On successive coefficients of univalent functions

Abstract:

Let $f(z) \in S$, that is, $f(z)$ is analytic and univalent in the unit disk $\left | z \right | < 1$, normalized by $f(0) = f’(0) - 1 = 0$. Let $p$ be real and \[ {\{ f(z)/z\} ^p} = 1 + \sum \limits _{n = 1}^\infty {{D_n}(p){z^n}} .\] Lucas proved that \[ \left | {{D_n}(p)} \right | - \left | {{D_{n + 1}}(p)} \right |\left | { \leq A{n^{(t(p) - 1)/2}}{{\log }^{3/2}}n,\quad n = 2,3, \ldots ,} \right .\] for some absolute constant $A$ and $t(p) = {(2\sqrt p - 1)^2}$. In this paper we improve $t(p)$ as follows: \[ T(p) = \frac {{4p - 1}}{{2p + t(p)}}t(p).\]