In this paper we study the conformal mappings of some symmetric simply connected domains in the complex plane whose boundary are fractal trees. In particular we present, based on the alternative simplified approach of Carleson and Makarov to the Brennan conjecture, see [CM], some evidence towards the truth of this conjecture.

The generalized polynomials such as Chebyshev polynomial and Hermite polynomial are widely used in interpolations and numerical fittings and so on. Therefore, it is significant to study inclusion regions of the zeros for generalized polynomials. In this paper, several new inclusion sets of zeros for Chebyshev polynomials are presented by applying Brauer theorem about the eigenvalues of the comrade matrix of Chebyshev polynomial and applying the properties of ovals of Cassini. Some examples are given to show that the new inclusion sets are tighter than those provided by Melman (2014) in some cases.