Brennan’s Conjecture and Integral Means Spectrum

The paper proves that the bounded universal integral means spectrum satisfies (\mathcal B_b(t)\le t^2/4) for (-2\le t\le2), using superharmonic potentials, critical-point counting, signed-area estimates, and weighted integral means. Brennan’s conjecture follows at (t=-2), yielding conformal-derivative integrability for every exponent (3<p<4), while Makarov’s identity extends the improvement to the unrestricted spectrum on ((-2,6-4\sqrt2)). The paper shows that Kraetzer’s proposed equality (\mathcal B_b(t)=t^2/4) cannot hold throughout the interior, although the detailed proof of strictness and the characterization of extremal maps are deferred or remain open. Problem setting and principal result The paper studies the universal integral means spectrum of conformal maps and derives a strengthened form of the bounds associated with Brennan’s conjecture. For a univalent map f:D→Cf:\mathbb D\to\mathbb C, the integral means spectrum is defined by βf(t)=lim sup⁡r↑1log⁡∫02π∣f′(reiθ)∣t dθlog⁡(1/(1−r)).\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t,d\theta} {\log(1/(1-r))}. The bounded and unrestricted universal spectra are respectively Bb(t)=sup⁡fβf(t),B(t)=sup⁡fβf(t),\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t), where the first supremum is over bounded univalent maps and the second over all univalent maps. The paper’s central theorem is Bb(t)≤t24,−2≤t≤2,\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2, with strict inequality for t∈(−2,2)∖{0}t\in(-2,2)\setminus{0}. Via Makarov’s identity, B(t)=max⁡{Bb(t),3t−1},\mathcal B(t)=\max{\mathcal B_b(t),3t-1}, the authors obtain B(t)<t24\mathcal B(t)<\frac{t^2}{4} for t∈(−2, 6−42)∖{0}.t\in\left(-2,,6-4\sqrt 2\right)\setminus{0}. The endpoint t=−2t=-2 corresponds to Brennan’s conjecture. Under the inverse-map change of variables, the assertion B(−2)=1\mathcal B(-2)=1 yields the integrability range βf(t)=lim sup⁡r↑1log⁡∫02π∣f′(reiθ)∣t dθlog⁡(1/(1−r)).\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t,d\theta} {\log(1/(1-r))}.0 for conformal maps βf(t)=lim sup⁡r↑1log⁡∫02π∣f′(reiθ)∣t dθlog⁡(1/(1−r)).\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t,d\theta} {\log(1/(1-r))}.1. Thus the paper proves not merely the endpoint estimate required by Brennan’s conjecture but a strict sub-Kraetzerian upper bound for the bounded spectrum throughout the interior of the interval …