Vol. 84, No. 1, 1979

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Valence properties of the solution of a differential equation

Douglas Michael Campbell and Vikramaditya Singh

Vol. 84 (1979), No. 1, 29–33

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.

Mathematical Subject Classification 2000
Primary: 30C35
Received: 25 July 1978
Revised: 30 October 1978
Published: 1 September 1979
Douglas Michael Campbell
Vikramaditya Singh