Estimates for the coefficients of univalent functions in terms of the second coefficient

L. P. Il'ina

Matematičeskie zametki, Tome 13 (1973) no. 3, pp. 351-357.

Résumé

For the coefficients $b_n$ of an odd function $f(z)=z+\sum^\infty_{k=1}b_kz^{2k+1}$, regular in the unit disk, we obtain the estimate \begin{equation} |b_n|\le\frac1{\sqrt2}\sqrt{1+|b_1|^2}\exp\frac12(\delta+\frac12|b_1|^2), \quad\text{where}\;\delta=0,312, \tag{1} \end{equation} from which it follows that $|b_n|\le1$, if $|b_1|\le0,524$. It follows from (1) that the coefficients $c_n, n=3, 4\ldots$ of a regular function $f(z)=z+\sum^\infty_{k=2}c_kz^k$, univalent in the unit desk, satisfy \begin{equation} |b_n|\le\frac1{\sqrt2}\sqrt{1+|b_1|^2}\exp\frac12(\delta+\frac12|b_1|^2), \quad\text{where}\;\delta=0,312, \tag{2} \end{equation} in particular, $|c_n|\le n$, if $|c_2|\le1,046$.