Integral Estimates for the Derivatives of Univalent Functions

In this paper we prove Brennans's conjecture for the conformal mapping of the unit circle on the assumption that even the Taylor coefficients of the function $\ln f'$ satisfies a certain condition. We also prove Brennan's conjecture for the case when there is an expansion of the function $1/f'$ into a series of simple fractions, provided that this series converges absolutely to zero. In addition, we obtain an estimate for the approximation of the function $1/f'$ by simple fractions.