AbstractIn this paper we study some properties of functions f which are analytic and normalized (i.e. $$f(0)=0=f'(0)-1$$
f
(
0
)
=
0
=
f
′
(
0
)
-
1
) such that satisfy the following subordination relation $$\begin{aligned} \left( \frac{zf'(z)}{f(z)}-1\right) \prec \frac{z}{(1-pz)(1-qz)}, \end{aligned}$$
z
f
′
(
z
)
f
(
z
)
-
1
≺
z
(
1
-
p
z
)
(
1
-
q
z
)
,
where $$(p,q) \in [-1,1] \times [-1,1]$$
(
p
,
q
)
∈
[
-
1
,
1
]
×
[
-
1
,
1
]
. These types of functions are starlike related to the generalized Koebe function. Some of the features are: radius of starlikeness of order $$\gamma \in [0,1)$$
γ
∈
[
0
,
1
)
, image of $$f\left( \{z:|z|<r\}\right) $$
f
{
z
:
|
z
|
<
r
}
where $$r\in (0,1)$$
r
∈
(
0
,
1
)
, radius of convexity, estimation of initial and logarithmic coefficients, and Fekete–Szegö problem.
Starlike functions associated with the generalized Koebe function
