On Brennan's Conjecture for a Special Class of Functions

In this paper, we prove Brennan's conjecture for conformal mappings $f$ of the disk $\{z:|z|\lt 1\}$ assuming that the Taylor coefficients of the function $\log(zf'(z)/f(z))$ at zero are nonnegative. We also obtain inequalities for the integral means over the circle $|z|=r$ of the squared modulus of the function $zf'(z)/f(z)$.