General Mathematics Vol. 16, No. 3 (2008), 19-27
On the Fekete-Szegö inequality for a class of
analytic functions defined by using the
generalized Sălăgean operator1
Dorina Răducanu
Abstract
In this paper we obtain the Fekete-Szegö inequality for a class
of analytic functions f (z) defined in the open unit disk for which
β
α n+1
n+2
Dλ
f
Dλ
f
(α , β , λ ≥ 0) lies in a region starlike with
Dn f
Dn+1 f
λ
λ
respect to 1 and which is symmetric with respect to the real axis.
2000 Mathematics Subject Classification: 30C45
Key words:Analytic functions,generalized Sălăgean operator,
Fekete-Szegö inequality
1
Introduction
Let A denote the class of functions f (z) of the form
(1.1)
f (z) = z +
∞
X
ak z k
k=2
which are analytic in the open unit disk U = {z ∈ C : |z| < 1} and let S be
the subclass of A consisting of univalent functions.
1
Received 28 December, 2007
Accepted for publication (in revised form) 3 January, 2008
19
20
Dorina Răducanu
The generalized Sălăgean differential operator is defined in [2] by
Dλ0 f (z) = f (z) , Dλ1 f (z) = (1 − λ)f (z) + λzf ′ (z)
Dλn f (z) = Dλ1 (Dλn−1 f (z)) , λ ≥ 0.
If f is given by (1) we see that
(1.2)
Dλn f (z)
=z+
∞
X
[1 + (k − 1)λ]n ak z k .
k=2
When λ = 1 we get the classic Sălăgean differential operator [6].
Let Φ(z) be an analytic function with positive real part on U with Φ(0) =
1 , Φ′ (0) > 0 which maps the unit disk U onto a region starlike with respect
to 1 which is symmetric with respect to the real axis.
Denote by S ∗ (Φ) the class of functions f ∈ S for which
zf ′ (z)
≺ Φ(z) , z ∈ U
f (z)
and denote by C(Φ) the class of functions f ∈ S for which
1+
zf ′′ (z)
≺ Φ(z) , z ∈ U
f ′ (z)
where ” ≺ ” stands for the usual subordination.The classes S ∗ (Φ) and C(Φ)
where defined and studied by Ma and Minda [1]. They obtained the FeketeSzegö inequality for functions in the class S ∗ (Φ) and also for functions in
the class C(Φ).
By using the generalized Sălăgean differential operator we define the
following class of functions:
Definition 1.1. Let Φ(z) be a univalent stralike function with respect to
1 which maps the unit disk onto a region in the right halfplane symmetric
with respect to the real axis, Φ(0) = 1 and Φ′ (0) > 0.A function f ∈ A is in
n,λ
(Φ) if
the class Mα,β
n+1
β
α n+2
Dλ f (z)
Dλ f (z)
(1.3)
≺ Φ(z),
Dλn f (z)
Dλn+1 f (z)
0 ≤ α ≤ 1, 0 ≤ β ≤ 1, λ > 0.
21
On the Fekete-Szegö inequality...
It follows that
0,1
0,1
(Φ) ≡ S ∗ (Φ).
(Φ) ≡ C(Φ) and M1,0
M0,1
When n = 0 and λ = 1 we obtain the class Mα,β (Φ) studied by Ravichadran
et.al. [3].
In this paper we obtain the Fekete-Szegö inequality for functions in the
n,λ
class Mα,β
(Φ).
To prove oue results we shall need the following lemmas.
Lemma 1.1. [1] If p1 (z) = 1 + c1 z + c2 z 2 + . . . is an analytic function with
positive real part in U , then


 −4v + 2, if v ≤ 0
c2 − vc21 ≤
2,
if 0 ≤ v ≤ 1


4v − 2,
if v ≥ 1.
When v < 0 or v > 1 , the equality holds if and only if p1 (z) is (1+z)/(1−z)
or one of its rotations.If 0 < v < 1,then the equality holds if and only if
p1 (z) is (1 + z 2 )/(1 − z 2 ) or one of its rotations.If v = 0,the equality holds
if and only if
1+a 1+z
1−a 1−z
p1 (z) =
+
, 0≤a≤1
2
1−z
2
1+z
or one of its rotations.If v = 1,the equality holds if and only if p1 is the
reciprocal of one of the functions such that the equality holds in the case of
v = 0.
Also the above upper bound is sharp and it can be improved as follows
when 0 < v < 1 :
|c2 − vc21 | + v|c21 | ≤ 2 , 0 < v ≤
and
|c2 − vc21 | + (1 − v)|c21 | ≤ 2 ,
1
2
1
< v ≤ 1.
2
22
Dorina Răducanu
Lemma 1.2. [4] If p1 (z) = 1 + c1 z + c2 z 2 + . . . is an analytic function with
positive real part in U , then
|c2 − vc21 | ≤ 2 max {1; |2v − 1|} .
The result is sharp for the function
p1 (z) =
2
1+z
1 + z2
.
or
p
(z)
=
1
1 − z2
1−z
Fekete-Szegö problem
We prove our main result by making use of Lemma 1.1.
Theorem 2.1. Let Φ(z) = 1 + B1 z + B2 z 2 + . . ..If f (z) given by (1.1) is in
n,λ
the class Mα,β
(Φ) , then
a3 − µa22 ≤
h
i

B12
1

2B2 − λ(1+λ)2n [α+β(1+λ)]2 γ ,
if µ ≤ σ1

 4λ(1+2λ)n [α+β(1+2λ)]
B1
≤
,
if σ1 ≤ µ ≤ σ2
2λ(1+2λ)n [α+β(1+2λ)] h
i


B12

1
γ , if µ ≥ σ .
−2B +
4λ(1+2λ)n [α+β(1+2λ)]
2
2
λ(1+λ)2n [α+β(1+λ)]2
Further,if σ1 < µ ≤ σ3 ,then
|a3 − µa22 |+
B2
γB1
λ(1 + λ)2n [α + β(1 + λ)]2
|a2 |2
1−
+
+
2(1 + 2λ)n [α + β(1 + 2λ)]B1
B1 2λ(1 + λ)2n [α + β(1 + λ)]2
≤
2λ(1 +
B1
2n
2λ) [α +
β(1 + 2λ)]
.
If σ3 < µ ≤ σ2 ,then
|a3 − µa22 |+
B2
γB1
λ(1 + λ)2n [α + β(1 + λ)]2
|a2 |2
1+
−
+
n
2n
2
2(1 + 2λ) [α + β(1 + 2λ)]B1
B1 2λ(1 + λ) [α + β(1 + λ)]
23
On the Fekete-Szegö inequality...
≤
where
σ1 :=
−
,
2λ(1 + λ)2n [α + β(1 + λ)]2 (B2 − B1 )
−
4(1 + 2λ)n [α + β(1 + 2λ)]B12
2λ(1 + λ)2n [α + β(1 + λ)]2 (B2 + B1 )
−
4(1 + 2λ)n [α + β(1 + 2λ)]B12
B12 (1 + λ)2n [λ[α + β(1 + λ)]2 − (λ + 2)[α + β(1 + λ)2 ]]
4(1 + 2λ)n [α + β(1 + 2λ)]B12
σ3 :=
−
β(1 + 2λ)]
B12 (1 + λ)2n [λ[α + β(1 + λ)]2 − (λ + 2)[α + β(1 + λ)2 ]]
4(1 + 2λ)n [α + β(1 + 2λ)]B12
σ2 :=
−
2λ(1 +
B1
2n
2λ) [α +
2λ(1 + λ)2n [α + β(1 + λ)]2 B2
−
4(1 + 2λ)n [α + β(1 + 2λ)]B12
B12 (1 + λ)2n [λ[α + β(1 + λ)]2 − (λ + 2)[α + β(1 + λ)2 ]]
4(1 + 2λ)n [α + β(1 + 2λ)]B12
and
γ := λ(1 + λ)2n [α + β(1 + λ)]2 −
−(λ + 2)(1 + λ)2n [α + β(1 + λ)2 ] + 4µ(1 + 2λ)n [α + β(1 + 2λ)].
These results are sharp.
n,λ
Proof. Let f ∈ Mα,β
(Φ) and let
(2.1)
p(z) :=
Dλn+1 f (z)
Dλn f (z)
α Dλn+2 f (z)
Dλn+1 f (z)
β
= 1 + b1 z + b2 z 2 + . . .
Since the function Φ(z) = 1 + B1 z + B2 z 2 + . . . is univalent and p ≺ Φ then
the function
p1 (z) =
1 + Φ−1 (p(z))
= 1 + c1 z + c2 z 2 . . .
1 − Φ−1 (p(z))
is ananlytic and has positive real part in U .We also have
1
1 2
1
1
p1 (z) − 1
2
= 1 + B1 c1 z + B1 (c2 − c1 ) + B2 c1 z 2 + . . .
p(z) = Φ
p1 (z) + 1
2
2
2
4
24
Dorina Răducanu
From (2.1) we obtain
1
1
1
1
b1 = B1 c1 and b2 = B1 (c2 − c21 ) + B2 c21 .
2
2
2
4
By making use of (1.1) and (1.2) we obtain
Dλn+1 f (z)
= 1 + λ(1 + λ)n a2 z + [2λ(1 + 2λ)n a3 − λ(1 + λ)2n a22 ]z 2 + . . .
Dλn f (z)
and therefore we have
Dλn+1 f (z)
Dλn f (z)
α
=
λα2 − α(λ + 2)
2n 2
= 1+αλ(1+λ) a2 z+λ 2α(1 + 2λ) a3 +
(1 + λ) a2 z 2 +. . .
2
n
Similarly we obtain
Dλn+2 f (z)
Dλn+1 f (z)
n
β
= 1 + βλ(1 + λ)n+1 a2 z+
λβ 2 − β(λ + 2)
n+1
2n+2 2
(1 + λ)
a2 z 2 + . . .
+λ 2β(1 + 2λ) a3 +
2
Thus we have
β
α n+2
n+1
Dλ f (z)
Dλ f (z)
= 1 + λ(1 + λ)n [α + β(1 + λ)]a2 z+
n+1
Dλn f (z)
Dλ f (z)
+λ {2(1 + 2λ)n [α + β(1 + 2λ)]a3 +
λ[α + β(1 + λ)]2 − (λ + 2)[α + β(1 + λ)2 ]
2n 2
+
(1 + λ) a2 z 2 + . . .
2
In view of (2.1) it results
(2.2)
b1 = λ(1 + λ)n [α + β(1 + λ)]a2
and
b2 = 2λ(1 + 2λ)n [α + β(1 + 2λ)]a3 +
λ2 [α + β(1 + λ)]2 − λ(λ + 2)[α + β(1 + λ)2 ]
(2.3) +
(1 + λ)2n a22 .
2
25
On the Fekete-Szegö inequality...
Therefore we have
(2.4)
a3 − µa22 =
where
4λ(1 +
B1
[c2 − vc21 ]
+ β(1 + 2λ)]
2λ)n [α
γB1
B2
1
+
1−
.
v :=
2
B1 2λ(1 + λ)2n [α + β(1 + λ)]2
Our result follows now by an application of Lemma 1.1.To show that the
bounds are sharp,we consider the functions KΦ,m (m = 2, 3, . . .) defined by
Dλn+1 KΦ,m (z)
Dλn KΦ,m (z)
α Dλn+2 KΦ,m (z)
Dλn+1 KΦ,m (z)
β
= Φ(z m−1 ),
KΦ,m (0) = [KΦ,m ]′ (0) − 1 = 0
and the functions Fδ , Gδ (0 ≤ δ ≤ 1) defined by
Dλn+1 Fδ (z)
Dλn Fδ (z)
α Dλn+2 Fδ (z)
Dλn+1 Fδ (z)
β
=Φ
z(z + δ)
1 + δz
, Fδ (0) = Fδ′ (0) − 1 = 0
and
β
n+1
α n+2
z(z + δ)
Dλ Gδ (z)
Dλ Gδ (z)
=Φ −
, Gδ (0) = G′δ (0)−1 = 0.
Dλn Gδ (z)
1 + δz
Dλn+1 Gδ (z)
n,λ
It is clear that the functions KΦ,m , Fδ and Gδ belong to the class Mα,β
(Φ).
If µ < σ1 or µ > σ2 , then the equality holds if and only if f is KΦ,2 or one
of its rotations.When σ1 < µ < σ2 , the equality holds if and only if f is
KΦ,3 or one of its rotations.If µ = σ1 , then the equality holds if and only if
f is Fδ or one of its rotations.If µ = σ2 ,then the equality holds if and only
if f is Gδ or one of its rotations.
By making use of Lemma 1.2. we easely obtain the next theorem.
Theorem 2.2. Let Φ(z) = 1 + B1 z + B2 z 2 + . . . and let f (z) be in the class
n,λ
Mα,β
(Φ) .For a complex number µ we have:
|a3 − µa22 | ≤
26
Dorina Răducanu
B2
γB
B1
1
−
max
1,
+
≤
B1 2λ(1 + λ)2n [α + β(1 + λ)]2 .
2λ(1 + 2λ)n [α + β(1 + 2λ)]
The result is sharp.
References
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[2] F.M.Al-Oboudi,On univalent functions defined by a generalized
Sălăgean operator,Internat.J.Math.Math.Sci.,27(2004),1429-1436.
[3] V.Ravichandran,M.Darus,
M.Hussain
Khan,K.G.Subramanian,
Fekete-Szegö inequality for certain class of analytic functions,
Aust.J.Math.Anal.Appl.,1,2(2004)art.4.
[4] V.Ravichadran,Y.Polotoglu,M.Bolcal,A.Sen,Certain subclasses of starlike and convex functions of complex order,preprint
[5] H.M.Srivastava,A.K.Mishra,M.K.Das,The Fekete-Szegö problem for
a subclass of close-to-convex functions,Complex Variables,Theory
Appl.,44,(2001),145-163.
27
On the Fekete-Szegö inequality...
[6] G.S.Sălăgean,Subclasses of
Math.,1013,(1983),362-372.
univalent
functions,Lect.Notes
in
[7] D.Răducanu,The Fekete-Szegö problem for a class of multivalent functions,(to appear).
Department of Mathematics and Computer Science
”Transilvania” University of Braşov
50091,Iuliu Maniu,50,Braşov
Romania
E-mail: dorinaraducanu@yahoo.com