An entire transcendental function whose inverse takes sets of finite measure into sets of finite measure

Abstract

PROOF. LetZ>= {z = x+iy\x>0, \y <l/(l+x)} and let E be the complement in the plane of the closure of D. Also let c(z) be a conformal map from the outside of | z — 11 = e (e a small positive constant to be determined later) into the unit disk which takes the point at infinity into the origin. By a particular case of Runge's theorem [3] and Rouche's theorem there exists a rational function R\(z) with the following properties: (i) Ri(z) has a pole only at 2 = 2. (ii) Ri(z) so closely approximates c(z) on E that Ri(z) on E has the following properties: (a) measure Ri(E) less than one. (b) Ri (z)+z is 1-1 (this is always possible by choosing e small enough) on E. (c) X = | £ i ( 1 ) R ( 2 ) | ? * 0 . By the same argument there exists a rational function R%(z) which has the following properties: (i) R%(z) has a pole only at z = 3. (ii) R<L(Z) SO closely approximates R\{z) in EUG 3 ( = {z = x-\-iy\x <3 /2} ) that R2(z) o n EKJGz has the following properties: (a) measure R2(E) less than one. (b) R2(z)+z is 1-1 on E. (c) I R2(z) -.Ri(s) J <X/8 on EUG 3 . Continuing inductively there exists a rational function Rn(z) which has the following properties: