A uniqueness theorem for Beltrami equations

( 1 . 2 ) (z) -q(Z) (I Zz) 2, where q is analytic and subject to the condition (1.3) 1 q(t -Z)21 < k< 1. Elementary classical methods yield a particular solution of (t.1). It is the purpose of this note to show that tlis soluition is univalent, anid that q is uniiquely determined by the boutndary values of f. Related questions occur in a paper by L. Bers 2 which was written independently of the present authors. 2. If we rewrite (1.1) as (2.1) fz = q(z) (I zz) 2 it is classical that f may be determined from the genieral solution of the ordinary differential equation (2.2) dz/dz q(z) (1z)2 whiere z and z are regarded as independenit variables. By the general theory, if vi and v2 are linearly independent solutionSs of the associated linear equation (2.3) v = qv, then (2.1) admits the solution ( ZV2 + (1 -ZZ2 (v1 + (1 -Z)v We shall show thatf, as determined by (2.4), is univalent in I z I < t, from which it follows that all solutions of (1.1) can be expressed as analytic functions of f.