Level sets for functions convex in one direction

Abstract

Goodman and Saff conjectured that if f f is convex in the direction of the imaginary axis then so are the functions 1 r f ( r z ) \frac {1}{r}f(rz) for all 0 > r > 2 − 1 0 > r > \sqrt 2 - 1 , i.e., the level sets f ( | z | > r ) f(|z|\, > r) are convex in the direction of the imaginary axis for 0 > r > 2 − 1 0 > r > \sqrt 2 - 1 . A weak form of this conjecture is proved and a question of Brannan is answered negatively.