In the paper the following result is proved: Theorem. Let p(z) be a monic polynomial of degree N all of whose roots lie on the unit circle, and E be any subset of \([-\pi,\pi]\) of measure \(2\alpha\), \(0<\alpha \leq \pi\). Then \[ \int_{E}| p(e^{i\theta})| d\theta \geq 8(\sin \alpha /2)^{N-1}(1-\cos \alpha /2), \] moreover the lower bound is best possible. The proof of this result depends on the following: Lemma. Let \(\psi: [0,2^ N]\to {\mathbb{R}}^+\) be a continuous and nondecreasing function, and \[ J=J(\alpha,N,\psi)=\inf \int_{E}\psi (| p(e^{i\theta})|)d\theta, \] where the infimum is taken over the polynomials p and sets E specified in the theorem. Then among the extremal configurations there is one in which the closure of E is connected. Two interesting corollaries are also given.