Coefficients of functions with bounded boundary rotation

James W. Noonan
1971 Proceedings of the American Mathematical Society  
For k ê2 denote by Vk the class of normalized functions analytic in the unit disc which have boundary rotation at most kir. For fixed «e(¿+6)/4 we determine the maximum of the set of values of \a"\, where an is the nth Taylor coefficient of a function in Vk. For a fixed k 2:2 let Vk denote the class of functions (1) f(z) = 3 + a2z2 + a3z3 Hwhich are analytic in U= {z:\z\ <1{ and have an integral representation of the form (2) f'(z) = exp j-j log(l -ze-^-'d^tyl , where p(t) is real-valued and of
more » ... bounded variation on [0, 27r] with dß(t) = 2ir, | dß(t) | g kir. o ^o V. Paatero [7] showed that/(z) given by (1) belongs to Vk if and only if/'(2)^0 in U and/(z) maps U onto a domain with boundary rotation at most kir. (See [4] for a definition of this concept.) V2 is precisely the class of normalized univalent functions that map U onto a convex domain, and it is known [7] that for 2^&^4, Vk
doi:10.1090/s0002-9939-1971-0274738-5 fatcat:dtjd4ebc7rhxvkhowj4xaw23d4