Libera proved that the first
order linear differential equation F(z) + zF′(z) = 2f(z) has a convex, starlike or
close-to-convex solution in |z| < 1 if the driving term f(z) is convex, starlike, or
close-to convex in |z| < 1. It was an open question whether the solution would be
univalent if f(z) were spiral-like or univalent. The paper shows the relation of
Libera’s question to the Mandelbrojt-Schiffer conjecture and the class M defined by
S. Ruscheweyh. The paper proves there are spiral-like functions f(z) for which the
solution of the above differential equation is of infinite valence. The paper closes with
four open problems.
