A note on successive coefficients of spirallike functions

Ming Li

Even there were several facts to show that ||an+1( f )| − |an( f )|| ≤ 1 is not true for the whole class of normalised univalent functions in the unit disk with with the form f (z) = z + ∑∞ k=2 akzk. In 1978, Leung[7] proved ||an+1( f )| − |an( f )|| is actually bounded by 1 for starlike functions and by this result it is easy to get the conclusion |an| ≤ n for starlike functions. Since ||an+1( f )| − |an( f )|| ≤ 1 implies the Bieberbach conjecture (now the de Brange theorem), so it is still interesting to investigate the bound of ||an+1( f )| − |an( f )|| for the class of spirallike functions as this class of functions is closely related to starlike functions. In this article we prove that this functional is bounded by 1 and equality occurs only for the starlike case. We are also able to give a precise form of extremal functions. Furthermore we also try to find the sharp bound of ||an+1( f )| − |an( f )|| for non-starlike spirallike functions. By using the Carathe´odory-Toeplitz theorem, we obtain the sharp lower and upper bounds of |an+1( f )| − |an( f )| for n = 1 and n = 2. These results disprove the expected inequality ||an+1( f )| − |an( f )|| ≤ cosα for α-spirallike functions