  We establish surprising improved Schauder regularity properties for solutions
to the Leray-Lions divergence type equation in the plane. The results are
achieved by studying the nonlinear Beltrami equation and making use of special
new relations between these two equations. In particular, we show that
solutions to an autonomous Beltrami equation enjoy a quantitative improved
degree of H\"older regularity, higher than what is given by the classical
exponent $1/K$.
