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