Sheil-Small's (1974) covering problem has a positive answer, and other miscellanea

T. Sheil-Small asked whether every self-inversive polynomial maps the unit disc onto a domain that contains a disc whose radius is the largest modulus of the polynomial's coefficients (Problem 4.24 in Hayman and Lingham's problem book). We show that the answer is yes. More generally, every non-constant polynomial whose coefficients have the same moduli when read in reverse order maps the unit disc onto a domain that contains an open disc of this radius. This class contains both the self-inversive polynomials, whose zeros are symmetric with respect to the unit circle, and the polynomials whose zeros, with multiplicity, are symmetric under taking reciprocals (the literal reading of the problem as printed), so the theorem answers the question under either reading. The radius cannot be increased. For a polynomial with all its zeros on the unit circle, the largest disc in the image has radius exactly half the maximum modulus of the polynomial on the circle. For these polynomials the question is therefore equivalent to asking whether every coefficient has modulus at most half this maximum. O'Hara and Rodriguez and, independently, Saff and Sheil-Small proved this in 1974 for every coefficient except the middle one, and Kristiansen proved the middle case in 1979. The new case is that of polynomials with zeros off the unit circle. We treat this case with the Schur–Cohn Hermitian form of the polynomial minus a value that it does not take in the disc. This expanded version adds a description by rays of the image of a self-inversive polynomial, two examples, and the trigonometric form of the middle case. The results of Sections 2–5 are also proved in Lean 4, with the coefficient bound taken as given.