General Mathematics Vol. 17, No. 3 (2009), 63–70
Subordination by p-valent convex functions
H. A. Alkharsani, S. S. Alhajry
Abstract
We obtain an interesting subordination relation for analytic pvalent functions by using subordinating factor sequence (bk )∞
1 .
2000 Mathematics Subject Classification:30C45
Key words and phrases:p-valent functions, subordination,
subordinating factor sequence.
1
Introduction
Let A(p) denote the class of functions of the form
f (z) = z p +
(1)
∞
X
ak z p+k−1 ,
k=2
1
Received 29 June, 2008
Accepted for publication (in revised form) 15 September, 2008
63
1
64
H. A. Alkharsani, S. S. Alhajry
which are analytic in the open unit disk D = {z : z ∈ C; |z| < 1} and p ∈ N .
Consider also its subclasses C(p), S ∗ (p) consisting of p−valent convex and
starlike functions respectivly, where C(1) ≡ C, S ∗ (1) ≡ S ∗ , the classes of
univalent convex and starlike functions .
For f (z) given by (1) and g(z) given by
p
(2)
g(z) = z +
∞
X
gk z p+k−1 ,
k=2
the convolution (or Hadamard product) of f and g, denoted by f ∗ g, is
defined by
(f ∗ g)(z) = z p +
∞
X
ak gk z p+k−1 .
k=2
Let g(z) given by (2) be a fixed function, with gk ≥ g2 > 0 (k ≥ 2), γ < 1,
and let
Tg (γ) := {f (z) ∈ A(p) :
∞
X
|ak gk | < p(p − γ)}.
k=2
Definition 1 A sequence (bk )∞
1 of complex numbers is said to be a subordinating factor sequence if for every convex p-valent function f (z) given by
(1)
∞
X
ak bk z p+k−1 ≺ f (z)
(≺ means subordinate).
k=2
Theorem 1 A sequence (bk )∞
1 of complex numbers is a subordinating factor
sequence if and only if
(
(3)
Re 1 + 2
∞
X
k=2
)
bk z k+p−1
> 0.
Subordination by p-valent convex functions
65
Proof. Assume that the sequence (bk )∞
1 of complex numbers is a subordinating factor sequence. Then
∞
X
bk z
p+k−1
≺
k=1
then
Re
∞
X
z p+k−1 = z p
k=1
̰
X
!
bk z
p+k−1
µ
≥ Re
k=1
and since
µ
Re
zp
(1 − z)
¶
≥ (−
z
,
(1 − z)
zp
(1 − z)
¶
,
cos pπ
), −1 ≤ cos pπ ≤ 1
2
which is equivalent to
(
Re
∞
X
)
bk z p+k−1
k=2
2
1
> − , (|z| < 1).
2
Subordination with convex functions
We begin with the following subordination result.
Theorem 2 If f (z) ∈ Tg (γ) and h(z) ∈ C(p), then
g2
(f ∗ h)(z) ≺ h(z),
2(g2 + p(p − γ))
(4)
(5)
Re(f (z)) > −
(g2 + p(p − γ))
, z ∈ D.
g2
The constant factor
g2
2(g2 + p(p − γ))
in the subordination result (4) cannot be replaced by a larger number.
66
H. A. Alkharsani, S. S. Alhajry
Proof. Let G(z) = z +
P∞
k=2
g2 z p+k−1 . Since Tg (γ) ⊆ TG (γ), our result
follows if we prove the result for the class TG (γ). Let f (z) ∈ TG (γ) and
suppose that
p
h(z) = z +
∞
X
ck z p+k−1 ∈ C(p).
k=2
In this case,
g2
g2
(f ∗ h)(z) =
2(g2 + p(p − γ))
2(g2 + p(p − γ))
Ã
zp +
∞
X
!
ak ck z p+k−1
.
k=2
Observe that the subordination result (4) holds true if
µ
g2
ak
2(g2 + p(p − γ))
¶∞
1
is a subordinating factor sequence (with a1 = 1). In view of Theorem 2 ,
this is equivalent to the condition that
Ã
!
∞
X
g2
(6)
Re 1 +
ak z p+k−1 > 0.
(g
2 + p(p − γ))
k=1
Since gk ≥ g2 > 0 for k ≥ 2, we have
Ã
!
∞
X
g2
Re 1 +
ak z p+k−1 =
(g
2 + p(p − γ))
k=1
Ã
!
∞
X
g2
1
p
p+k−1
Re 1 +
z +
g2 ak z
g2 + p(p − γ)
g2 + p(p − γ) k=2
Ã
!
∞
X
1
g2
≥1−
rp +
|g2 ak | rp+k−1
g2 + p(p − γ)
g2 + p(p − γ) k=2
µ
¶
p(p − γ)
g2
p
p
r +
r > 0, (|z| = r < 1).
>1−
g2 + p(p − γ)
g2 + p(p − γ)
Subordination by p-valent convex functions
67
Thus(6) holds true in D, and proves (4). The inequality (5) follows by
taking
Zz
h(z) = p
0
∞
X
tp−1
(2p)k−p
p
dt = z + 2
Bk z p+k−1 , Bk =
2p
(1 − t)
k(k − p)!
k=2
in (4).
Now consider the function
F (z) = z p −
p(p − γ) 2
z , (γ < 1).
g2
Clearly, F (z) ∈ Tg (γ). For this function F (z), (4) becomes
g2
F (z) ≺ h(z) = p
2(g2 + p(p − γ))
It is easily verified that
½ µ
min Re
Zz
tp−1
dt,
(1 − t)2p
0
¶¾
g2
−1
F (z)
=
, z ∈ D.
2(g2 + p(p − γ))
2
Therefore,
g2
2(g2 + p(p − γ))
cannot be replaced by any larger constant.
Corollary 1 If f (z) ∈ Tp (j, λ, α, n), (Alkharsani and Alhajry [1]); and
h(z) ∈ C(p), then
(7)
2n (2p − α) (λ (2p − 1) + 1)
(f ∗ h)(z) ≺ h(z),
2 (2n (2p − α) (λ (2p − 1) + 1) + (p − α) (λ (p − 1) + 1))
68
H. A. Alkharsani, S. S. Alhajry
Re(f (z)) > −
(2n (2p − α) (λ (2p − 1) + 1) + (p − α) (λ (p − 1) + 1))
, z ∈ D.
2n (2p − α) (λ (2p − 1) + 1)
The constant factor
2n (2p − α) (λ (2p − 1) + 1)
2 (2n (2p − α) (λ (2p − 1) + 1) + (p − α) (λ (p − 1) + 1))
in the subordination result (7) cannot be replaced by a larger number.
Corollary 2 If f (z) ∈ T (j, p, λ, α) (Altintas et al. [2]); and h(z) ∈ C(p),
then
(8)
(2p − α) (λ (2p − 1) + 1)
(f ∗ h)(z) ≺ h(z),
2 ((2p − α) (λ (2p − 1) + 1) + (p − α) (λ (p − 1) + 1))
Re(f (z)) > −
(2p − α) (λ (2p − 1) + 1) + (p − α) (λ (p − 1) + 1)
, z ∈ D.
(2p − α) (λ (2p − 1) + 1)
The constant factor
(2p − α) (λ (2p − 1) + 1)
2 ((2p − α) (λ (2p − 1) + 1) + (p − α) (λ (p − 1) + 1))
in the subordination result (8) cannot be replaced by a larger number.
Remark 1 The case λ = 1 and p = 1 in Corollary 2 was obtained by
Rosihan et al. [6]
Corollary 3 If f (z) ∈ T ∗ (p, j, α) (Owa [5]); and h(z) ∈ C(p), then
(9)
(2p − α)
(f ∗ h)(z) ≺ h(z),
2 (3p − 2α)
Re(f (z)) > −
(3p − 2α)
, z ∈ D.
(2p − α)
The constant factor
(2p − α)
2 (3p − 2α)
in the subordination result (9) cannot be replaced by a larger number.
Subordination by p-valent convex functions
69
Remark 2 The case α = 0 and p = 1 in Corollary 3 was obtained by
Singh [7].
Corollary 4 If f (z) ∈ p(j, λ, α, n), (Aouf and Srivastava [3]); and h(z) ∈
C, then
(10)
2n−1 (2 − α) (λ + 1)
(f ∗ h)(z) ≺ h(z),
(2n (2 − α) (λ + 1) + (1 − α))
Re(f (z)) > −
(2n (2 − α) (λ + 1) + (1 − α))
, z ∈ D.
2n−1 (2 − α) (λ + 1)
The constant factor
2n−1 (2 − α) (λ + 1)
(2n (2 − α) (λ + 1) + (1 − α))
in the subordination result (10) cannot be replaced by a larger number.
Remark 3 The case λ = 0 in Corollary 4 was obtained by Eker et al. [4].
References
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30(2),1995, 9-15.
[3] M. K. Aouf and H.M. Srivastava, Some families of starlike functions
with negative coefficients, J. Math. Anal. Appl. 203(411),1996, 762-790.
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H. A. Alkharsani, S. S. Alhajry
[4] S. S. Eker , Bilal S¸ eker, and Shigeyoshi Owa, On Subordination Result Associated with Certain Subclass of Analytic Functions Involving
Salagean Operator, Int. J. Math. Math. Sci. Article ID 48294, 2007, 1–6.
[5] S. Owa, Some properties of certain multivalent functions, Appl. Math.
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S. S. Alhajry
King Faisal University
Department of Mathematics
Dammam, Saudi Arabia.
e-mail: ssmh69@gmail.com
H. A. Alkharsani
King Faisal University
Department of Mathematics
Dammam, Saudi Arabia.
e-mail: hakh73@hotmail.com