On polynomials with constraints on circular arcs

For polynomials with prescribed minimal and maximal values of their moduli on a collection of circular arcs it is shown that new covering and distortion theorems and a modulus estimates for a product of leading and free coefficients follow from a majorization principle for meromorphic functions proved by the authors earlier. As corollaries, recent results on polynomials with additional constraints on zeros established by other mathematicians are obtained.