A Note on Successive Coefficients of Spirallike Functions

Abstract

Even there were several facts to show that $||a_{n+1}(f)|-|a_{n}(f)||\leq 1$ is not true for the whole class of normalised univalent functions in the unit disk with with the form $f(z)=z+\sum_{k=2}^{\infty}a_{k}z^k$. In 1978, Leung\cite{Leung} proved $||a_{n+1}(f)|-|a_{n}(f)||$ is actually bounded by 1 for starlike functions and by this result it is easy to get the conclusion $|a_n|\leq n$ for starlike functions. Since $||a_{n+1}(f)|-|a_{n}(f)||\leq 1$ implies the Bieberbach conjecture (now the de Brange theorem), so it is still interesting to investigate the bound of $||a_{n+1}(f)|-|a_{n}(f)||$ for the class of spirallike functions as this class of functions is closely related to starlike functions.
In this article we
%deal with the functional $||a_{n+1}(f)|-|a_{n}(f)||$ for the class of spirallike functions with the form $f(z)=z+\sum_{k=2}^{\infty}a_{k}z^k$. By using Leung's method \cite{Leung} 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 $||a_{n+1}(f)|-|a_{n}(f)||$ for non-starlike spirallike functions. By using the Carath\'eodory-Toeplitz theorem, we obtain the sharp lower and upper bounds of $|a_{n+1}(f)|-|a_{n}(f)|$ for $n=1$ and $n=2$. These results disprove the expected inequality $||a_{n+1}|-|a_{n}||\leq\cos\alpha$ for $\alpha$-spirallike functions.