Remarks on results by Müger and Tuset on the moments of polynomials

Let f(x) be a non-zero polynomial with complex coefficients, and M_p=∫₀¹f(x)^pdx for p a positive integer. In a recent paper, Müger and Tuset showed that lim sup_p→∞|M_p|^1∕p>0, and conjectured that this limit is equal to the maximum amongst the critical values of f together with the values |f(0)| and |f(1)|. We give an example that shows that this conjecture is false. It also may be natural to guess that lim sup_p→∞|M_p|^1∕p is equal to the maximum of |f(x)| on [0,1]. However, we give a counterexample to this as well. We also provide a few more guesses as to the behavior of the quantity lim sup_p→∞|M_p|^1∕p.