We establish sharp bounds for simultaneous local rotation and Hölder-distortion of planar quasiconformal maps. In addition, we give sharp estimates for the corresponding joint quasiconformal multifractal spectrum, based on new estimates for Burkholder functionals with complex parameters. As a consequence, we obtain optimal rotation estimates also for bi-Lipschitz maps.2010 Mathematics Subject Classification. Primary 30C62; Secondary 37C45.

Inspired by Morrey's Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals Bp, p 2, are quasiconcave, when tested on deformations of identity f ∈ Id + C ∞ • (Ω) with Bp (Df (x)) 0 pointwise, or equivalently, deformations such that |Df | 2 p p−2 J f . In particular, this holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible L pestimates for the gradient of a principal solution to the Beltrami equation fz = µ(z)fz , for any p in the critical interval 2 p 1 + 1/ µ f ∞. Examples of local maxima lacking symmetry manifest the intricate nature of the problem.

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We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a k-quasicircle is at most 1 + k 2 , it is natural to expect that the maximum asymptotic variance Σ 2 = 1. In this paper, we prove 0.87913 Σ 2 1.For the lower bound, we give examples of polynomial Julia sets which are k-quasicircles with dimensions 1 + 0.87913 k 2 for k small, thereby showing that Σ 2 0.87913. The key ingredient in this construction is a good estimate for the distortion k, which is better than the one given by a straightforward use of the λ-lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating Σ 2 in terms of nearly circular polynomial Julia sets.