Regions without complex zeros for chromatic polynomials on graphs with bounded degree

We prove that the chromatic polynomial $P_\mathbb{G}(q)$ of a finite graph $\mathbb{G}$ of maximal degree $\D$ is free of zeros for $\card q\ge C^*(\D)$ with $$C^*(\D) = \min_{0<x<2^{1\over \D}-1} {(1+x)^{\D-1}\over x [2-(1+x)^\D]}$$ This improves results by Sokal (2001) and Borgs (2005). Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.