Remarks on a multiplier conjecture for univalent functions

Abstract:

In this paper we study some aspects of a conjecture on the convolution of univalent functions in the unit disk $\mathbb {D}$, which was recently proposed by Grünberg, Rønning, and Ruscheweyh (Trans. Amer. Math. Soc. 322 (1990), 377-393) and is as follows: let $\mathcal {D}: = \{ f{\text { analytic in }}\mathbb {D}:\left | {f''(z)} \right | \leq \operatorname {Re} f’(z),z \in \mathbb {D}\}$ and $g,h \in \mathcal {S}$ (the class of normalized univalent functions in $\mathbb {D}$. Then $\operatorname {Re} (f*g*h)(z)/z > 0$ in $\mathbb {D}$. We discuss several special cases, which lead to interesting, more specific statements about functions in $\mathcal {S}$, determine certain extreme points of $\mathcal {D}$, and note that the former conjectures of Bieberbach and Sheil-Small are contained in this one. It is an interesting matter of fact that the functions in $\mathcal {D}$, which are "responsible" for the Bieberbach coefficient estimates are not extreme points in $\mathcal {D}$.