Univalent functions and nonanalytic curves

The two examples z+ (1 +e)z2/2 and z+(1 +e)z2/4 with e>0, show that the right sides of (2) and (3) cannot be increased without destroying respectively the starlikeness and convexity of the image regions. This theorem was first discovered by Alexander [I], but according to Remak [6], Alexander's proof contains an error. Remak supplied a correct proof, and we give here a second proof of this theorem which is somewhat different and yields a little more information about the mapping function, than the one given by Remak. The condition (2) on the power series has been investigated further by Yan'-Sin Cin [2], Mullender [3; 4], and Schild [7]. It is well known that the condition