An extremal problem for polynomials

Abstract

For the polynomials F ( z ) = ∑ j = 1 N a j z j with real coefficients and normalization a 1 = 1 we solve the extremal problem sup a 2 , … , a N ⁡ ( inf z ∈ D ⁡ { Re ( F ( z ) ) : Im ( F ( z ) ) = 0 } ) . We show that the solution is − 1 4 sec 2 ⁡ π N + 2 , and the extremal polynomial 1 U N ′ ( cos ⁡ π N + 2 ) ∑ j = 1 N U N − j + 1 ′ ( cos ⁡ π N + 2 ) U j − 1 ( cos ⁡ π N + 2 ) z j is unique and univalent, where the U j ( x ) are the Chebyshev polynomials of the second kind, j = 1 , … , N . As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.