A well-known conjecture
states that for polynomials having all their zeros on the unit circle C half the
maximum modulus on C bounds the modulus of all the coefficients. This has been
established in all cases except for the middle coefficient of even degree polynomials
greater than four. In this note this conjecture is verified for all even degree
polynomials having simple zeros in a set of arcs dividing the circle into equal
parts and related classes of polynomials. The local extremal polynomials are
identified.