Abstract
Let a fractional operator
D
λ
n
,
α
(
n
∈
N
0
=
{
0
,
1
,
2
,
…
}
,
0
⩽
α
<
1
,
λ
⩾
0
) be defined by
D
λ
0
,
0
=
f
(
z
)
,
D
λ
1
,
α
f
(
z
)
=
(
1
−
λ
)
Ω
α
f
(
z
)
+
λ
z
(
Ω
α
f
(
z
)
)
′
=
D
λ
α
(
f
(
z
)
)
,
D
λ
2
,
α
f
(
z
)
=
D
λ
α
(
D
λ
1
,
α
f
(
z
)
)
,
⋮
D
λ
n
,
α
f
(
z
)
=
D
λ
α
(
D
λ
n
−
1
,
α
f
(
z
)
)
,
where
Ω
α
f
(
z
)
=
Γ
(
2
−
α
)
z
α
D
z
α
f
(
z
)
,
and
D
z
α
is the known fractional derivative. In this paper, several interesting subordination results are derived for certain classes of analytic functions related to conic domains defined by the operator
D
λ
n
,
α
, which yield sharp distortion, rotation theorems and Koebe domain. These results extended corresponding previously known results.