Journal of Inequalities in Pure and
Applied Mathematics
SUBORDINATION RESULTS FOR A CLASS OF ANALYTIC
FUNCTIONS DEFINED BY A LINEAR OPERATOR
volume 7, issue 4, article 134,
2006.
B.A. FRASIN
Department of Mathematics
Al al-Bayt University
P.O. Box: 130095
Mafraq, Jordan
Received 21 June, 2005;
accepted 01 June, 2006.
Communicated by: A. Sofo
EMail: bafrasin@yahoo.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper, we derive several interesting subordination results for certain
class of analytic functions defined by the linear operator L(a, c)f(z) which introduced and studied by Carlson and Shaffer [2].
2000 Mathematics Subject Classification: Primary 30C45; Secondary 30A10,
30C80.
Key words: Analytic functions, Hadamard product, Subordinating factor sequence.
Subordination Results for a
Class of Analytic Functions
Defined by a Linear Operator
Contents
1
Introduction and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
B.A. Frasin
3
8
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1.
Introduction and Definitions
Let A denote the class of functions of the form:
∞
X
(1.1)
f (z) = z +
an z n
n=2
which are analytic in the open unit disc ∆ = {z : |z| < 1} . For two functions
f (z) and g(z) given by
(1.2)
f (z) = z +
∞
X
an z
n
and g(z) = z +
n=2
∞
X
cn z n
n=2
their Hadamard product (or convolution) is defined by
(f ∗ g)(z) := z +
(1.3)
∞
X
an c n z n .
n=2
Define the function φ(a, c; z) by
(1.4) φ(a, c; z) :=
∞
X
(a)n
n=0
(c)n
z n+1
(c ∈
/ Z−
0 := {0, −1, −2, . . .}, z ∈ ∆),
where (λ)n is the Pochhammer symbol given, in terms of Gamma functions,
(1.5)
Γ(λ + n)
(λ)n :=
Γ(λ)
(
1,
n = 0,
=
λ(λ + 1)(λ + 2) . . . (λ + n − 1), n ∈ N : {1, 2, . . .}.
Subordination Results for a
Class of Analytic Functions
Defined by a Linear Operator
B.A. Frasin
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Corresponding to the function φ(a, c; z), Carlson and Shaffer [2] introduced
a linear operator L(a, c) : A → A by
L(a, c)f (z) := φ(a, c; z) ∗ f (z),
(1.6)
or, equivalently, by
L(a, c)f (z) := z +
∞
X
(a)n
n=1
(c)n
an+1 z n+1
(z ∈ ∆).
Note that L(1, 1)f (z) = f (z), L(2, 1)f (z) = zf 0 (z) and L(3, 1)f (z) = zf 0 (z)+
1 2 00
z f (z).
2
For −1 ≤ α < 1, β ≥ 0, we let L(a, c; α, β) consist of functions f in A
satisfying the condition
(1.7) Re
aL(a + 1, c)f (z)
− (a − 1)
L(a, c)f (z)
aL(a + 1, c)f (z)
>β
− a + α,
L(a, c)f (z)
Subordination Results for a
Class of Analytic Functions
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B.A. Frasin
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(z ∈ ∆)
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The family L(a, c; α, β) is of special interest for it contains many wellknown as well as many new classes of analytic univalent functions. For L(1, 1; α, 0),
we obtain the family of starlike functions of order α (0 ≤ α < 1) and L(2, 1; α, 0)
is the family of convex functions of order α (0 ≤ α < 1). For L(1, 1; 0, β) and
L(2, 1; 0, β), we obtain the class of uniformly β- starlike functions and uniformly β- convex functions, respectively, introduced by Kanas and Winsiowska
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([3],[4]) (see also the work of Kanas and Srivastava [5], Goodman ([7],[8]),
Rønning ([10],[11]), Ma and Minda [9] and Gangadharan et al. [6]).
Before we state and prove our main result we need the following definitions
and lemmas.
Definition 1.1 (Subordination Principle). Let g(z) be analytic and univalent
in ∆. If f (z) is analytic in ∆, f (0) = g(0), and f (∆) ⊂ g(∆), then we see that
the function f (z) is subordinate to g(z) in ∆, and we write f (z) ≺ g(z).
Definition 1.2 (Subordinating Factor Sequence). A sequence {bn }∞
n=1 of complex numbers is called a subordinating factor sequence if, whenever f (z) is
analytic , univalent and convex in ∆, we have the subordination given by
(1.8)
∞
X
bn an z n ≺ f (z)
(z ∈ ∆, a1 = 1).
n=2
Lemma 1.1 ([14]). The sequence {bn }∞
n=1 is a subordinating factor sequence if
and only if
(
)
∞
X
(1.9)
Re 1 + 2
bn z n > 0
(z ∈ ∆).
Subordination Results for a
Class of Analytic Functions
Defined by a Linear Operator
B.A. Frasin
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n=1
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Lemma 1.2. If
(1.10)
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∞
X
σn (a, c; α, β) |an | ≤ 1 − α
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n=2
J. Ineq. Pure and Appl. Math. 7(4) Art. 134, 2006
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where, for convenience,
(1.11)
(1 + β)(a)n + [1 − α − a(1 + β)](a)n−1
(c)n−1
(−1 ≤ α < 1; β ≥ 0, n ≥ 2),
σn (a, c; α, β) :=
then f (z) ∈ L(a, c; α, β).
Proof. It suffices to show that
aL(a + 1, c)f (z)
aL(a
+
1,
c)f
(z)
β − a − Re
− a ≤ 1 − α.
L(a, c)f (z)
L(a, c)f (z)
We have
aL(a + 1, c)f (z)
aL(a + 1, c)f (z)
β
− a − Re
−a
L(a, c)f (z)
L(a, c)f (z)
aL(a + 1, c)f (z)
≤ (1 + β) − a
L(a, c)f (z)
P∞ a(a+1)n−1 −a(a)n−1 (1 + β) n=2
|an | |z|n−1
(c)n−1
≤
P
n−1
(a)n−1
1− ∞
n=2 (c)n−1 |an | |z|
P
(a)n −a(a)n−1
(1 + β) ∞
|an |
n=2
(c)n−1
≤
.
P
(a)n−1
1− ∞
|a
|
n
n=2 (c)n−1
The last expression is bounded above by 1 − α if
∞
X
(1 + β)(a)n + [1 − α − a(1 + β)](a)n−1
n=2
(c)n−1
Subordination Results for a
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B.A. Frasin
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|an | ≤ 1 − α
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and the proof is complete.
Let L? (a, c; α, β) denote the class of functions f (z) ∈ A whose coefficients
satisfy the condition (1.10). We note that L? (a, c; α, β) ⊆ L(a, c; α, β).
Subordination Results for a
Class of Analytic Functions
Defined by a Linear Operator
B.A. Frasin
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2.
Main Theorem
Employing the techniques used earlier by Srivastava and Attiya [13], Attiya [1]
and Singh [12], we state and prove the following theorem.
Theorem 2.1. Let the function f (z) defined by (1.1) be in the class L? (a, c; α, β)
where −1 ≤ α < 1 ; β ≥ 0; a > 0; c > 0. Also let K denote the familiar class
of functions f (z) ∈ A which are also univalent and convex in ∆. Then
(2.1)
σ2 (a, c; α, β)
(f ∗ g)(z) ≺ g(z)
2[1 − α + σ2 (a, c; α, β)]
( z ∈ ∆; g ∈ K),
and
(2.2)
1 − α + σ2 (a, c; α, β)
,
Re(f (z)) > −
σ2 (a, c; α, β)
The constant
σ2 (a,c;α,β)
2[1−α+σ2 (a,c;α,β)]
(2.3)
B.A. Frasin
(z ∈ ∆).
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is the best estimate.
Proof. Let f (z) ∈ L? (a, c; α, β) and let g(z) = z +
Contents
P∞
n=2 cn z
n
∈ K. Then
σ2 (a, c; α, β)
(f ∗ g)(z)
2[1 − α + σ2 (a, c; α, β)]
=
Subordination Results for a
Class of Analytic Functions
Defined by a Linear Operator
σ2 (a, c; α, β)
2[1 − α + σ2 (a, c; α, β)]
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z+
∞
X
!
an c n z n .
n=2
Thus, by Definition 1.2, the assertion of our theorem will hold if the sequence
∞
σ2 (a, c; α, β)
(2.4)
an
2[1 − α + σ2 (a, c; α, β)]
n=1
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is a subordinating factor sequence, with a1 = 1. In view of Lemma 1.1, this will
be the case if and only if
(
)
∞
X
σ2 (a, c; α, β)
(2.5)
Re 1 + 2
an z n > 0
(z ∈ ∆).
2[1
−
α
+
σ
(a,
c;
α,
β)]
2
n=1
Now
(
Re 1 +
∞
X
)
σ2 (a, c; α, β)
an z n
1 − α + σ2 (a, c; α, β) n=1
σ2 (a, c; α, β)
z
= Re 1 +
1 − α + σ2 (a, c; α, β)
)
∞
X
1
n
+
σ2 (a, c; α, β)an z
1 − α + σ2 (a, c; α, β) n=1
σ2 (a, c; α, β)
≥1−
r
1 − α + σ2 (a, c; α, β)
)
∞
X
1
−
σn (a, c; α, β)an rn .
1 − α + σ2 (a, c; α, β) n=1
Since σn (a, c; α, β) is an increasing function of n (n ≥ 2)
σ2 (a, c; α, β)
r
1−
1 − α + σ2 (a, c; α, β)
)
∞
X
1
n
−
σn (a, c; α, β)an r
1 − α + σ2 (a, c; α, β) n=1
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B.A. Frasin
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>1−
σ2 (a, c; α, β)
1−α
r−
r
1 − α + σ2 (a, c; α, β)
1 − α + σ2 (a, c; α, β)
(|z| = r)
> 0.
Thus (2.5) holds true in ∆. This proves the inequality (2.1).PThe inequality (2.2)
z
n
follows by taking the convex function g(z) = 1−z
=z+ ∞
in (2.1). To
n=2 z
σ2 (a,c;α,β)
prove the sharpness of the constant 2[1−α+σ2 (a,c;α,β)] , we consider the function
f0 (z) ∈ L? (a, c; α, β) given by
(2.6)
1−α
f0 (z) = z −
z2
σ2 (a, c; α, β)
Subordination Results for a
Class of Analytic Functions
Defined by a Linear Operator
(−1 ≤ α < 1; β ≥ 0).
B.A. Frasin
Thus from (2.1), we have
(2.7)
z
σ2 (a, c; α, β)
f0 (z) ≺
.
2[1 − α + σ2 (a, c; α, β)]
1−z
It can easily verified that
1
σ2 (a, c; α, β)
f0 (z)
=−
(2.8)
min Re
2[1 − α + σ2 (a, c; α, β)]
2
This shows that the constant
σ2 (a,c;α,β)
2[1−α+σ2 (a,c;α,β)]
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(z ∈ ∆),
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Corollary 2.2. Let the function f (z) defined by (1.1) be in the class L (1, 1; α, β)
and satisfy the condition
(2.9)
n=2
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is best possible.
?
∞
X
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[n(1 + β) − (α + β)] |an | ≤ 1 − α
J. Ineq. Pure and Appl. Math. 7(4) Art. 134, 2006
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then
β+2−α
(f ∗ g)(z) ≺ g(z)
2(β + 3 − 2α)
(−1 ≤ α < 1; β ≥ 0; z ∈ ∆; g ∈ K)
(2.10)
and
Re(f (z)) > −
(2.11)
The constant
β+2−α
2(β+3−2α)
β + 3 − 2α
,
β+2−α
(z ∈ ∆).
Subordination Results for a
Class of Analytic Functions
Defined by a Linear Operator
is the best estimate.
Corollary 2.3. Let the function f (z) defined by (1.1) be in the class L? (1, 1; α, 0)
and satisfy the condition
∞
X
(2.12)
(n − α) |an | ≤ 1 − α,
then
2−α
(f ∗ g)(z) ≺ g(z)
6 − 4α
(z ∈ ∆; g ∈ K)
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3 − 2α
Re(f (z)) > −
,
2−α
(2.14)
The constant
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n=2
(2.13)
B.A. Frasin
2−α
6−4α
is the best estimate.
(z ∈ ∆).
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Putting α = 0 in Corollary 2.3, we obtain
Corollary 2.4 ([12]). Let the function f (z) defined by (1.1) be in the class
L? (1, 1; 0, 0) and satisfy the condition
∞
X
(2.15)
n |an | ≤ 1
n=2
then
(2.16)
1
(f ∗ g)(z) ≺ g(z)
3
(z ∈ ∆; g ∈ K)
and
B.A. Frasin
3
Re(f (z)) > − ,
2
(2.17)
(z ∈ ∆).
The constant 1/3 is the best estimate.
Corollary 2.5. Let the function f (z) defined by (1.1) be in the class L? (2, 1; α, β)
and satisfy the condition
(2.18)
∞
X
n[n(1 + β) − (α + β)] |an | ≤ 1 − α,
n=2
then
(2.19)
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β+2−α
(f ∗ g)(z) ≺ g(z)
2β + 5 − 3α
(−1 ≤ α < 1; β ≥ 0; z ∈ ∆; g ∈ K)
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and
(2.20)
The constant
Re(f (z)) > −
β+2−α
2β+5−3α
2β + 5 − 3α
,
2(β + 2 − α)
(z ∈ ∆).
is the best estimate.
Corollary 2.6. Let the function f (z) defined by (1.1) be in the class L? (2, 1; α, 0)
and satisfy the condition
∞
X
(2.21)
Subordination Results for a
Class of Analytic Functions
Defined by a Linear Operator
n(n − α) |an | ≤ 1 − α,
n=2
B.A. Frasin
then
(2.22)
2−α
(f ∗ g)(z) ≺ g(z)
5 − 3α
(z ∈ ∆; g ∈ K)
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and
Re(f (z)) > −
(2.23)
The constant
2−α
5−3α
5 − 3α
,
2(2 − α)
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(z ∈ ∆).
is the best estimate.
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Putting α = 0 in Corollary 2.6, we obtain
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?
Corollary 2.7. Let the function f (z) defined by (1.1) be in the class L (2, 1; 0, 0)
and satisfy the condition
(2.24)
∞
X
n=2
n2 |an | ≤ 1
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then
(2.25)
2
(f ∗ g)(z) ≺ g(z)
5
(z ∈ ∆; g ∈ K)
and
(2.26)
Re(f (z)) >
−5
,
4
The constant 2/5 is the best estimate.
(z ∈ ∆).
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B.A. Frasin
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References
[1] A.A. ATTIYA, On some application of a subordination theorems, J. Math.
Anal. Appl., 311 (2005), 489–494.
[2] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15(4) (1984), 737–745.
[3] S. KANAS AND A. WISNIOWSKA, Conic regions and k- uniform convexity, J. Comput. Appl. Math., 105 (1999), 327–336.
[4] S. KANAS AND A. WISNIOWSKA, Conic regions and k- starlike functions, Rev. Roumaine Math. Pures Appl., 45(4) (2000), 647–657.
[5] S. KANAS AND H.M. SRIVASTAVA, Linear operators associated with kuniformly convex functions, Integral Transform. Spec. Funct., 9 (2000),
121–132.
[6] A. GANGADHARAN, T.N. SHANMUGAN AND H.M. SRIVASTAVA,
Generalized Hypergeometric functions associated with k−uniformly convex functions, Comput. Math. App., 44 (2002), 1515–1526.
Subordination Results for a
Class of Analytic Functions
Defined by a Linear Operator
B.A. Frasin
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[7] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 56
(1991), 87–92.
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[8] A.W. GOODMAN, On uniformly starlike functions, J. Math. Anal. Appl.,
155 (1991), 364–370.
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[9] W.C. MA AND D. MINDA, Uniformly convex functions, Ann. Polon.
Math., 57(2) (1992), 165–175.
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[10] F. RØNNING, Uniformly convex functions and a corresponding class of
starlike functions, Proc. Amer. Math. Soc., 118(1) (1993), 189–196.
[11] F. RØNNING, On starlike functions associated with parabolic regions,
Ann Univ. Mariae Curie-Sklodowska Sect. A, 45 (1991), 117–122.
[12] S. SINGH, A subordination theorems for spirallike functions, IJMMS,
24(7) (2000), 433–435.
[13] H.M. SRIVASTAVA AND A.A. ATTIYA, Some subordination results associated with certain subclasses of analytic functions, J. Inequal. Pure Appl.
Math., 5(4) (2004), Art. 82. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=434].
[14] H.S. WILF, Subordinating factor sequence for convex maps of the unit
circle, Proc. Amer. Math. Soc., 12 (1961), 689–693, MR0125214 (23
#A2519).
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