Let $p$ be a monic complex polynomial of degree $n$, and let $K$ be a measurable subset of the complex plane. Then the area of $p(K)$, counted with multiplicity, is at least $\pi n ( \Area (K)/\pi )^n$, and the area of the pre-image of $K$ under $p$ is at most $\pi^{1-1/n} (\Area (K) )^{1/n}$. Both bounds are sharp. The special case of the pre-image result in which $K$ is a disc is a classical result, due to Pólya. The proof is based on Carleman's classical isoperimetric inequality for plane condensers.

This paper addresses a number of questions concerning the size of factors of polynomials. Let p be a monic algebraic polynomial of degree n and suppose q1q2 … qi is a monic factor of p of degree m. Then we can, in many cases, exactly determine

Here ‖ . ‖ is the supremum norm either on [—1, 1] or on {|z| ≤ 1}. We do this by showing that, in the interval case, for each m and n, the n-th Chebyshev polynomial is extremal. This extends work of Gel'fond, Mahler, Granville, Boyd and others. A number of variants of this problem are also considered.

Let be the class of all polynomials/? of degree at most n such that In view of the example zn it follows from Bernstein's inequality for polynomials that at each point z0 of the unitcirle. It was shown by A. Giroux and Q. I. Rahman [2] that if denotes the subclass of polynomials in which vanish at 1, then

where c1 and c2 are constants not depending on n. Here we find the exact value of which has some special significance and also at certain other points of the unit circle.