We consider the class $S_{p}$ of functions \[ f(z) = z + a_{p+1}z^{p+1} +\cdots \] univalent in the unit disk $\triangle$. We show that, if $f\in S_{p}$ and $p$ is large, \[ \alpha (f) = \lim\limits_{n{\to}\infty}\frac{|a_{n}|}{n} \le \alpha (p) = \frac{2(\log p\log\log p)^2}{p^4}\]. We also obtain estimates for $|a_{n}|$ when $n\in S_p$.

In a companion paper [1] it will be shown that there exists $f$ in $S_{p}$ for $p=1,2,\ldots$ such that \[ \alpha (f) \ge \frac{C_{0}}{p^4},\] where $C_{0}$ is a positive absolute constant.