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In this paper we obtain estimates for the first nontrivial eigenvalue of the$p$-Laplace Neumann operator in bounded simply connected planar domains$\Omega\subset\mathbb R^2$. This study is based on a quasiconformal version ofthe universal weighted Poincar\'e-Sobolev inequalities obtained in our previouspapers for conformal weights. The suggested weights in the present paper areJacobians of quasiconformal mappings. The main technical tool is the theory ofcomposition operators in relation with the Brennan's Conjecture for(quasi)conformal mappings.

Spectral Estimates of the $p$-Laplace Neumann operator and Brennan's Conjecture