ON A CERTAIN SUBCLASS OF ANALYTIC
FUNCTIONS INVOLVING THE AL-OBOUDI
DIFFERENTIAL OPERATOR
S.M. KHAIRNAR AND MEENA MORE
Department of Mathematics
Maharashtra Academy of Engineering
Alandi -412 105, Pune (M.S.), India
EMail: smkhairnar2007@gmail.com meenamores@gmail.com
Received:
26 November, 2008
Accepted:
28 May, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
30C45.
Key words:
Close-to-convex function, Close-to-starlike function, Ruscheweyh derivative operator, Al-Oboudi differential operator and subordination relationship.
Abstract:
In this paper we introduce a new subclass of normalized analytic functions in the
open unit disc which is defined by the Al-Oboudi differential operator. A coefficient inequality, extreme points and integral mean inequalities of a differential
operator for this class are given. We investigate various subordination results
for the subclass of analytic functions and obtain sufficient conditions for univalent close-to-starlikeness. We also discuss the boundedness properties associated
with partial sums of functions in the class. Several interesting connections with
the class of close-to-starlike and close-to-convex functions are also pointed out.
Acknowledgements:
The paper presented here is a part of our research project funded by the Department of Science and Technology (DST), New Delhi, Ministry of Science and
Technology, Government of India (No.SR/S4/MS:544/08), and BCUD, University of Pune (UOP), Pune (Ref No BCUD/14/Engg.10). The authors are thankful
to DST and UOP for their financial support. We also express our sincere thanks
to the referee for his valuable suggestions.
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
JJ
II
J
I
Page 1 of 22
Go Back
Full Screen
Close
Contents
1
Introduction and Preliminaries
3
2
Coefficient Inequalities, Growth and Distortion Theorems
7
3
Extreme Points
9
Subclass Of Analytic Functions
4
Integral Mean Inequalities for a Differential Operator
11
5
Subordination Results for the Class T (η, f )
14
6
Partial Sums
18
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
JJ
II
J
I
Page 2 of 22
Go Back
Full Screen
Close
1.
Introduction and Preliminaries
Let A denote the class of normalized functions f defined by
f (z) = z +
(1.1)
∞
X
ak z k
k=2
which are analytic in the open unit disc U = {z ∈ C : |z| < 1}. For f ∈ A, [1] has
introduced the following differential operator.
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
D0 f (z) = f (z)
(1.2)
Subclass Of Analytic Functions
Title Page
(1.3)
(1.4)
0
1
D f (z) = (1 − δ)f (z) + δzf (z) = Dδ f (z),
Dn f (z) = Dδ (Dn−1 f (z)),
δ≥0
(n ∈ N).
For f (z) given by (1.1), we notice from (1.3) and (1.4) that
(1.5)
Dn f (z) = z +
∞
X
[1 + (k − 1)δ]n ak z k
(n ∈ N0 = N ∪ {0}).
k=2
For δ = 1 we obtain the Sălăgean operator [11].
Definition 1.1. A function f in A is said to be starlike of order α (0 ≤ α < 1) in
U , that is, f ∈ S ∗ (α), if and only if
0 zf (z)
(1.6)
Re
>α
(z ∈ U ).
f (z)
Contents
JJ
II
J
I
Page 3 of 22
Go Back
Full Screen
Close
Definition 1.2. A function f in A is said to be convex of order α (0 ≤ α < 1) in U ,
that is, f ∈ K(α), if and only if
zf 00 (z)
(1.7)
Re 1 + 0
>α
(z ∈ U ).
f (z)
Definition 1.3. A function f in A is said to be close-to-convex in U , of order α, that
is, f ∈ C(α), if and only if
(1.8)
0
Re{f (z)} > α
(z ∈ U ).
Definition 1.4. A function f in A is said to be close-to-starlike of order α (0 ≤ α <
1) in U , that is, f ∈ CS ∗ (α), if and only if
f (z)
(1.9)
Re
>α
(z ∈ U \ {0}).
z
We note that the classes S, S ∗ (0) = S ∗ , K(0) = K, C(0) = C, CS ∗ (0) = CS ∗ are
the well known classes of univalent, starlike, convex, close-to-convex and close-tostarlike functions in U , respectively. It is also clear that
Definition 1.5. For two functions f and g analytic in U , we say that the function
f (z) is subordinate to g(z) in U , and write
(z ∈ U )
if there exists a Schwarz function w(z), analytic in U with w(0) = 0 and |w(z)| < 1
such that
(1.11)
f (z) = g(w(z))
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
JJ
II
J
I
Go Back
(ii) K(α) ⊂ S ∗ (α) ⊂ C(α) ⊂ S.
f (z) ≺ g(z)
S.M. Khairnar and Meena More
Page 4 of 22
(i) f ∈ K(α) if and only if zf 0 ∈ S ∗ (α);
(1.10)
Subclass Of Analytic Functions
(z ∈ U ).
Full Screen
Close
In particular, if the function g is univalent in U , the above subordination is equivalent
to
f (0) = g(0),
(1.12)
f (U ) ⊂ g(U ).
Littlewood [7] in 1925 has proved the following subordination theorem which we
state as a lemma.
Lemma 1.6. Let f and g be analytic in the unit disc, and suppose g ≺ f . Then for
0 < p < ∞,
Z 2π
Z 2π
iθ p
(1.13)
|g(re )| dθ ≤
|f (reiθ )|p dθ
(0 ≤ r < 1, p > 0).
0
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
0
Strict inequality holds for 0 < r < 1 unless f is constant or w(z) = αz, |α| = 1.
Definition 1.7. Let n ∈ N ∪ {0} and λ ≥ 0. Let Dλn f denote the operator defined
by Dλn : A → A such that
(1.14)
Subclass Of Analytic Functions
Dλn f (z) = (1 − λ)S n f (z) + λRn f (z)
z ∈ U,
where S n f is the Sălăgean differential operator and Rn f is the Ruscheweyh differential operator defined by Rn : A → A such that
R0 f (z) = f (z), R1 f (z) = zf 0 (z),
Title Page
Contents
JJ
II
J
I
Page 5 of 22
Go Back
Full Screen
with recurrence relation given by
(1.15)
(n + 1)Rn+1 f (z) = z[Rn f (z)]0 + nRn f (z) (z ∈ U ).
For f ∈ A given by (1.1)
(1.16)
n
R f (z) = z +
∞
X
k=2
n
Cn+k−1 ak z k
(z ∈ U ).
Close
Notice that Dλn is a linear operator and for f ∈ A defined by (1.1), we have
Dλn f (z)
(1.17)
=z+
∞
X
[(1 − λ)k n + λ n Cn+k−1 ]ak z k .
k=2
It is observed that for n = 0,
Dλ0 f (z) = (1 − λ)S 0 f (z) + λR0 f (z) = f (z) = S 0 f (z) = R0 f (z),
Subclass Of Analytic Functions
and for n = 1
S.M. Khairnar and Meena More
Dλ1 f (z) = (1 − λ)S 1 f (z) + λR1 f (z) = zf 0 (z) = S 1 f (z) = R1 f (z).
Definition 1.8. Let K(γ, µ, m, β) denote the subclass of A consisting of functions f
which satisfy the inequality
m
1
D
f
m
0
< β,
(1.18)
(1
−
µ)
+
µ(D
f
)
−
1
γ
z
where z ∈ U, γ ∈ C \ {0}, 0 < β ≤ 1, 0 ≤ µ ≤ 1, m ∈ N0 and Dm is as defined in
(1.5).
Remark 1. For γ = 1, µ = 1, m = 0, we obtain the class of close-to-convex
functions of order (1 − β). For the values γ = 1, µ = 0, m = 0, we obtain the class
of close-to-starlike functions of order (1 − β).
Let
f (z)
T (η, f ) = (1 − η)
+ η f 0 (z) (z ∈ U \ {0})
z
for η real and f ∈ A. Define
Tη := {f ∈ A : Re{T (η, f )} > 0}.
We note that Tη can be derived from the class K(γ, µ, m, β) by replacing µ by η and
Dm f by f .
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
JJ
II
J
I
Page 6 of 22
Go Back
Full Screen
Close
2.
Coefficient Inequalities, Growth and Distortion Theorems
Here we first give a sufficient condition for f ∈ A to belong to the class K(γ, µ, m, β).
Theorem 2.1. Let f (z) ∈ A satisfy
(2.1)
∞
X
(1 + (k − 1)µ)(1 + (k − 1)δ)m |ak | ≤ |γ|β,
k=2
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
where γ ∈ C \ {0}, 0 < β ≤ 1, 0 ≤ µ ≤ 1, m ∈ N0 , δ ≥ 0. Then f (z) ∈
K(γ, µ, m, β).
Proof. Suppose that (2.1) is true for γ (γ ∈ C\{0}), β (0 < β ≤ 1), µ (0 ≤ µ ≤ 1),
m ∈ N0 , and δ (δ ≥ 0) for f (z) ∈ A.
Using (1.5) for |z| = 1, we have
∞
m
X
(1 − µ) D f + µ(Dm f )0 − 1 ≤
(1 + (k − 1)µ)(1 + (k − 1)δ)m |ak |
z
k=2
≤ |γ|β.
Thus by Definition 1.8 f (z) ∈ K(γ, µ, m, β).
Notice that the function given by
(2.2)
f (z) = z +
∞
X
k=2
|γ|β
zk
(1 + (k − 1)µ)(1 + (k − 1)δ)m
belongs to the class K(γ, µ, m, β) and plays the role of extremal function for the
result (2.1).
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
JJ
II
J
I
Page 7 of 22
Go Back
Full Screen
Close
We denote by K̃(γ, µ, m, β) ⊆ K(γ, µ, m, β) the functions
f (z) = z +
∞
X
ak z k ,
k=2
where the Taylor-Maclaurin coefficients satisfy inequality (2.1).
Next we state the growth and distortion theorems for the class K̃(γ, µ, m, β). The
results follow easily on applying Theorem 2.1, therefore, we omit the proof.
Theorem 2.2. Let the function f (z) defined by (1.1) be in the class K̃(γ, µ, m, β).
Then
(2.3)
|z| −
|γ|β
|γ|β
|z|2 ≤ |f (z)| ≤ |z| +
|z|2 .
m
(1 + µ)(1 + δ)
(1 + µ)(1 + δ)m
The equality in (2.3) is attained for the function f (z) given by
f (z) = z +
(2.4)
|γ|β
z2.
(1 + µ)(1 + δ)m
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
JJ
II
J
I
Page 8 of 22
Theorem 2.3. Let the function f (z) defined by (1.1) be in the class K̃(γ, µ, m, β).
Then
(2.5)
1−
2|γ|β
2|γ|β
|z| ≤ |f 0 (z)| ≤ 1 +
|z|.
m
(1 + µ)(1 + δ)
(1 + µ)(1 + δ)m
The equality in (2.5) is attained for the function f (z) given by (2.4).
In view of Remark 1, Theorem 2.2 and Theorem 2.3 would yield the corresponding distortion properties for the class of close-to-convex and close-to-starlike functions.
Go Back
Full Screen
Close
3.
Extreme Points
Now we determine the extreme points of the class K̃(γ, µ, m, β).
Remark 2. For γ ∈ C\{0}, 0 < β ≤ 1, 0 ≤ µ ≤ 1, m ∈ N0 and δ ≥ 0 the following
functions are in the class K̃(γ, µ, m, β)
β|γ|
z2
(z ∈ U );
m
(1 + µ)(1 + δ)
β|γ|
z3
(z ∈ U );
f2 (z) = z +
m
(1 + 2µ)(1 + 2δ)
(|γ|β − 1)
1
z2 +
z3
f3 (z) = z +
m
m
(1 + µ)(1 + δ)
(1 + 2µ)(1 + 2δ)
f1 (z) = z +
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
(z ∈ U ).
Title Page
Contents
Theorem 3.1. Let f1 (z) = z and
(3.1)
|γ|β
fk (z) = z +
zk
m
(1 + (k − 1)µ)(1 + (k − 1)δ)
(k ≥ 2).
Then f (z) ∈ K̃(γ, µ, m, β), if and only if it can be expressed in the form
f (z) =
(3.2)
∞
X
P∞
k=1
λk fk (z)
f (z) =
k=1
λk fk (z) = z +
J
I
Page 9 of 22
Full Screen
Close
λk = 1.
Proof. Suppose that
∞
X
II
Go Back
k=1
where λk ≥ 0 and
JJ
∞
X
k=2
λk
|γ|β
zk .
m
(1 + (k − 1)µ)(1 + (k − 1)δ)
Then
∞
X
(1 + (k − 1)µ)(1 + (k − 1)δ)m
k=2
= |γ|β
∞
X
|γ|β
λk
(1 + (k − 1)µ)(1 + (k − 1)δ)m
λk
k=2
Subclass Of Analytic Functions
≤ |γ|β (1 − λ1 )
≤ |γ|β.
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
Thus, in view of Theorem 2.1, f (z) ∈ K̃(γ, µ, m, β).
Conversely, suppose that f (z) ∈ K̃(γ, µ, m, β). Setting
Title Page
∞
X
(1 + (k − 1)µ)(1 + (k − 1)δ)m
ak and λ1 = 1 −
λk ,
λk =
|γ|β
k=2
we obtain
f (z) =
∞
X
Contents
JJ
II
J
I
Page 10 of 22
λk fk (z).
k=1
Go Back
Full Screen
Corollary 3.2. The extreme points of K̃(γ, µ, m, β) are the functions f1 (z) = z and
|γ|β
zk
fk (z) = z +
m
(1 + (k − 1)µ)(1 + (k − 1)δ)
(k = 2, 3, . . . ).
Close
4.
Integral Mean Inequalities for a Differential Operator
Theorem 4.1. Let f (z) ∈ K̃(γ, µ, m, β) and suppose that
(4.1)
∞
X
[(1 − λ)k n + λ n Cn+k−1 ]|ak | ≤
k=2
|γ|β[(1 − λ)j n + λ n Cn+j−1 ]
.
(1 + µ(j − 1))(1 + δ(j − 1))m
Also, let the function
(4.2)
fj (z) = z +
Subclass Of Analytic Functions
|γ|β
zj
m
(1 + µ(j − 1))(1 + δ(j − 1))
S.M. Khairnar and Meena More
(j ≥ 2).
vol. 10, iss. 2, art. 57, 2009
If there exists an analytic function w(z) given by
w(z)j−1
∞
(1 + µ(j − 1))(1 + δ(j − 1))m X
[(1 − λ)k n + λ n Cn+k−1 ]ak z k−1 ,
=
|γ|β[(1 − λ)j n + λ n Cn+j−1 ] k=2
then for z = reiθ with 0 < r < 1,
Z 2π
Z 2π
n
p
|Dλ f (z)| dθ ≤
|Dλn fj (z)|p dθ
0
(0 ≤ λ ≤ 1, p > 0)
0
for the differential operator defined in (1.17).
Proof. By Definition 1.7 and by virtue of relation (1.17), we have
(4.3)
Dλn f (z) = z +
∞
X
[(1 − λ)k n + λ n Cn+k−1 ]ak z k .
k=2
Likewise,
(4.4)
Dλn fj (z) = z +
|γ|β[(1 − λ)j n + λ n Cn+j−1 ] j
z .
(1 + µ(j − 1))(1 + δ(j − 1))m
Title Page
Contents
JJ
II
J
I
Page 11 of 22
Go Back
Full Screen
Close
For z = reiθ , 0 < r < 1, we need to show that
p
Z 2π ∞
X
(4.5)
[(1 − λ)k n + λ n Cn+k−1 ]ak z k−1 dθ
1 +
0
k=2
Z 2π n
n
|γ|β[(1
−
λ)j
+
λ
C
]
n+j−1
j−1
1 +
dθ (p > 0).
≤
z
(1 + µ(j − 1))(1 + δ(j − 1))m
0
By applying Littlewood’s subordination theorem, it would be sufficient to show that
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
(4.6) 1 +
∞
X
[(1 − λ)k n + λ n Cn+k−1 ]ak z k−1
Title Page
k=2
n
≺1+
n
|γ|β[(1 − λ)j + λ Cn+j−1 ] j−1
z .
(1 + µ(j − 1))(1 + δ(j − 1))m
Set
∞
X
|γ|β[(1 − λ)j n + λ n Cn+j−1 ]
1+ [(1−λ)k n +λ n Cn+k−1 ]ak z k−1 = 1+
w(z)j−1 .
m
(1
+
µ(j
−
1))(1
+
δ(j
−
1))
k=2
Contents
JJ
II
J
I
Page 12 of 22
Go Back
We note that
Full Screen
j−1
(4.7) (w(z))
∞
(1 + µ(j − 1))(1 + δ(j − 1)) X
=
[(1 − λ)k n + λ n Cn+k−1 ]ak z k−1 ,
n
n
|γ|β[(1 − λ)j + λ Cn+j−1 ] k=2
m
and w(0) = 0. Moreover, we prove that the analytic function w(z) satisfies |w(z)| <
Close
1, z ∈ U
|w(z)|j−1
∞
(1 + µ(j − 1))(1 + δ(j − 1))m X
n
n
k−1 [(1
−
λ)k
+
λ
C
]a
z
≤
n+k−1 k
|γ|β[(1 − λ)j n + λ n Cn+j−1 ]
k=2
∞
(1 + µ(j − 1))(1 + δ(j − 1))m X
≤
[(1 − λ)k n + λ n Cn+k−1 ]|ak ||z|k−1
|γ|β[(1 − λ)j n + λ n Cn+j−1 ] k=2
∞
(1 + µ(j − 1))(1 + δ(j − 1)) X
≤ |z|
[(1 − λ)k n + λ n Cn+k−1 ]|ak |
n
n
|γ|β[(1 − λ)j + λ Cn+j−1 ] k=2
m
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
≤ |z| < 1 by hypothesis (4.1).
Title Page
This completes the proof of Theorem 4.1.
Contents
As a particular case of Theorem 4.1, we can derive the following result when
n = 0. That is, for Dλ0 f (z) = f (z).
Corollary 4.2. Let f (z) ∈ K̃(γ, µ, m, β) be given by (1.1), then for z = reiθ (0 <
r < 1)
Z 2π
Z 2π
iθ p
|f (re )| dθ ≤
|fj (reiθ )|p dθ
(p > 0),
0
where
JJ
II
J
I
Page 13 of 22
Go Back
0
Full Screen
|γ|β
fj (z) = z +
zj
m
(1 + µ(j − 1))(1 + δ(j − 1))
(j ≥ 2).
We conclude this section by observing that by specializing the parameters in Theorem 4.1, several integral mean inequalities can be deduced for S n f (z), Rn f (z),
the class of close-to-convex functions and the class of close-to-starlike functions as
mentioned in Remark 1.
Close
5.
Subordination Results for the Class T (η, f )
In proving the main subordination results we need the following lemma due to [8, p.
132].
Lemma 5.1. Let q be univalent in U and θ and φ be analytic in a domain D containing q(U ), with φ(w) 6= 0, when w ∈ q(U ). Set
Q(z) = zq 0 (z) · φ[q(z)],
h(z) = θ[q(z)] + Q(z)
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
and suppose that either:
vol. 10, iss. 2, art. 57, 2009
(i) Q is starlike or
(ii) h is convex.
In addition, assume that
0 0
zh (z)
θ (q(z))
Q0 (z)
(iii) Re Q(z) = Re φ(q(z)) + z Q(z) > 0.
Title Page
Contents
If P is analytic in U , with P (0) = q(0), P (U ) ⊂ D and
0
0
θ[P (z)] = zP (z) · φ[P (z)] ≺ θ[q(z)] + zq (z)φ[q(z)] = h(z)
JJ
II
J
I
Page 14 of 22
then P ≺ q, and q is the best dominant.
Lemma 5.2. Let q ∈ H = {f ∈ A : f (z) = 1 + b1 z + b2 z 2 + · · · } be univalent and
satisfy the following conditions: q(z) is convex and
1
zq 00 (z)
(5.1)
Re
+1 + 0
>0
η
q (z)
for η 6= 0 and all z ∈ U . For P ∈ H in U if
(5.2)
P (z) + ηzP 0 (z) ≺ q(z) + ηzq 0 (z),
then P ≺ q and q is the best dominant.
Go Back
Full Screen
Close
Proof. For η 6= 0 a real number, we define θ and φ by
θ(w) := w,
(5.3)
φ(w) := η,
D = {w : w 6= 0}
in Lemma 5.1. Then the functions
Q(z) = zq 0 (z)φ(q(z)) = ηzq 0 (z)
h(z) = θ(q(z)) + Q(z) = q(z) + ηzq 0 (z).
0 (z)
Using (5.1), we notice that Q(z) is starlike in U and Re zh
> 0 for all z ∈ U
Q(z)
and η 6= 0.
Thus the hypotheses of Lemma 5.1 are satisfied. Therefore, from (5.2) it follows
that P ≺ q and q is the best dominant.
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
Theorem 5.3. Let q ∈ H be univalent and satisfy the condition (5.1) in Lemma 5.2.
For Dm f if
JJ
II
T (η, Dm f ) ≺ q(z) + ηzq 0 (z),
J
I
(5.4)
then
Dm f (z)
z
≺ q(z) and q(z) is the best dominant.
Proof. Substituting P (z) =
Dm f (z)
,
z
where P (0) = 1, we have
P (z) + ηzP 0 (z) = T (η, Dm f ).
Thus using (5.4) and Lemma 5.2, we get the required result.
Corollary 5.4. Let q ∈ H be univalent and satisfy the conditions (5.1) in Lemma
5.2. For f ∈ A, if T (η, f ) ≺ q(z) + ηzq 0 (z), then f (z)
≺ q(z) and q is the best
z
dominant.
Proof. By substituting m = 0 in Theorem 5.3 we obtain Corollary 5.4.
Page 15 of 22
Go Back
Full Screen
Close
Corollary 5.5. Let q ∈ H be univalent and convex for all z ∈ U . For P ∈ H in U
if
P (z) + zP 0 (z) ≺ q(z) + zq 0 (z),
(5.5)
then P ≺ q, and q is the best dominant.
Proof. Take η = 1 in Lemma 5.2.
Corollary 5.6. Let q ∈ S be convex. For f ∈ A if
f 0 (z) ≺ q(z) + zq 0 (z),
then
f (z)
z
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
≺ q(z) and q is the best dominant.
Proof. Take η = 1 in Corollary 5.4.
Title Page
Contents
Corollary 5.7. Let q ∈ S satisfy
T (η, f ) ≺
where f ∈ A. Then
f (z)
z
Proof. Take q(z) =
1+(1−2α)z
1−z
1 + 2(η − α − ηα)z − (1 − 2α)z 2
(1 − z)2
∈ CS ∗ (α) and q is the best dominant.
in Corollary 5.4. Then it follows that
f (z)
1 + (1 − 2α)z
≺
,
z
1−z
n
o
> α. Therefore
which is equivalent to Re f (z)
z
f (z)
∈ CS ∗ (α).
z
JJ
II
J
I
Page 16 of 22
Go Back
Full Screen
Close
Corollary 5.8. Let q ∈ S satisfy
f 0 (z) ≺
where f ∈ A. Then
f (z)
z
1 + 2(1 − 2α)z − (1 − 2α)z 2
,
(1 − z)2
∈ CS ∗ (α) and q is the best dominant.
Proof. Substituting η = 1 in Corollary 5.7, we get the desired result.
Subclass Of Analytic Functions
Corollary 5.9. Let q ∈ S satisfy
f 0 (z) ≺
S.M. Khairnar and Meena More
2
1 + 2z − z
,
(1 − z)2
vol. 10, iss. 2, art. 57, 2009
where f ∈ A. Then f (z) ∈ CS ∗ and q is the best dominant.
Title Page
Proof. Take α = 0 in Corollary 5.8.
Contents
JJ
II
J
I
Page 17 of 22
Go Back
Full Screen
Close
6.
Partial Sums
In line with the earlier works of Silverman [12] and Silvia [13] on the partial sums
of analytic functions, we investigate in this section the partial sums of functions in
the class K(γ, µ, m, β). We obtain sharp lower bounds for the ratios of the real part
of f (z) to fN (z) and f 0 (z) to fN0 (z).
Theorem 6.1. Let f (z) of the form (1.1) belong to K(γ, µ, m, β) and h(N +1, γ, µ, m, β)
≥ 1. Then
f (z)
1
(6.1)
Re
≥1−
fN (z)
h(N + 1, γ, µ, m, β)
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
Title Page
and
(6.2)
Re
fN (z)
f (z)
Contents
h(N + 1, γ, µ, m, β)
≥
,
h(N + 1, γ, µ, m, β) + 1
where
(6.3)
h(k, γ, µ, m, β) =
(1 + (k − 1)µ)(1 + (k − 1)δ)m
.
|γ|β
f (z) = z +
1
z N +1
h(N + 1, γ, µ, m, β)
J
I
Go Back
Full Screen
Close
(N ∈ N \ {1}).
Proof. To prove (6.1), it is sufficient to show that
f (z)
1
1+z
h(N + 1, γ, µ, m, β)
− 1−
≺
fN (z)
h(N + 1, γ, µ, m, β)
1−z
II
Page 18 of 22
The result is sharp for every N , with extremal functions given by
(6.4)
JJ
(z ∈ U ).
By the subordination property (1.11), we can write
"
P
#
k−1
1+ ∞
a
z
1
k=2 k
− 1−
h(N + 1, γ, µ, m, β)
PN
h(N + 1, γ, µ, m, β)
1 + k=2 ak z k−1
=
1 + w(z)
.
1 − w(z)
Subclass Of Analytic Functions
Notice that w(0) = 0 and
S.M. Khairnar and Meena More
P∞
|w(z)| ≤
h(N + 1, γ, µ, m, β) k=N +1 |ak |
PN
P
2 − 2 k=2 |ak | − h(N + 1, γ, µ, m, β) ∞
k=N +1 |ak |
Title Page
|w(z)| < 1 if and only if
N
X
vol. 10, iss. 2, art. 57, 2009
|ak | + h(N + 1, γ, µ, m, β)
k=2
∞
X
Contents
|ak | ≤ 1.
k=N +1
In view of (2.1), we can equivalently show that
JJ
II
J
I
Page 19 of 22
N
X
(h(k, γ, µ, m, β) − 1)|ak |
k=2
+
∞
X
Go Back
Full Screen
((h(k, γ, µ, m, β) − h(N + 1, γ, µ, m, β))|ak | ≥ 0.
k=N +1
The above inequality holds because h(k, γ, µ, m, β) is a non-decreasing sequence.
This completes the proof of (6.1). Finally, it is observed that equality in (6.1) is
attained for the function given by (6.4) when z = re2πi/N as r → 1− . The proof of
(6.2) is similar to that of (6.1), and is hence omitted.
Close
Using a similar method, we can prove the following theorem.
Theorem 6.2. Let f (z) of the form (1.1) belong to K(γ, µ, m, β), and h(N +
1, γ, µ, m, β) ≥ N + 1. Then
0
f (z)
N +1
Re
≥1−
0
fN (z)
h(N + 1, γ, µ, m, β)
Subclass Of Analytic Functions
and
Re
fN0 (z)
f 0 (z)
≥
h(N + 1, γ, µ, m, β)
,
N + 1 + h(N + 1, γ, µ, m, β)
where h(N + 1, γ, µ, m, β) is given by (6.3). The result is sharp for every N , with
extremal functions given by (6.4).
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
JJ
II
J
I
Page 20 of 22
Go Back
Full Screen
Close
References
[1] F.M. AL-OBOUDI, On univalent functions defined by a generalized Sălăgean
operator, Internat. J. Math. and Mathematical Sciences, 27 (2004), 1429–1436.
[2] P.L. DUREN, Univalent functions, Grundlehren der Mathematischen
Wissenchaften, 259, Springer-Verlag, New York, USA, 1983.
[3] S.S. EKER AND H.Ö. GÜNEY, A new subclass of analytic functions involving Al-Oboudi differential operator, Journal of Inequalities and Applications,
(2008), Art. ID. 452057.
[4] S.M. KHAIRNAR AND M. MORE, A class of analytic functions defined by
Hurwitz-Lerch Zeta function, Internat. J. Math. and Computation, 1(8) (2008),
106–123.
[5] S.M. KHAIRNAR AND M. MORE, A new class of analytic and multivalent
functions defined by subordination property with applications to generalized
hypergeometric functions, Gen. Math., (to appear in 2009).
[6] Ö.Ö. KIHC, Sufficient conditions for subordination of multivalent functions,
Journal of Inequalities and Applications, (2008), Art. ID. 374756.
[7] J.E. LITTLEWOOD, On inequalities in the theory of functions, Proc. London
Math. Soc., 23 (1925), 481–519.
[8] S.S. MILLER AND P.T. MOCANU, Differential Subordinations, Theory and
Applications, Vol. 225, of Monographs and Textbooks in Pure and Applied
Mathematics, Marcel Dekker, New York, NY, USA, 2000.
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
JJ
II
J
I
Page 21 of 22
Go Back
Full Screen
Close
[9] G. MURUGUSUNDARAMOORTHY, T. ROSY AND K. MUTHUNAGAI,
Carlson-Shaffer operator and their applications to certain subclass of uniformly
convex functions, Gen. Math., 15(4) (2007), 131–143.
[10] R.K. RAINA AND D. BANSAL, Some properties of a new class of analytic functions defined in terms of a Hadamard product, J. Inequal. Pure and
Appl. Math., 9(1) (2008), Art. 22. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=957]
Subclass Of Analytic Functions
S.M. Khairnar and Meena More
[11] G.S. SĂLĂGEAN, Subclass of univalent functions in Complex Analysis, 5th
Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), Vol. 1013 of Lecture
Notes in Mathematics, Springer, Berlin, Germany, (1983), 362–372.
[12] H. SILVERMAN, Partial sums of starlike and convex functions, J. Math. Anal.
and Appl., 209 (1997), 221–227.
[13] E.M. SILVIA, Partial sums of convex functions of order α, Houston J. Math.,
Math. Soc., 11(3) (1985), 397–404.
vol. 10, iss. 2, art. 57, 2009
Title Page
Contents
JJ
II
J
I
Page 22 of 22
Go Back
Full Screen
Close