A CERTAIN CLASS OF ANALYTIC AND
MULTIVALENT FUNCTIONS DEFINED BY MEANS
OF A LINEAR OPERATOR
DING-GONG YANG
N-ENG XU
Department of Mathematics
Suzhou University
Suzhou, Jiangsu 215006,
P.R. China.
Department of Mathematics
Changshu Institute of Technology
Changshu, Jiangsu 215500,
P.R. China.
EMail: xuneng11@pub.sz.jsinfo.net
SHIGEYOSHI OWA
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502, Japan
EMail: owa@math.kindai.ac.jp
Received:
13 June, 2007
Accepted:
01 November, 2007
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
Primary 30C45.
Key words:
Analytic function; Multivalent function; Linear operator; Convex univalent function; Hadamard product (or convolution); Subordination; Integral operator.
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
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Abstract:
Making use of a linear operator, which is defined here by means of the
Hadamard product (or convolution), we introduce a class Qp (a, c; h) of
analytic and multivalent functions in the open unit disk. An inclusion relation and a convolution property for the class Qp (a, c; h) are presented.
Some integral-preserving properties are also given.
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
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Contents
1
Introduction and Preliminaries
4
2
Main Results
9
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
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1.
Introduction and Preliminaries
Let the functions
f (z) =
∞
X
ak z
p+k
and g(z) =
k=0
∞
X
bk z p+k (p ∈ N = {1, 2, 3, . . . })
k=0
be analytic in the open unit disk U = {z : |z| < 1}. Then the Hadamard product (or
convolution) (f ∗ g)(z) of f (z) and g(z) is defined by
(1.1)
(f ∗ g)(z) =
∞
X
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
ak b k z
p+k
= (g ∗ f )(z).
k=0
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Let Ap denote the class of functions f (z) normalized by
(1.2)
f (z) = z p +
∞
X
ak z p+k (p ∈ N),
k=1
which are analytic in U . A function f (z) ∈ Ap is said to be in the class Sp∗ (α) if it
satisfies
(1.3)
zf 0 (z)
> pα (z ∈ U )
Re
f (z)
for some α(α < 1). When 0 ≤ α < 1, Sp∗ (α) is the class of p-valently starlike
functions of order α in U . Also we write A1 = A and S1∗ (α) = S ∗ (α). A function
f (z) ∈ A is said to be prestarlike of order α(α < 1) in U if
(1.4)
z
∗ f (z) ∈ S ∗ (α).
2(1−α)
(1 − z)
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We denote this class by R(α) (see [9]). It is clear that a function f (z) ∈ A is in the
class R(0) if and only if f (z) is convex univalent in U and
1
1
∗
R
=S
.
2
2
We now define the function ϕp (a, c; z) by
ϕp (a, c; z) = z p +
(1.5)
∞
X
k=1
(a)k p+k
z
(z ∈ U ),
(c)k
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
where
c∈
/ {0, −1, −2, . . . } and (x)k = x(x + 1) · · · (x + k − 1) (k ∈ N).
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Corresponding to the function ϕp (a, c; z), Saitoh [10] introduced and studied a linear
operator Lp (a, c) on Ap by the following Hadamard product (or convolution):
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Lp (a, c)f (z) = ϕp (a, c; z) ∗ f (z) (f (z) ∈ Ap ).
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(1.6)
For p = 1, L1 (a, c) on A was first defined by Carlson and Shaffer [1]. We remark
in passing that a much more general convolution operator than the operator Lp (a, c)
was introduced by Dziok and Srivastava [2].
It is known [10] that
(1.7) z(Lp (a, c)f (z)) = aLp (a + 1, c)f (z) − (a − p)Lp (a, c)f (z) (f (z) ∈ Ap ).
Setting a = n + p > 0 and c = 1 in (1.6), we have
Lp (n + p, 1)f (z) =
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0
(1.8)
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zp
∗ f (z) = Dn+p−1 f (z) (f (z) ∈ Ap ).
(1 − z)n+p
The operator Dn+p−1 when p = 1 was first introduced by Ruscheweyh [8], and
Dn+p−1 was introduced by Goel and Sohi [3]. Thus we name Dn+p−1 as the Ruscheweyh
derivative of (n + p − 1)th order.
For functions f (z) and g(z) analytic in U , we say that f (z) is subordinate to g(z)
in U , and write f (z) ≺ g(z), if there exists an analytic function w(z) in U such that
|w(z)| ≤ |z| and f (z) = g(w(z)) (z ∈ U ).
Furthermore, if the function g(z) is univalent in U , then
f (z) ≺ g(z) ⇔ f (0) = g(0) and f (U ) ⊂ g(U ).
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
Let P be the class of analytic functions h(z) with h(0) = p, which are convex
univalent in U and for which
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Re h(z) > 0 (z ∈ U ).
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In this paper we introduce and investigate the following subclass of Ap .
Definition 1.1. A function f (z) ∈ Ap is said to be in the class Qp (a, c; h) if it satisfies
(1.9)
Lp (a + 1, c)f (z)
p h(z)
≺1− +
,
Lp (a, c)f (z)
a
a
a 6= 0,
c∈
/ {0, −1, −2, . . . } and h(z) ∈ P.
It is easy to see that, if f (z) ∈ Qp (a, c; h), then Lp (a, c)f (z) ∈ Sp∗ (0).
For a = n + p (n > −p), c = 1 and
(1.11)
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where
(1.10)
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h(z) = p +
(A − B)z
(−1 ≤ B < A ≤ 1),
1 + Bz
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Yang [12] introduced and studied the class
Qp (n + p, 1; h) = Sn,p (A, B).
For h(z) given by (1.11), the class
(1.12)
Qp (a, c; h) = Ha,c,p (A, B)
has been considered by Liu and Owa [5].
For p = 1, A = 1 − 2α (0 ≤ α < 1) and B = −1, Kim and Srivastava [4] have
shown some properties of the class Ha,c,1 (1 − 2α, −1).
In the present paper, we shall establish an inclusion relation and a convolution
property for the class Qp (a, c; h). Integral transforms of functions in this class are
also discussed. We observe that the proof of each of the results in [5] is much akin to
that of the corresponding assertion made by Yang [12] in the case of a = n + p and
c = 1. However, the methods used in [5, 12] do not work for the general function
class Qp (a, c; h).
We need the following lemmas in order to derive our main results for the class
Qp (a, c; h).
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
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Lemma 1.2 (Ruscheweyh [9]). Let α < 1, f (z) ∈ S ∗ (α) and g(z) ∈ R(α). Then,
for any analytic function F (z) in U ,
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g ∗ (f F )
(U ) ⊂ co(F (U )),
g∗f
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where co(F (U )) denotes the closed convex hull of F (U ).
Lemma 1.3 (Miller and Mocanu [6]). Let β (β 6= 0) and γ be complex numbers
and let h(z) be analytic and convex univalent in U with
Re(βh(z) + γ) > 0 (z ∈ U ).
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If q(z) is analytic in U with q(0) = h(0), then the subordination
q(z) +
zq 0 (z)
≺ h(z)
βq(z) + γ
implies that q(z) ≺ h(z).
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
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2.
Main Results
Theorem 2.1. Let h(z) ∈ P and
Re h(z) > β (z ∈ U ; 0 ≤ β < p).
(2.1)
If
0 < a1 < a2 and a2 ≥ 2(p − β),
(2.2)
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
then
vol. 9, iss. 2, art. 50, 2008
Qp (a2 , c; h) ⊂ Qp (a1 , c; h).
Proof. Define
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g(z) = z +
∞
X
(a1 )k
k=1
(a2 )k
Then
ϕp (a1 , a2 ; z)
= g(z) ∈ A,
z p−1
(2.3)
where ϕp (a1 , a2 ; z) is defined as in (1.5), and
z
z
∗ g(z) =
.
a
2
(1 − z)
(1 − z)a1
(2.4)
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z k+1 (z ∈ U ; 0 < a1 < a2 ).
From (2.4) we have
z
a1 a2 ∗
∗
∗
g(z)
∈
S
1
−
⊂
S
1
−
(1 − z)a2
2
2
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for 0 < a1 < a2 , which implies that
a2 g(z) ∈ R 1 −
.
2
(2.5)
Since
Lp (a1 , c)f (z) = ϕp (a1 , a2 ; z) ∗ Lp (a2 , c)f (z) (f (z) ∈ Ap ),
(2.6)
Analytic and Multivalent Functions
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Ding-Gong Yang, N-eng Xu
we deduce from (1.7) and (2.6) that
a1 Lp (a1 + 1, c)f (z)
and Shigeyoshi Owa
0
(2.7)
= z (Lp (a1 , c)f (z)) + (a1 − p)Lp (a1 , c)f (z)
= ϕp (a1 , a2 ; z) ∗ (z(Lp (a2 , c)f (z))0 + (a1 − p)Lp (a2 , c)f (z))
= ϕp (a1 , a2 ; z) ∗ (a2 Lp (a2 + 1, c)f (z) + (a1 − a2 )Lp (a2 , c)f (z)).
vol. 9, iss. 2, art. 50, 2008
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By using (2.3), (2.6) and (2.7), we find that
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Lp (a1 + 1, c)f (z)
Lp (a1 , c)f (z)
(z
p−1
g(z)) ∗
=
(2.8)
a2
L (a
a1 p 2
+ 1, c)f (z) + 1 −
(z p−1 g(z))
g(z) ∗
=
a2 Lp (a2 +1,c)f (z)
a1
z p−1
Lp (a2 , c)f (z)
∗ Lp (a2 , c)f (z)
,c)f (z)
+ 1 − aa21 Lp (az2p−1
g(z) ∗
g(z) ∗ (q(z)F (z))
=
(f (z) ∈ Ap ),
g(z) ∗ q(z)
q(z) =
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Lp (a2 ,c)f (z)
z p−1
where
a2
a1
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Lp (a2 , c)f (z)
∈A
z p−1
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and
a2 Lp (a2 + 1, c)f (z)
a2
+1− .
a1 Lp (a2 , c)f (z)
a1
Let f (z) ∈ Qp (a2 , c; h). Then
a2
p
h(z)
a2
F (z) ≺
1−
+
+1−
a1
a2
a2
a1
p
h(z)
(2.9)
=1−
+
= h1 (z) (say),
a1
a1
F (z) =
where h1 (z) is convex univalent in U , and, by (1.7),
zq 0 (z)
z(Lp (a2 , c)f (z))0
=
+1−p
q(z)
Lp (a2 , c)f (z)
Lp (a2 + 1, c)f (z)
= a2
+ 1 − a2
Lp (a2 , c)f (z)
(2.10)
≺ 1 − p + h(z).
By using (2.1), (2.2) and (2.10), we get
a2
zq 0 (z)
Re
>1−p+β ≥1−
(z ∈ U ),
q(z)
2
that is,
a2 (2.11)
q(z) ∈ S ∗ 1 −
.
2
Consequently, in view of (2.5), (2.8), (2.9) and (2.11), an application of Lemma 1.2
yields
Lp (a1 + 1, c)f (z)
≺ h1 (z).
Lp (a1 , c)f (z)
Thus f (z) ∈ Qp (a1 , c; h) and the proof of Theorem 2.1 is completed.
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
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By carefully selecting the function h(z) involved in Theorem 2.1, we can obtain
a number of useful consequences.
Corollary 2.2. Let
(2.12)
h(z) = p − 1 +
1 + Az
1 + Bz
If
γ
(z ∈ U ; 0 < γ ≤ 1; −1 ≤ B < A ≤ 1).
0 < a1 < a2 and a2 ≥ 2 1 −
1−A
1−B
γ ,
then
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
Qp (a2 , c; h) ⊂ Qp (a1 , c; h).
Proof. The analytic function h(z) defined by (2.12) is convex univalent in U (cf.
[11]), h(0) = p, and h(U ) is symmetric with respect to the real axis. Thus h(z) ∈ P
γ
and
1−A
≥ 0 (z ∈ U ).
Re h(z) > β = h(−1) = p − 1 +
1−B
Hence the desired result follows from Theorem 2.1 at once.
If we let γ = 1, then Corollary 2.2 yields the following.
Corollary 2.3. Let h(z) be given by (1.11). If a, A and B(−1 ≤ B < A ≤ 1) satisfy
either
1−A
(i) a ≥ 1 − 2 1−B
>0
or
(ii) a > 0 ≥ 1 − 2
then
Analytic and Multivalent Functions
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Ding-Gong Yang, N-eng Xu
1−A
1−B
,
Qp (a + 1, c; h) ⊂ Qp (a, c; h).
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Using Jack’s Lemma, Liu and Owa [5, Theorem 1] proved that, if a ≥
A−B
,
1−B
then
Ha+1,c,p (A, B) ⊂ Ha,c,p (A, B).
Since
A−B
≥1−2
1−B
1−A
1−B
(−1 ≤ B < A ≤ 1)
and the equality occurs only when A = 1, we see that Corollary 2.3 is better than
the result of [5].
Corollary 2.4. Let
(2.13)
h(z) = p +
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
∞ X
γ+1
k=1
γ+k
δ k z k (z ∈ U ; 0 < δ ≤ 1; γ ≥ 0).
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If
0 < a1 < a2 and a2 ≥ 2
∞
X
k+1
(−1)
k=1
γ+1
γ+k
k
δ ,
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then
Qp (a2 , c; h) ⊂ Qp (a1 , c; h).
Proof. The function h(z) defined by (2.13) is in the class P (cf. [8]) and satisfies
h(z) = h(z). Thus
∞
X
γ+1 k
k
Re h(z) > β = h(−1) = p +
(−1)
δ > p − δ ≥ 0 (z ∈ U ).
γ
+
k
k=1
Therefore we have the corollary by using Theorem 2.1.
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Corollary 2.5. Let
(2.14)
2
h(z) = p + 2
π
log
√ 2
1 + γz
√
1 − γz
If
0 < a1 < a2 and a2 ≥
(z ∈ U ; 0 < γ ≤ 1).
16
√
(arctan γ)2 ,
2
π
then
and Shigeyoshi Owa
Qp (a2 , c; h) ⊂ Qp (a1 , c; h).
vol. 9, iss. 2, art. 50, 2008
Proof. The function h(z) defined by (2.14) belongs to the class P (cf. [7]) and
satisfies h(z) = h(z). Thus
1
8
√
(arctan γ)2 ≥ p − > 0 (z ∈ U ).
2
2
π
Re h(z) > β = h(−1) = p −
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Hence an application of Theorem 2.1 yields the desired result.
For γ = 1, Corollary 2.5 leads to
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Corollary 2.6. Let
h(z) = p +
2
π2
log
√ 2
1+ z
√
1− z
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(z ∈ U ).
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Then, for a > 0,
Qp (a + 1, c; h) ⊂ Qp (a, c; h).
Theorem 2.7. Let h(z) ∈ P and
(2.15)
Re h(z) > p − 1 + α (z ∈ U ; α < 1).
If f (z) ∈ Qp (a, c; h),
(2.16)
g(z)
∈ R(α) (α < 1),
z p−1
g(z) ∈ Ap and
then
(f ∗ g)(z) ∈ Qp (a, c; h).
Analytic and Multivalent Functions
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Ding-Gong Yang, N-eng Xu
Proof. Let f (z) ∈ Qp (a, c; h) and suppose that
q(z) =
(2.17)
Lp (a, c)f (z)
.
z p−1
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
Then
(2.18)
F (z) =
p h(z)
Lp (a + 1, c)f (z)
≺1− +
,
Lp (a, c)f (z)
a
a
q(z) ∈ A and
0
(2.19)
zq (z)
≺ 1 − p + h(z)
q(z)
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(see (2.10) used in the proof of Theorem 2.1). By (2.15) and (2.19), we see that
(2.20)
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q(z) ∈ S ∗ (α).
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For g(z) ∈ Ap , it follows from (2.17) and (2.18) that
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Lp (a + 1, c)(f ∗ g)(z)
g(z) ∗ Lp (a + 1, c)f (z)
=
Lp (a, c)(f ∗ g)(z)
g(z) ∗ Lp (a, c)f (z)
(2.21)
=
g(z)
z p−1
∗ (q(z)F (z))
g(z)
z p−1
∗ q(z)
(z ∈ U ).
Now, by using (2.16), (2.18), (2.20) and (2.21), an application of Lemma 1.2 leads
to
Lp (a + 1, c)(f ∗ g)(z)
p h(z)
≺1− +
.
Lp (a, c)(f ∗ g)(z)
a
a
This shows that (f ∗ g)(z) ∈ Qp (a, c; h).
For α = 0 and α = 12 , Theorem 2.7 reduces to
Corollary 2.8. Let h(z) ∈ P and g(z) ∈ Ap satisfy either
(i)
g(z)
z p−1
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
is convex univalent in U and
Re h(z) > p − 1 (z ∈ U )
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or
(ii)
g(z)
z p−1
∈ S ∗ ( 12 ) and
Re h(z) > p −
1
(z ∈ U ).
2
If f (z) ∈ Qp (a, c; h), then
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(f ∗ g)(z) ∈ Qp (a, c; h).
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Theorem 2.9. Let h(z) ∈ P and
(2.22)
Re h(z) > − Re λ (z ∈ U ),
where λ is a complex number such that Re λ > −p. If f (z) ∈ Qp (a, c; h), then the
function
Z
λ + p z λ−1
(2.23)
g(z) =
t f (t)dt
zλ
0
is also in the class Qp (a, c; h).
Proof. For f (z) ∈ Ap and Re λ > −p, it follows from (1.7) and (2.23) that g(z) ∈
Ap and
(λ + p)Lp (a, c)f (z) = λLp (a, c)g(z) + z(Lp (a, c)g(z))0
(2.24)
= aLp (a + 1, c)g(z) + (λ + p − a)Lp (a, c)g(z).
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
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If we let
(2.25)
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Lp (a + 1, c)g(z)
q(z) =
,
Lp (a, c)g(z)
then (2.24) and (2.25) lead to
(2.26)
Lp (a, c)f (z)
aq(z) + λ + p − a = (λ + p)
.
Lp (a, c)g(z)
Differentiating both sides of (2.26) logarithmically and using (1.7) and (2.25), we
obtain
zq 0 (z)
1 z(Lp (a, c)f (z))0 z(Lp (a, c)g(z))0
=
−
aq(z) + λ + p − a
a
Lp (a, c)f (z)
Lp (a, c)g(z)
Lp (a + 1, c)f (z)
(2.27)
=
− q(z).
Lp (a, c)f (z)
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Let f (z) ∈ Qp (a, c; h). Then it follows from (2.27) that
(2.28)
q(z) +
zq 0 (z)
p h(z)
≺1− +
.
aq(z) + λ + p − a
a
a
Also, in view of (2.22), we have
p h(z)
(2.29) Re a 1 − +
+ λ + p − a = Re h(z) + Re λ > 0 (z ∈ U ).
a
a
Therefore, it follows from (2.28), (2.29) and Lemma 1.3 that
q(z) ≺ 1 −
p h(z)
+
.
a
a
This proves that g(z) ∈ Qp (a, c; h).
Analytic and Multivalent Functions
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Ding-Gong Yang, N-eng Xu
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vol. 9, iss. 2, art. 50, 2008
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From Theorem 2.9 we have the following corollaries.
Corollary 2.10. Let h(z) be defined as in Corollary 2.2. If f (z) ∈ Qp (a, c; h) and
γ
1−A
Re λ ≥ 1 − p −
(0 < γ ≤ 1; −1 ≤ B < A ≤ 1),
1−B
then the function g(z) given by (2.23) is also in the class Qp (a, c; h).
In the special case when γ = 1, Corollary 2.10 was obtained by Liu and Owa [5,
Theorem 2] using Jack’s Lemma.
Corollary 2.11. Let h(z) be defined as in Corollary 2.4. If f (z) ∈ Qp (a, c; h) and
∞
X
γ+1 k
k+1
Re λ ≥
(−1)
δ − p (0 < δ ≤ 1; γ ≥ 0),
γ
+
k
k=1
then the function g(z) given by (2.23) is also in the class Qp (a, c; h).
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Corollary 2.12. Let h(z) be defined as in Corollary 2.5. If f (z) ∈ Qp (a, c; h) and
Re λ ≥
8
√
(arctan γ)2 − p (0 < γ ≤ 1),
2
π
then the function g(z) given by (2.23) is also in the class Qp (a, c; h).
Theorem 2.13. Let h(z) ∈ P and
(2.30)
Re λ
Re h(z) > −
(z ∈ U ),
β
where β > 0 and λ is a complex number such that Re λ > −pβ. If f (z) ∈
Qp (a, c; h), then the function g(z) ∈ Ap defined by
β1
Z
λ + pβ z λ−1
β
t
(Lp (a, c)f (t)) dt
(2.31)
Lp (a, c)g(z) =
zλ
0
is also in the class Qp (a, c; h).
Proof. Let f (z) ∈ Qp (a, c; h). From the definition of g(z) we have
Z z
λ
β
(2.32)
z (Lp (a, c)g(z)) = (λ + pβ)
tλ−1 (Lp (a, c)f (t))β dt.
0
Differentiating both sides of (2.32) logarithmically and using (1.7), we get
β
Lp (a, c)f (z)
(2.33)
λ + β(aq(z) + p − a) = (λ + pβ)
,
Lp (a, c)g(z)
where
(2.34)
q(z) =
Lp (a + 1, c)g(z)
.
Lp (a, c)g(z)
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
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Also, differentiating both sides of (2.33) logarithmically and using (1.7), we arrive
at
(2.35)
q(z) +
zq 0 (z)
Lp (a + 1, c)f (z)
p h(z)
=
≺1− +
.
aβq(z) + λ + β(p − a)
Lp (a, c)f (z)
a
a
Noting that (2.30) and β > 0, we see that
p h(z)
(2.36)
Re aβ 1 − +
+ λ + β(p − a)
a
a
= β Re h(z) + Re λ > 0
Analytic and Multivalent Functions
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Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
(z ∈ U ).
Now, in view of (2.34), (2.35) and (2.36), an application of Lemma 1.3 yields
Lp (a + 1, c)g(z)
p h(z)
≺1− +
,
Lp (a, c)g(z)
a
a
that is, g(z) ∈ Qp (a, c; h).
Corollary 2.14. Let h(z) be defined as in Corollary 2.2. If f (z) ∈ Qp (a, c; h) and
γ 1−A
(0 < γ ≤ 1; −1 ≤ B < A ≤ 1; β > 0),
Re λ ≥ β 1 − p −
1−B
then the function g(z) ∈ Ap defined by (2.31) is also in the class Qp (a, c; h).
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References
[1] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.
[2] J. DZIOK AND H.M. SRIVASTAVA, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput., 103
(1999), 1–13.
[3] R.M. GOEL AND N.S. SOHI, A new criterion for p-valent functions, Proc.
Amer. Math. Soc., 78 (1980), 353–357.
[4] Y.C. KIM AND H.M. SRIVASTAVA, Fractional integral and other linear operators associated with the Gaussian hypergeometric function, Complex Variables
Theory Appl., 34 (1997), 293–312.
Analytic and Multivalent Functions
Defined by Linear Operators
Ding-Gong Yang, N-eng Xu
and Shigeyoshi Owa
vol. 9, iss. 2, art. 50, 2008
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[5] JIN-LIN LIU AND S. OWA, On a class of multivalent functions involving certain linear operator, Indian J. Pure Appl. Math., 33 (2002), 1713–1722.
[6] S.S. MILLER AND P.T. MOCANU, On some classes of first order differential
subordinations, Michigan Math. J., 32 (1985), 185–195.
[7] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196.
[8] S. RUSCHEWEYH, New criteria for univalent functions, Proc. Amer. Math.
Soc., 49 (1975), 109–115.
[9] S. RUSCHEWEYH, Convolutions in Geometric Function Theory, Les Presses
de 1’Université de Montréal, Montréal, 1982.
[10] H. SAITOH, A linear operator and its applications of first order differential
subordinations, Math. Japon., 44 (1996), 31–38.
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[11] N-ENG XU AND DING-GONG YANG, An application of differential subordinations and some criteria for starlikeness, Indian J. Pure Appl. Math., 36
(2005), 541–556.
[12] DING-GONG YANG, On p-valent starlike functions, Northeast. Math. J., 5
(1989), 263–271.
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and Shigeyoshi Owa
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