On Brennan's conjecture in conformal mapping

by Bertilsson, Daniel

Let f be a one-to-one analytic function in the unit disc with f 0(0) = 1. Brennan's conjecture states that for every ">0 Z 2 0 jf 0(rei )j;2 d = O ;(1 ; r);1;" We do some work on the following reformulations, which we prove are equivalent. Z 2 jf 0 0(rei )j;2 C d ? where C is an absolute constant. (1) 1 ; r We propose the stronger conjecture that (for xed r) the integral is maximized when f is the Koebe function z(1+z);2. To support this, we showthat the Koebe function is a local maximum in the sense that analytic variations of the omitted arc decrease the integral. If p 2, the MacLaurin coe cients of (f 0);p grow like O(np;1). (2) We show that if n 2p +1,then the nth coe cient is maximized when f is the Koebe function. The proof is similar to de Branges' proof of the Bieberbach conjecture. As a consequence we get sharp estimates for certain higher-order Schwarzian derivatives of f. These are used to show that (*) holds with " =0:547. Earlier it was known that one can take " =0:601. The Carleson-Makarov conjecture about -numbers: X 2 j 1: (3) We show the existence of extremal domains for the sum Pn p 1 j , and use the second variation to prove that the boundary of consists of trajectories of a quadratic di erential which has no multiple zeros on the boundary of . We also prove some estimates of extremal length, that give geometric criteria for a point to have positive -number. This is related to the angular derivative problem. (*) Key words and phrases: geometric function theory, conformal mappings, univalent functions, beta numbers, extremal length estimates, the angular derivative problem, harmonic measure, integral means spectrum of the derivative, Brennan's conjecture, variational methods, second variation, the Koebe function, coe cient problems, Lowner's equation, de Branges' proof of the Bieberbach conjecture, higher-order Schwarzian derivatives. Mathematics Subject Classi cation (MSC 2000): Primary 30Cxx. Secondary 30C35, 30C50, 30C55, 30C70, 30C75, 30C85, 31A15.