OAI.Brennan.main_theorem

The theorem states that the defined proposition MainStatement holds, and it is a conjunction of four claims about the unit disk 𝔻 and schlicht functions (functions f : ℂ → ℂ that are differentiable and injective on 𝔻 with f(0)=0 and f′(0)=1). Here integralMean(f,t,r) is the average over θ in [−π,π] of |f′(re^{iθ})|^t, namely (2π)⁻¹ times the integral. First, for every ε>0 there is a constant C such that every schlicht f and every r with 1/2 ≤ r < 1 satisfy integralMean(f,−2,r) ≤ C(1−r)^(−1−ε). Second, the spectrum at t=−2 equals 1, where beta(f,t) is the limsup as r→1⁻ of log integralMean(f,t,r) divided by log(1/(1−r)), taken in the extended reals, and spectrum(t) is the supremum of beta(f,t) over all schlicht f. Third, let W be any open, connected, simply connected subset of ℂ whose image in the one-point compactification of ℂ has a frontier with more than one point, and let φ be differentiable on W and a bijection from W onto 𝔻. Then for every s with 4/3 < s < 4, the function |φ′(z)|^s is Lebesgue integrable over W. Fourth, for every function f that is differentiable and injective on 𝔻, with no normalization of f(0) or f′(0) required, and every t with −2 < t < 2/3, the function |f′(z)|^t is Lebesgue integrable over 𝔻.