Inequalities for Rational Functions Original Research Article

A new hyperbolic area estimate for the level sets of finite Blaschke products is presented. The following inversion of the usual Sobolev embedding theorem is proved:‖r‖W1q(D)⩽Cn ‖r‖Lp(D), p>2, 1/q=1/p+1/2.Hereris a rational function of degreenwith poles outside D. This estimate implies a new inverse theorem for rational approximation of analytic functions with respect to areaLpnorm.