Coefficients of univalent functions which assume no pair of values $W$ and $-W$

In this paper we study the behavior of the coefficients of functions $\varphi(z)=1+\sum_{k=1}^\infty b_kz^k$, univalent in the disk $|z|<1$ and assuming there are no pair of values $W$ and $-W$. In particular, we establish the asymptotic behavior of $b_n$ ($n\to\infty$); for the coefficients we obtain the estimate $|b_n|<2,34\exp\{1/4n\}$ ($n=2,3,\dots$) and for each function of the class indicated we prove, subject to a certain condition, the relationship $||b_{n+1}|-|b_n||=O(n^{-1/2})$.