We consider conformal maps of the unit disk onto a bounded domain in the complex plane. The average growth of the derivative of such functions is described by the integral means spectrum. There are only partial results about finding sharp estimates. We use Fatou sets of quadratic polynomials as image domain to get an indication for better estimates. The results of a computer search lead to a conjecture abouth the shape of the universal integral means spectrum. This includes some previous and other contecture the Brennan conjecture and the Carleson Jones contecture concerning the coefficient problem.