Logarithmic coefficients of starlike functions connected with k-Fibonacci numbers
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Date
2021
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Ankara Üniversitesi
Abstract
Let
A
denote the class of analytic functions in the open unit disc
U
normalized by
f
(
0
)
=
f
′
(
0
)
−
1
=
0
,
and let
S
be the class of all functions
f
∈
A
which are univalent in
U
. For a function
f
∈
S
, the logarithmic coefficients
δ
n
(
n
=
1
,
2
,
3
,
…
)
are defined by
log
f
(
z
)
z
=
2
∑
∞
n
=
1
δ
n
z
n
(
z
∈
U
)
.
and it is known that
|
δ
1
|
≤
1
and
|
δ
2
|
≤
1
2
(
1
+
2
e
−
2
)
=
0
,
635
⋯
.
The problem of the best upper bounds for
|
δ
n
|
of univalent functions for
n
≥
3
is still open. Let
S
L
k
denote the class of functions
f
∈
A
such that
z
f
′
(
z
)
f
(
z
)
≺
1
+
τ
2
k
z
2
1
−
k
τ
k
z
−
τ
2
k
z
2
,
τ
k
=
k
−
√
k
2
+
4
2
(
z
∈
U
)
.
In the present paper, we determine the sharp upper bound for
|
δ
1
|
,
|
δ
2
|
and
|
δ
3
|
for functions
f
belong to the class
S
L
k
which is a subclass of
S
. Furthermore, a general formula is given for
|
δ
n
|
(
n
∈
N
)
as a conjecture.
Description
Keywords
Analytic function, univalent function, shell-like function, logarithmic coefficients, k-Fibonacci number, subordination