Let
R
R
denote the class of functions
f
(
z
)
=
z
+
a
2
z
2
+
⋯
f(z) = z + {a_2}{z^2} + \cdots
that are analytic in the unit disc
E
=
{
z
:
|
z
|
>
1
}
E = \{ z:\left | z \right | > 1\}
and satisfy the condition
Re
(
f
′
(
z
)
+
z
f
(
z
)
)
>
0
,
z
∈
E
\operatorname {Re} (f’(z) + zf(z)) > 0,z \in E
. It is known that
R
R
is a subclass of
S
t
{S_t}
, the class of univalent starlike functions in
E
E
. In the present paper, among other things, we prove (i) for every
n
≥
1
n \geq 1
, the
n
n
th partial sum of
f
∈
R
,
s
n
(
z
,
f
)
f \in R,{s_n}(z,f)
, is univalent in
E
E
, (ii)
R
R
is closed with respect to Hadamard convolution, and (iii) the Hadamard convolution of any two members of
R
R
is a convex function in
E
E
.