On quasi-conformality and pseudo-analyticity

Abstract

1. Introduction.There exist various definitions of quasi-conformality and pseudo-analyticity.The following definition seems to be most basic: A topological mapping w= T(z) =u(x, y)+iv(x, y) of a plane region D of the z = x+iyplane onto a region in the w = u+iv-plane is called quasi-conformal if the following conditions are satisfied:(i) the partial derivatives ux, uy, vx, and vv exist and are continuous; (ii) J(z) = uxvv -uuvx > 0; and max I Dtw \ (Hi) Q(z) = | , , ^ K min | Dew \ e for a constant K^.1, where Dew = (ux cos 6 + uy sin t?) + i(vx cos 6 + vy sin 0), 0 ^ d < 2ir.