On the coefficients of functions with bounded boundary rotation

Abstract

Let Vk be the class of normalised functions of bounded boundary rotation.For/6 Vt define M(r,f) = max |/(z)|, |s|_r and let L(r,f) denote the length of/(|z|=r).Then if/(z)=z+ 2n=2anz", it is shown that (i) 2M(r,f)<L(r,f)^knM(r,f), and(ii) n-\at\-^{T¡k\rn-'í)M(r,f'), n^.2.The class At of meromorphic functions of boundary rotation is also studied and estimates for the coefficients are given.(1) ReW dd < ÍC7T. Jo f'(z)V2 is the class of normalised convex functions, and it is well known that for 2^k^4, Vk contains only univalent functions.Suppose/e Vk and has Taylor expansion (2) f(z) = z + 2 «^"-fl=2 Then the problem An(k)=ma\ \a"\ has been extensively studied, but remains largely unsolved.It is known that, for k^.2, (3) A2(k) = kj2, A3(k) = (k2 + 2)/6, At(k) = (k* + 8*)/24, and, for n^.2, that (4) |c.| ^ c(k)nm~\

Locations

• Proceedings of the American Mathematical Society - View - PDF