Journal of Inequalities in Pure and
Applied Mathematics
A UNIFIED TREATMENT OF CERTAIN SUBCLASSES OF
PRESTARLIKE FUNCTIONS
volume 6, issue 5, article 132,
2005.
MASLINA DARUS
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
Bangi 43600 Selangor D.E., Malaysia.
EMail: maslina@pkrisc.cc.ukm.my
URL: http://www.webspawner.com/users/maslinadarus/
Received 15 March, 2005;
accepted 04 October, 2005.
Communicated by: H.M. Srivastava
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper we introduce and study some properties of a unified class
U[µ, α, β, γ, λ, η, A, B] of prestarlike functions with negative coefficients in a unit
disk U. These properties include growth and distortion, radii of convexity, radii
of starlikeness and radii of close-to-convexity.
2000 Mathematics Subject Classification: 30C45.
Key words: Analytic functions, Prestarlike functions, radii of starlikeness, convexity
and close-to-convexity, Cauchy-Schwarz inequality.
This paper is based on the talk given by the author within the “International
Conference of Mathematical Inequalities and their Applications, I”, December 0608, 2004, Victoria University, Melbourne, Australia [http://rgmia.vu.edu.au/
conference]
The work presented here is supported by IRPA 09-02-02-10029 EAR.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Coefficient Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Growth and Distortion Theorem . . . . . . . . . . . . . . . . . . . . . . . . 9
4
Radii Convexity and Starlikeness . . . . . . . . . . . . . . . . . . . . . . . 12
References
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
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1.
Introduction
Let A denote the class of normalized analytic functions of the form:
f (z) = z +
(1.1)
∞
X
an z n ,
n=2
in the unit disk U = {z : |z| < 1}. Further let S denote the subclass of A
consisting of analytic and univalent functions f in the unit disk U . A function
f in S is said to be starlike of order α if and only if
0 zf (z)
(1.2)
Re
>α
f (z)
for some α (0 ≤ α < 1). We denote by S ∗ (α) the class of all starlike functions
of order α. It is well-known that S ∗ (α) ⊆ S ∗ (0) ≡ S ∗ .
Let the function
z
(1.3)
Sα (z) =
, (z ∈ U ; 0 ≤ α < 1)
(1 − z)2(1−α)
which is the extremal function for the class S ∗ (α). We also note that Sα (z) can
be written in the form:
∞
X
(1.4)
Sα (z) = z +
|cn (α)|z n ,
n=2
where
(1.5)
A Unified Treatment of Certain
Subclasses of Prestarlike
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cn (α) =
Πnj=2 (j
− 2α)
(n − 1)!
(n ∈ N {1}, N := {1, 2, 3, . . .}).
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We note that cn (α) is decreasing in α and satisfies


∞ if α < 12 ,



(1.6)
lim cn (α) = 1
if α = 12 ,
n→∞




0
if α > 12 .
Also a function f in S is said to be convex of order α if and only if
zf 00 (z)
>α
(1.7)
Re 1 + 0
f (z)
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
for some α (0 ≤ α < 1). We denote by K(α) the class of all convex functions
of order α. It is a fact that f ∈ K(α) if and only if zf 0 (z) ∈ S ∗ (α).
The well-known Hadamard product (or convolution)
of two functions f (z)
P
n
given by (1.1) and g(z) given by g(z) = z + ∞
b
z
is
defined by
n=2 n
(1.8)
(f ∗ g)(z) = z +
∞
X
n
an b n z ,
(z ∈ U ).
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Let R[µ, α, β, γ, λ, A, B] denote the class of prestarlike functions satisfying
the following condition
zHλ0 (z)
−1
H
(z)
λ
< β,
(1.9)
0
0
zHλ (z)
λ (z)
2γ(B − A) zH
− µ − B Hλ (z) − 1 Hλ (z)
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where Hλ (z) = (1 − λ)h(z) + λzh0 (z), λ ≥ 0, h = f ∗ Sα , 0 < β ≤ 1,
0 ≤ µ < 1, and

 B , µ=
6 0,
B
2(B−A)µ
<γ≤
1,
2(B − A)
µ=0
for fixed −1 ≤ A ≤ B ≤ 1 and 0 < B ≤ 1.
We also note that a function f is a so-called α-prestarlike (0 ≤ α < 1)
function if, and only if, h = f ∗ Sα ∈ S ∗ (α) which was first introduced by
Ruscheweyh [3], and was rigorously studied by Silverman and Silvia [4], Owa
and Ahuja [5] and Uralegaddi and Sarangi [6]. Further, a function f ∈ A is in
the class C[µ, α, β, γ, λ, A, B] if and only if, zf 0 (z) ∈ R[µ, α, β, γ, λ, A, B].
Let T denote the subclass of A consisting of functions of the form
f (z) = z −
(1.10)
∞
X
an z n ,
(an ≥ 0).
n=2
Let us write
RT [µ, α, β, γ, λ, A, B] = R[µ, α, β, γ, λ, A, B] ∩ T
A Unified Treatment of Certain
Subclasses of Prestarlike
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Maslina Darus
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and
CT [µ, α, β, γ, λ, A, B] = C[µ, α, β, γ, λ, A, B] ∩ T
where T is the class of functions of the form (1.10) that are analytic and univalent in U . The idea of unifying the study of classes RT [µ, α, β, γ, λ, A, B]
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and CT [µ, α, β, γ, λ, A, B] thus, forming a new class U[µ, α, β, γ, λ, η, A, B] is
somewhat or rather motivated from the work of [1] and [2].
In this paper, we will study the unified presentation of prestarlike functions
belonging to U[µ, α, β, γ, λ, η, A, B] which include growth and distortion theorem, radii of convexity, radii of starlikeness and radii of close-to-convexity.
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
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2.
Coefficient Inequality
Our main tool in this paper is the following result, which can be easily proven,
and the details are omitted.
Lemma 2.1. Let the function f be defined by (1.10). Then f ∈ RT [µ, α, β,
γ, λ, A, B] if and only if
(2.1)
∞
X
Λ(n, λ)D[n, β, γ, A, B]|an |cn (α) ≤ E[β, γ, µ, A, B]
n=2
where
Λ(n, λ) = (1 + (n − 1)λ),
D[n, β, γ, A, B] = n − 1 + 2βγ(n − µ)(B − A) − Bβ(n − 1),
E[β, γ, µ, A, B] = 2βγ(1 − µ)(B − A).
A Unified Treatment of Certain
Subclasses of Prestarlike
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Maslina Darus
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The result is sharp.
Next, by observing that
(2.2)
f ∈ CT [µ, α, β, γ, λ, A, B] ⇔ zf 0 (z) ∈ RT [µ, α, β, γ, λ, A, B],
we gain the following Lemma 2.2.
Lemma 2.2. Let the function f be defined by (1.10). Then f ∈ CT [µ, α, β,
γ, λ, A, B] if and only if
(2.3)
∞
X
n=2
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nΛ(n, λ)D[n, β, γ, A, B]|an |cn (α) ≤ E[β, γ, µ, A, B]
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where
Λ(n, λ) = (1 + (n − 1)λ),
D[n, β, γ, A, B] = n − 1 + 2βγ(n − µ)(B − A) − Bβ(n − 1),
E[β, γ, µ, A, B] = 2βγ(1 − µ)(B − A)
and cn (α) given by (1.5).
In view of Lemma 2.1 and Lemma 2.2, we unified the classes RT [µ, α, β,
γ, λ, A, B] and CT [µ, α, β, γ, λ, A, B] and so a new class U[µ, α, β, γ, λ, η, A, B]
is formed. Thus we say that a function f defined by (1.10) belongs to
U[µ, α, β, γ, λ, η, A, B] if and only if,
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
(2.4)
∞
X
(1 − η + nη)Λ(n, λ)D[n, β, γ, A, B]|an |cn (α) ≤ E[β, γ, µ, A, B],
n=2
(0 ≤ α < 1; 0 < β ≤ 1; η ≥ 0; λ ≥ 0; −1 ≤ A ≤ B ≤ 1 and 0 < B ≤ 1),
where Λ(n, λ), D[n, β, γ, A, B] , E[β, γ, µ, A, B] and cn (α) are given in (Lemma
2.1 and Lemma 2.2) and given by (1.5), respectively.
Clearly, we obtain
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U[µ, α, β, γ, λ, η, A, B] = (1 − η)RT [µ, α, β, γ, A, B] + ηCT [µ, α, β, γ, A, B],
so that
U[µ, α, β, γ, λ, 0, A, B] = RT [µ, α, β, γ, A, B],
and
U[µ, α, β, γ, λ, 1, A, B] = CT [µ, α, β, γ, A, B].
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3.
Growth and Distortion Theorem
A distortion property for function f in the class U[µ, α, β, γ, λ, η, A, B] is given
as follows:
Theorem 3.1. Let the function f defined by (1.10) be in the class U[µ, α, β,
γ, λ, η, A, B], then
(3.1) r −
E[β, γ, µ, A, B]
r2
2(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
E[β, γ, µ, A, B]
≤ |f (z)| ≤ r +
r2 ,
2(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
(η ≥ 0;
0 ≤ α < 1;
0 < β ≤ 1;
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
z ∈ U)
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(3.2) 1 −
E[β, γ, µ, A, B]
r
(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
E[β, γ, µ, A, B]
≤ |f 0 (z)| ≤ 1 +
r,
(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
(η ≥ 0;
0 ≤ α < 1;
0 < β ≤ 1;
z ∈ U ).
The bounds in (3.1) and (3.2) are attained for the function f given by
E[β, γ, µ, A, B]
f (z) = z −
z2.
2(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
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Proof. Observing that cn (α) defined by (1.5) is nondecreasing for (0 ≤ α < 1),
we find from (2.4) that
(3.3)
∞
X
|an | ≤
n=2
E[β, γ, µ, A, B]
.
2(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
Using (1.10) and (3.3), we readily have (z ∈ U )
|f (z)| ≥ |z| −
∞
X
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
|an |cn (α)|z n |
n=2
≥ |z| − |z 2 |
∞
X
Maslina Darus
|an |cn (α),
n=2
≥r−
E[β, γ, µ, A, B]
r2 ,
2(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
|z| = r < 1
Contents
and
|f (z)| ≤ |z| +
∞
X
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n=2
≤ |z| + |z 2 |
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X
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|an |cn (α),
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n=2
E[β, γ, µ, A, B]
≤r+
r2 ,
2(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
which proves the assertion (3.1) of Theorem 3.1.
|z| = r < 1,
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Also, from (1.10), we find for z ∈ U that
0
|f (z)| ≥ 1 −
∞
X
n|an |cn (α)|z n−1 |
n=2
≥ 1 − |z|
∞
X
n|an |cn (α),
n=2
≥1−
E[β, γ, µ, A, B]
r,
(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
|z| = r < 1
and
0
|f (z)| ≤ 1 +
∞
X
Maslina Darus
n|an |cn (α)|z
n−1
|
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n=2
≤ 1 + |z|
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
∞
X
Contents
n|an |cn (α),
n=2
E[β, γ, µ, A, B]
r,
≤1+
2(1 + η)Λ(2, λ)D[2, β, γ, A, B](1 − α)
which proves the assertion (3.2) of Theorem 3.1.
|z| = r < 1,
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4.
Radii Convexity and Starlikeness
The radii of convexity for class U[µ, α, β, γ, λ, η, A, B] is given by the following
theorem.
Theorem 4.1. Let the function f be in the class U[µ, α, β, γ, λ, η, A, B]. Then
the function f is convex of order ρ (0 ≤ ρ < 1) in the disk |z| < r1 (µ, α, β,
γ, λ, η, A, B) = r1 , where
1
2(1 − α)(1 − ρ)Λ(n, λ)D[n, β, γ, A, B](1 − η + nη) n−1
.
(4.1) r1 = inf
n
n(n − ρ)E[β, γ, µ, A, B]
Proof. It sufficient to show that
00 P∞
zf (z) − n=2 n(n − 1)an z n−1 f 0 (z) = 1 − P∞ nan z n−1 n=2
P∞
n(n − 1)an |z|n−1
n=2P
(4.2)
≤
1−n
1− ∞
n=2 nan |z|
which implies that
(4.3)
00 P∞
zf (z) − 1)|an ||z|n−1
n=2 n(n
P
(1 − ρ) − 0
≥
(1
−
ρ)
−
n−1
f (z) 1− ∞
n=2 nan z
P∞
(1 − ρ) − n=2 n(n − ρ)an |z|n−1
P
.
=
n−1
1− ∞
n=2 nan |z|
Hence from (4.1), if
(4.4)
|z|n−1 ≤
(1 − ρ) 2(1 − α)Λ(n, λ)D[n, β, γ, A, B](1 − η + nη)
·
,
n(n − ρ)
E[β, γ, µ, A, B]
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
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and according to (2.4)
∞
X
(4.5)
1−ρ−
n(n − ρ)an |z|n−1 > 1 − ρ − (1 − ρ) = ρ.
n=2
Hence from (4.3), we obtain
00 zf (z) f 0 (z) < 1 − ρ
Therefore
zf 00 (z)
Re 1 + 0
> 0,
f (z)
which shows that f is convex in the disk |z| < r1 (µ, α, β, γ, λ, η, ρ, A, B).
By setting η = 0 and η = 1, we have the Corollary 4.2 and the Corollary
4.3, respectively.
Corollary 4.2. Let the function f be in the class RT (µ, α, β, γ, λ, ρ, A, B).
Then the function f is convex of order ρ (0 ≤ ρ < 1) in the disk |z| <
r2 (µ, α, β, γ, λ, ρ, A, B) = r2 , where
1
2(1 − α)(1 − ρ)Λ(n, λ)D[n, β, γ, A, B] n−1
(4.6)
r2 = inf
.
n
n(n − ρ)E[β, γ, µ, A, B]
Corollary 4.3. Let the function f be in the class CT (µ, α, β, γ, λ, ρ, A, B).
Then the function f is convex of order ρ (0 ≤ ρ < 1) in the disk |z| <
r3 (µ, α, β, γ, λ, ρ, A, B) = r3 , where
1
2(1 − α)(1 − ρ)Λ(n, λ)D[n, β, γ, A, B] n−1
(4.7)
r3 = inf
.
n
(n − ρ)E[β, γ, µ, A, B]
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
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Theorem 4.4. Let the function f be in the class U[µ, α, β, γ, λ, η, A, B]. Then
the function f is starlike of order ρ (0 ≤ ρ < 1) in the disk |z| < r4 (µ, α, β, γ,
λ, η, A, B) = r4 , where
1
2(1 − α)(1 − ρ)Λ(n, λ)D[n, β, γ, A, B](1 − η + nη) n−1
(4.8) r4 = inf
.
n
(n − ρ)E[β, γ, µ, A, B]
Proof. It sufficient to show that
0
zf (z)
<1−ρ
−
1
f (z)
Using a similar method to Theorem 4.1 and making use of (2.4), we get (4.8).
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
Letting η = 0 and η = 1, we have the Corollary 4.5 and the Corollary 4.6,
respectively.
Corollary 4.5. Let the function f be in the class RT (µ, α, β, γ, λ, ρ, A, B).
Then the function f is starlike of order ρ (0 ≤ ρ < 1) in the disk |z| <
r5 (µ, α, β, γ, λ, ρ, A, B) = r5 , where
1
2(1 − α)(1 − ρ)Λ(n, λ)D[n, β, γ, A, B] n−1
(4.9)
r5 = inf
.
n
(n − ρ)E[β, γ, µ, A, B]
Corollary 4.6. Let the function f be in the class CT (µ, α, β, γ, λ, ρ, A, B).
Then the function f is starlike of order ρ (0 ≤ ρ < 1) in the disk |z| <
r6 (µ, α, β, γ, λ, ρ, A, B) = r6 , where
1
2n(1 − α)(1 − ρ)Λ(n, λ)D[n, β, γ, A, B] n−1
(4.10)
r6 = inf
.
n
(n − ρ)E[β, γ, µ, A, B]
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Last, but not least we give the following result.
Theorem 4.7. Let the function f be in the class U[µ, α, β, γ, λ, η, A, B]. Then
the function f is close-to-convex of order ρ (0 ≤ ρ < 1) in the disk |z| <
r7 (µ, α, β, γ, λ, η, A, B) = r7 , where
(4.11) r7 = inf
n
2(1 − α)(1 − ρ)Λ(n, λ)D[n, β, γ, A, B](1 − η + nη)
nE[β, γ, µ, A, B]
1
n−1
.
Proof. It sufficient to show that
|f 0 (z) − 1| < 1 − ρ.
Using a similar technique to Theorem 4.1 and making use of (2.4), we get
(4.11).
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
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References
[1] H.M. SRIVASTAVA, H.M. HOSSEN AND M.K. AOUF, A unified presentation of some classes meromorphically multivalent functions, Computers
Math. Applic., 38 (1999), 63–70.
[2] R.K. RAINA AND H.M. SRIVASTAVA, A unified presentation of certain subclasses of prestarlike functions with negative functions, Computers
Math. Applic., 38 (1999), 71–78.
[3] S. RUSCHEWEYH, Linear operator between classes of prestarlike functions, Comm. Math. Helv., 52 (1977), 497–509.
[4] H. SILVERMAN AND E.M. SILVIA, Prestarlike functions with negative
coefficients, Internat. J. Math. Math. Sci., 2 (1979), 427–439.
A Unified Treatment of Certain
Subclasses of Prestarlike
Functions
Maslina Darus
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[5] S. OWA AND O.P. AHUJA, An application of the fractional calculus, Math.
Japon., 30 (1985), 947–955.
[6] B.A. URALEGADDI AND S.M. SARANGI, Certain generalization of
prestarlike functions with negative coefficients, Ganita, 34 (1983), 99–105.
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