPECIAL CLASSES OF ANALYTIC FUNCTIONS IN A DISK I. M. Kotsur and M. F. Kotsur UDC 517.54 . :(n-l) We introduce the class Oa, O_<c~_< i, of functions w=f(z),f(O)=O, f'(O)=O .... J(0) =0, f(n)(O) =(n - I ) ! analytic in the disk Izl < i and satisfying the condition + Z2n ) 1 - 2z ~ cos| Re f'(z) > co, 0<| n=1,2,3 ..... zn-I We establish the radius of convexity in the class Oa and the radius of starlikeness in the class Uc~ of functions c0(z) = zf" ( z), f( z) C 0 a. rt By za, 0 _ c~ < 1, n = 1, 2, 3 ..... we denote the class of functions ,,(n-l) :,(n) = (n-- 1)[, w = g(z), g(O) = O, g'(O) = O, g"(O) = 0 ..... 6(0) = O, 6(0) regular in the disk E = { z I z ] < 1 } and satisfying the condition Re( 1-2znc~174 ) zn_l g'(z) > c~, 0 <- 6) < ~z. (1) Since the function Z q~n(Z) = f (l-2~ncos| 0 II belongs to the class of convex functions, functions of the class Zc~ form a subclass of almost convex functions [1] and, hence, they are schlicht for n = 1. For n = 1, the function gt(z) = g(iz)/i satisfies the inequality Re[(1-2izcos| > co, whence, for O = n / 2, we obtain the subclass Qc~ of almost convex functions w = W(z), W (0) = 0, W'(0) = 1, defined by the condition Re [( 1 -z2)W'(z)] > c~. Note that Qc~ c Qo and the class Qo coincides with the class of functions convex along the direction of the imaginary axis [2]. Consider the class Uc~ of functions w= F(z), F(0)= 0, F'(0)= 1, regular in E and such that F(z)= zgt'(z), where W(z) ~ Qc~- Note that U a c U 0 and U 0 coincides with the class of typically real functions whose radius of starlikeness was found by Libera by the Robertson method [3, 4]. Zaporozh'e Technical University, Zaporozh'e. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 7, pp. 954-957, July, 1996. Original article submitted January 3, 1995. 0041-5995/96/4807-1079 $15.00 9 1997 Plenum Publishing Corporation 1079