OAI.StrictInverseFirstPower.main

The theorem states a conjunction of three claims about the integral means of the derivative to the power -1 for normalized univalent functions on the unit disk. Here a function f:ℂ→ℂ is normalized univalent if it is complex-differentiable on the open unit disk, injective there, and satisfies f(0)=0 and f'(0)=1. The integral mean of order p at radius r is (2π)⁻¹ times the integral over θ from −π to π of |f'(re^{iθ})|^p. First, there exist real numbers ε and C with 0<ε<1/4 such that, for every normalized univalent f and every r with 1/2≤r<1, the order −1 integral mean of f at r is at most C(1−r)^(−1/4+ε). Second, the bounded spectrum at p=−1 is strictly less than 1/4 in the extended reals; this spectrum is the supremum, over normalized univalent f that are also bounded on the disk, of the growth exponent, which is the limsup as r→1⁻ of log(integral mean of order p at r) divided by log(1/(1−r)). Third, it is not the case that the bounded spectrum equals the prediction function kraetzerPrediction for every real p, where that function is p²/4 when |p|≤2 and |p|−1 otherwise.