MATEMATIQKI VESNIK
UDK 517.546
originalni nauqni rad
research paper
60 (2008), 87–94
ON A CLASS OF MULTIVALENT FUNCTIONS
DEFINED BY A MULTIPLIER TRANSFORMATION
Sukhwinder Singh, Sushma Gupta and Sukhjit Singh
Abstract. In the present paper, the authors investigate starlikeness and convexity of a
class of multivalent functions defined by multiplier transfomation. As a consequence, a number
of sufficient conditions for starlikeness and convexity of analytic functions are also obtained.
1. Introduction
P∞
Let Ap denote the class of functions of the form f (z) = z p + k=p+1 ak z k ,
p ∈ N = {1, 2, . . . }, which are analytic in the open unit disc E = {z : |z| < 1}.
We write A1 = A. A function f ∈ Ap is said to be p-valent starlike of order α
(0 ≤ α < p) in E if
µ 0 ¶
zf (z)
Re
> α, z ∈ E.
f (z)
We denote by Sp∗ (α), the class of all such functions. A function f ∈ Ap is said to
be p-valent convex of order α (0 ≤ α < p) in E if
µ
¶
zf 00 (z)
Re 1 + 0
> α, z ∈ E.
f (z)
Let Kp (α) denote the class of all those functions f ∈ Ap which are multivalently
convex of order α in E. Note that S1∗ (α) and K1 (α) are, respectively, the usual
classes of univalent starlike functions of order α and univalent convex functions of
order α, 0 ≤ α < 1, and will be denoted here by S ∗ (α) and K(α), respectively. We
shall use S ∗ and K to denote S ∗ (0) and K(0), respectively which are the classes of
univalent starlike (w.r.t. the origin) and univalent convex functions.
For f ∈ Ap , we define the multiplier transformation Ip (n, λ) as
µ
¶n
∞
P
k+λ
Ip (n, λ)f (z) = z p +
ak z k , (λ ≥ 0, n ∈ Z).
(1)
p
+
λ
k=p+1
AMS Subject Classification: 30C45
Keywords and phrases: Multivalent function; starlike function; convex function; multiplier
transformation.
87
88
S. Singh, S. Gupta and S. Singh
The operator Ip (n, λ) has been recently studied by Aghalary et. al. [1]. Earlier,
the operator I1 (n, λ) was investigated by Cho and Srivastava [2] and Cho and Kim
[3], whereas the operator I1 (n, 1) was studied by Uralegaddi and Somanatha [9].
I1 (n,P0) is the well-known Sălăgean [7] derivative operator Dn , defined as: Dn f (z) =
∞
z + k=2 k n an z n , n ∈ N0 = N ∪ {0} and f ∈ A.
A function f ∈ Ap is said to be in the class Sn (p, λ, α) for all z in E if it
satisfies
·
¸
Ip (n + 1, λ)f (z)
α
> ,
Re
(2)
Ip (n, λ)f (z)
p
for some α (0 ≤ α < p, p ∈ N). We note that S0 (1, 0, α) and S1 (1, 0, α) are the
usual classes S ∗ (α) and K(α) of starlike functions of order α and convex functions
of order α, respectively.
In 1989, Owa, Shen and Obradović [6] obtained a sufficient condition for a
function f ∈ A to belong to the class Sn (1, 0, α) = Sn (α), say. In fact, they proved
the following result:
Theorem A. For n ∈ N0 , if f ∈ A satisfies
¯ n+1
¯1−β ¯ n+2
¯β
¯D
¯
¯D
¯
f (z)
3
f (z)
1−2β
¯
¯
¯
¯
(1 − α + α2 )β , z ∈ E,
¯ Dn f (z) − 1¯
¯ Dn+1 f (z) − 1¯ < (1 − α)
2
for some real numbers α(0 ≤ α ≤
· n+1
¸
D
f (z)
Re
> α in E.
Dn f (z)
1
2)
and β(0 ≤ β ≤ 1), then f ∈ Sn (α), i.e.
This result was, later on, extended by Li and Owa [5] for all α, 0 ≤ α < 1 and
β ≥ 0. They proved
Theorem B. If f ∈ A satisfies
¢β
¯ n+1
¯γ ¯ n+2
¯β (
γ¡
¯D
¯ ¯D
¯
(1 − α) 32 − α ,
f (z)
f (z)
¯
¯
¯
¯
¯ Dn f (z) − 1¯ ¯ Dn+1 f (z) − 1¯ <
β+γ
2β (1 − α)
,
0 ≤ α ≤ 1/2,
1/2 ≤ α < 1.
for some reals α (0 ≤ α < 1), β ≥ 0 and γ ≥ 0 with β + γ > 0, then f ∈ Sn (α),
n ∈ N0 .
In the present paper, our aim is to determine sufficient conditions for a function
f ∈ Ap to be a member of the class Sn (p, λ, α). As a consequence of our main result,
we get a number of sufficient conditions for starlikeness and convexity of analytic
functions.
2. Main result
To prove our result, we shall make use of the famous Jack’s lemma which we
state below.
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On a class of multivalent functions
Lemma 2.1. (Jack [4]) Suppose w(z) be a nonconstant analytic function in E
with w(0) = 0. If |w(z)| attains its maximum value at a point z0 ∈ E on the circle
|z| = r < 1, then z0 w0 (z0 ) = mw(z0 ), where m, m ≥ 1, is some real number.
We, now, state and prove our main result.
Theorem 2.1. If f ∈ Ap satisfies
¯
¯γ ¯
¯β
¯ Ip (n + 1, λ)f (z)
¯ ¯ Ip (n + 2, λ)f (z)
¯
¯
¯
¯
¯
¯ Ip (n, λ)f (z) − 1¯ ¯ Ip (n + 1, λ)f (z) − 1¯ < M (p, λ, α, β, γ), z ∈ E,
(3)
for some real numbers α, β and γ such that 0 ≤ α < p, β ≥ 0, γ ≥ 0, β + γ > 0,
then f ∈ Sn (p, λ, α), where n ∈ N0 and
³
´γ ³
´β
1

 1 − αp
1 − αp + 2(p+λ)
, 0 ≤ α ≤ p/2,
M (p, λ, α, β, γ) =
´
³
´
³
γ+β
β

1
 1− α
1 + p+λ
,
p/2 ≤ α < p.
p
Proof. Case (i). Let 0 ≤ α ≤
Define a function w(z) as
p
2.
Writing
α
p
1
2.
= µ, we see that 0 ≤ µ ≤
Ip (n + 1, λ)f (z)
1 + (1 − 2µ)w(z)
=
,
Ip (n, λ)f (z)
1 − w(z)
z ∈ E.
(4)
Then w is analytic in E, w(0) = 0 and w(z) 6= 1 in E. By a simple computation,
we obtain from (4),
0
z(Ip (n + 1, λ)f (z))0
z(Ip (n, λ)f (z))0
2(1 − µ)zw (z)
−
=
Ip (n + 1, λ)f (z)
Ip (n, λ)f (z)
(1 − w(z))(1 + (1 − 2µ)w(z))
(5)
By making use of the identity
(p + λ)Ip (n + 1, λ)f (z) = z(Ip (n, λ)f (z))0 + λIp (n, λ)f (z)
(6)
we obtain from (5)
0
1 + (1 − 2µ)w(z)
2(1 − µ)zw (z)
Ip (n + 2, λ)f (z)
=
+
Ip (n + 1, λ)f (z)
1 − w(z)
(p + λ)(1 − w(z))(1 + (1 − 2µ)w(z))
Thus, we have
¯
¯γ ¯
¯β
¯ Ip (n + 1, λ)f (z)
¯ ¯ Ip (n + 2, λ)f (z)
¯
¯
¯
¯
¯
¯ Ip (n, λ)f (z) − 1¯ ¯ Ip (n + 1, λ)f (z) − 1¯
¯β
¯ ¯
¯
0
¯
¯ 2(1 − µ)w(z) ¯γ ¯¯ 2(1 − µ)w(z)
2(1
−
µ)zw
(z)
¯
¯ ¯
= ¯¯
+
¯
¯
¯ 1 − w(z)
1 − w(z)
(p + λ)(1 − w(z))(1 + (1 − 2µ)w(z)) ¯
¯
¯β
¯
¯
0
¯
¯ 2(1 − µ)w(z) ¯γ+β ¯¯
zw
(z)
¯
¯
= ¯¯
¯1 +
¯ .
¯
¯
1 − w(z)
(p + λ)w(z)(1 + (1 − 2µ)w(z)) ¯
90
S. Singh, S. Gupta and S. Singh
Suppose that there exists a point z0 ∈ E such that max|z|≤|z0 | |w(z)| = |w(z0 )| = 1.
Then by Lemma 2.1, we have w(z0 ) = eiθ , 0 < θ ≤ 2π and z0 w0 (z0 ) = mw(z0 ),
m ≥ 1. Therefore
¯
¯γ ¯
¯β
¯ Ip (n + 1, λ)f (z0 )
¯ ¯ Ip (n + 2, λ)f (z0 )
¯
¯
¯
¯
¯
¯ Ip (n, λ)f (z0 ) − 1¯ ¯ Ip (n + 1, λ)f (z0 ) − 1¯
¯
¯β
¯
¯
¯
¯ 2(1 − µ)w(z0 ) ¯γ+β ¯
m
¯
¯
¯
¯
=¯
¯1 + (p + λ)(1 + (1 − 2µ)w(z0 )) ¯
1 − w(z0 ) ¯
¯β
¯
¯
m
2γ+β (1 − µ)γ+β ¯¯
¯
1
+
=
¯
β+γ
iθ
(p + λ)(1 + (1 − 2µ)e ¯
|1 − eiθ |
µ
µ
¶β
¶β
m
1
≥ (1 − µ)β+γ 1 +
≥ (1 − µ)β+γ 1 +
2(p + λ)(1 − µ)
2(p + λ)(1 − µ)
µ
¶β
1
γ
= (1 − µ) 1 − µ +
2(p + λ)
which contradicts (3) for 0 ≤ α ≤ p2 . Therefore, we must have |w(z)| < 1 for all
z ∈ E, and hence f ∈ Sn (p, λ, α).
Case (ii). When p2 ≤ α < p. In this case, we must have 21 ≤ µ < 1, where
µ = αp . Let w be defined by
Ip (n + 1, λ)f (z)
µ
=
,
Ip (n, λ)f (z)
µ − (1 − µ)w(z)
z ∈ E,
µ
where w(z) 6= 1−µ
in E. Then w is analytic in E and w(0) = 0. Proceeding as in
Case (i) and using identity (6), we obtain
¯
¯γ ¯
¯β
¯ Ip (n + 1, λ)f (z)
¯ ¯ Ip (n + 2, λ)f (z)
¯
¯
¯
¯
¯
¯ Ip (n, λ)f (z) − 1¯ ¯ Ip (n + 1, λ)f (z) − 1¯
¯β
¯ ¯
¯
0
¯
¯ (1 − µ)w(z) ¯γ ¯¯ (1 − µ)w(z)
(1
−
µ)zw
(z)
¯
¯ ¯
+
= ¯¯
¯
¯
µ − (1 − µ)w(z) ¯ µ − (1 − µ)w(z) (p + λ)(µ − (1 − µ)w(z)) ¯
¯γ+β
¯β
¯
¯
¯
¯
1−µ
zw0 (z) ¯¯
γ ¯¯
¯
¯
=¯
|w(z)| ¯w(z) +
.
µ − (1 − µ)w(z) ¯
p+λ ¯
Suppose that there exists a point z0 ∈ E such that max|z|≤|z0 | |w(z)| = |w(z0 )| =
1, then by Lemma 2.1, we obtain w(z0 ) = eiθ and z0 w0 (z0 ) = mw(z0 ), m ≥ 1.
Therefore
¯γ ¯
¯β
¯
¯ ¯ Ip (n + 2, λ)f (z0 )
¯
¯ Ip (n + 1, λ)f (z0 )
¯
¯
¯
¯
¯ Ip (n, λ)f (z0 ) − 1¯ ¯ Ip (n + 1, λ)f (z0 ) − 1¯
µ
¶γ+β µ
¶β
m γ+β
(1 − µ)γ+β (1 + p+λ
)
α
1
≥
1
−
=
1
+
γ+β
p
p+λ
|µ − (1 − µ)eiθ |
which contradicts (3) for p2 ≤ α ≤ p. Therefore we must have |w(z)| < 1 for all
z ∈ E, and hence f ∈ Sn (p, λ, α). This completes the proof of our theorem.
91
On a class of multivalent functions
3. Deductions
For p = 1, Theorem 2.1 reduces to the following result:
Corollary 3.1. If, for all z ∈ E, a function f ∈ A satisfies
¯
¯γ ¯
¯β
¯ I1 (n + 1, λ)f (z)
¯ ¯ I1 (n + 2, λ)f (z)
¯
¯
¯
¯
¯
¯ I1 (n, λ)f (z) − 1¯ ¯ I1 (n + 1, λ)f (z) − 1¯

³
´β
1

 (1 − α)γ 1 − α + 2(1+λ)
, 0 ≤ α ≤ 1/2,
<
³
´β

 (1 − α)γ+β 1 + 1
,
1/2 ≤ α < 1,
1+λ
for some reals α (0 ≤ α < 1), β ≥ 0 and γ ≥ 0 with β + γ > 0, then f ∈ Sn (1, λ, α),
where n ∈ N0 .
Remark 3.1. Setting λ = 0 in Corollary 3.1, we obtain Theorem B.
Recently, Sivaprasad Kumar et. al. [8] proved the following result:
Theorem C. Let ψ be univalent in E, ψ(0) = 1, Re ψ(z) > 0 and
starlike in E. Suppose f ∈ Ap satisfies
zψ 0 (z)
ψ(z)
be
0
Ip (n + 2, λ)f (z)
zψ (z)
≺ ψ(z) +
.
Ip (n + 1, λ)f (z)
(p + λ)ψ(z)
Ip (n + 1, λ)f (z)
Then,
≺ ψ(z).
Ip (n, λ)f (z)
1+(1− 2α )z
p
Set ψ(z) =
, 0 ≤ α < p, p ∈ N, in Theorem C. Clearly ψ satisfies all
1−z
the conditions of above theorem. Thus, we obtain the following result:
If f ∈ Ap satisfies
1 + (1 − 2α
(1 − 2α
Ip (n + 2, λ)f (z)
p )z
p )z
≺
+
Ip (n + 1, λ)f (z)
1−z
(p + λ)(1 + (1 −
then
i.e.
,
2α
p )z)
1 + (1 − 2α
Ip (n + 1, λ)f (z)
p )z
≺
,
Ip (n, λ)f (z)
1−z
µ
¶
Ip (n + 1, λ)f (z)
α
Re
> .
Ip (n, λ)f (z)
p
Compare this result with the result below, which we get by writing γ = 0 and
β = 1 in Theorem 2.1:
Corollary 3.2. If, for all z ∈ E, a function f ∈ Ap satisfies
¯
¯ (
1
,
0 ≤ α ≤ p/2,
1 − αp + 2(p+λ)
¯ Ip (n + 2, λ)f (z)
¯
¯
¯<
−
1
¯ Ip (n + 1, λ)f (z)
¯
α
1
(1 − p )(1 + p+λ ), p/2 ≤ α < p,
µ
¶
Ip (n + 1, λ)f (z)
α
then Re
> , z ∈ E.
Ip (n, λ)f (z)
p
92
S. Singh, S. Gupta and S. Singh
Setting p = 1, λ = 1 and n = 0 in Theorem 2.1, we obtain the following result:
Corollary 3.3. If f ∈ A satisfies
¯ 0
¯γ ¯
¯β µ ¶
β
¯ zf (z)
¯ ¯ 2zf 0 (z) + z 2 f 00 (z)
¯
3
¯
¯ ¯
¯
− 1¯ ¯
− 1¯ <
(1 − α)γ+β ,
¯
0
¯ f (z)
¯ ¯ f (z) + zf (z)
¯
2
0 ≤ α < 1, z ∈ E,
for some β ≥ 0 and γ ≥ 0 with β + γ > 0, then f ∈ S ∗ (α).
Setting α = 0 in Corollary 3.3, we obtain the following criterion for starlikeness:
Corollary 3.4. For some non-negative real numbers β and γ with β + γ > 0,
if f ∈ A satisfies
¯ 0
¯γ ¯
¯β µ ¶
β
¯ zf (z)
¯ ¯ 2zf 0 (z) + z 2 f 00 (z)
¯
3
¯
¯ ¯
¯
<
− 1¯ ¯
−
1
, z ∈ E,
¯
¯
¯ f (z)
¯ ¯ f (z) + zf 0 (z)
¯
2
then f ∈ S ∗ .
In particular, for β = 1 and γ = 1, we obtain the following interesting criterion
for starlikeness:
Corollary 3.5. If f ∈ A satisfies
¯ 0
¯¯
¯
¯ zf (z)
¯ ¯ 2zf 0 (z) + z 2 f 00 (z)
¯ 3
¯
¯¯
¯
− 1¯ ¯
− 1¯ < ,
¯
0
¯ f (z)
¯ ¯ f (z) + zf (z)
¯ 2
z ∈ E,
then f ∈ S ∗ .
Setting λ = 0 and n = 0 in Theorem 2.1, we obtain the following sufficient
condition for a function f ∈ Ap to be a p-valent starlike function of order α.
Corollary 3.6. For all z ∈ E, if f ∈ Ap satisfies the condition

´γ ³
´β
¯β  ³
¯ 0
¯γ ¯ Ã
!
α
1
α
00
¯
¯ zf (z)
¯ ¯1

1
−
+
, 0 ≤ α ≤ p/2,
1
−
p
p
2p
zf (z)
¯
¯
¯ ¯
1+ 0
− 1¯ <
− 1¯ ¯
¯
³
´
³
´
γ+β
β
¯

¯ pf (z)
¯ ¯p
f (z)
 1− α
1 + p1 ,
p/2 ≤ α < p,
p
for some real numbers α, β and γ with 0 ≤ α < p, β ≥ 0, γ ≥ 0, β + γ > 0, then
f ∈ Sp∗ (α).
The substitution p = 1 in Corollary 3.6, yields the following result:
Corollary 3.7. If f ∈ A satisfies
¯ 0
¯γ ¯ 00
¯
(
¢β
γ¡
¯ zf (z)
¯ ¯ zf (z) ¯β
(1 − α) 32 − α
¯
¯ ¯
¯
−
1
<
¯
¯ ¯ 0
¯
γ+β β
¯ f (z)
¯ ¯ f (z) ¯
(1 − α)
2
, 0 ≤ α ≤ 1/2,
, 1/2 ≤ α < 1,
where z ∈ E and α, β, γ are real numbers with 0 ≤ α < 1, β ≥ 0, γ ≥ 0, β + γ > 0,
then f ∈ S ∗ (α).
93
On a class of multivalent functions
In particular, writing β = 1,γ = 1 and α = 0 in Corollary 3.7, we obtain the
following result of Li and Owa [5]:
Corollary 3.8. If f ∈ A satisfies
¯ 00
Ã
!¯
¯ zf (z) zf 0 (z)
¯ 3
¯
¯
−1 ¯< ,
¯ 0
¯ f (z)
¯ 2
f (z)
z ∈ E,
then f ∈ S ∗ .
Taking λ = 0 and n = 1 in Theorem 2.1, we get the following interesting
criterion for convexity of multivalent functions:
Corollary 3.9. If, for all z ∈ E, a function f ∈ Ap satisfies
¯γ ¯ Ã
¯β
¯ Ã
!
!
00
00
000
¯ ¯1
¯
¯1
zf (z)
2zf (z) + z 2 f (z)
¯ ¯
¯
¯
1+ 0
− 1¯ ¯
1+
− 1¯
¯
0
00
¯ ¯p
¯
¯p
f (z)
f (z) + f (z)
³
´γ ³
´β
1

 1 − αp
1 − αp + 2p
, 0 ≤ α ≤ p/2,
< ³
´γ+β ³
´β

 1− α
1 + p1 ,
p/2 ≤ α < p,
p
for some real numbers α, β and γ with 0 ≤ α < p, β ≥ 0, γ ≥ 0, β + γ > 0, then
f ∈ Kp (α).
Taking p = 1 in Corollary 3.9, we obtain the following sufficient condition for
convexity of univalent functions.
Corollary 3.10. For some non-negative real numbers α, β and γ, with
β + γ > 0 and α < 1, if f ∈ A satisfies
¯ 00
¯ ¯
¯
(
¢β
¡
¯ zf (z) ¯γ ¯ 2zf 00 (z) + z 2 f 000 (z) ¯β
(1 − α)γ 1 − α + 12 , 0 ≤ α ≤ 1/2,
¯
¯ ¯
¯
<
¯ 0
¯ ¯
¯
¯ f (z) ¯ ¯ f 0 (z) + f 00 (z) ¯
(1 − α)γ+β 2β ,
1/2 ≤ α < 1,
for all z ∈ E, then f ∈ K(α).
In particular, writing β = 1, γ = 1 and α = 0 in Corollary 3.10, we obtain the
following sufficient conditon for convexity of analytic functions:
Corollary 3.11. If f ∈ A satisfies
¯ 00
Ã
!¯
¯ zf (z) 2zf 00 (z) + z 2 f 000 (z) ¯ 3
¯
¯
¯ 0
¯< ,
¯ f (z)
¯ 2
f 0 (z) + f 00 (z)
z ∈ E,
then f ∈ K.
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(received 15.01.2007, in revised form 19.03.2008)
Sukhwinder Singh, Department of Applied Sciences, B.B.S.B.Engineering College, Fatehgarh
Sahib-144 407 (Punjab), INDIA.
Sushma Gupta, Department of Mathematics, Sant Longowal Institute of Engineering & Technology, Longowal-148 106 (Punjab), INDIA, E-mail: sushmagupta1@yahoo.com
Sukhjit Singh, Department of Mathematics, Sant Longowal Institute of Engineering & Technology,
Longowal – 148 106 (Punjab), INDIA