Volume 9 (2008), Issue 2, Article 50, 10 pp.
A CERTAIN CLASS OF ANALYTIC AND MULTIVALENT FUNCTIONS DEFINED
BY MEANS OF A LINEAR OPERATOR
DING-GONG YANG, N-ENG XU, AND SHIGEYOSHI OWA
D EPARTMENT OF M ATHEMATICS
S UZHOU U NIVERSITY
S UZHOU , J IANGSU 215006, C HINA
D EPARTMENT OF M ATHEMATICS
C HANGSHU I NSTITUTE OF T ECHNOLOGY
C HANGSHU , J IANGSU 215500, C HINA
xuneng11@pub.sz.jsinfo.net
D EPARTMENT OF M ATHEMATICS
K INKI U NIVERSITY
H IGASHI -O SAKA , O SAKA 577-8502, JAPAN
owa@math.kindai.ac.jp
Received 13 June, 2007; accepted 01 November, 2007
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Analytic function; Multivalent function; Linear operator; Convex univalent function; Hadamard product (or convolution); Subordination; Integral operator.
2000 Mathematics Subject Classification. Primary 30C45.
1. I NTRODUCTION AND P RELIMINARIES
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
k=0
199-07
ak bk z p+k = (g ∗ f )(z).
2
D ING -G ONG YANG , N- ENG X U ,
AND
S HIGEYOSHI OWA
Let Ap denote the class of functions f (z) normalized by
(1.2)
p
f (z) = z +
∞
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)
Re
zf 0 (z)
> pα (z ∈ U )
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
z
(1.4)
∗ f (z) ∈ S ∗ (α).
(1 − z)2(1−α)
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
(1.5)
p
ϕp (a, c; z) = z +
∞
X
(a)k
k=1
(c)k
z p+k (z ∈ U ),
where
c∈
/ {0, −1, −2, . . . } and (x)k = x(x + 1) · · · (x + k − 1) (k ∈ N).
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):
(1.6)
Lp (a, c)f (z) = ϕp (a, c; z) ∗ f (z) (f (z) ∈ Ap ).
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))0 = 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
zp
(1.8)
Lp (n + p, 1)f (z) =
∗ f (z) = Dn+p−1 f (z) (f (z) ∈ Ap ).
n+p
(1 − z)
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 ).
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 50, 10 pp.
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A NALYTIC AND M ULTIVALENT F UNCTIONS D EFINED BY L INEAR O PERATORS
3
Let P be the class of analytic functions h(z) with h(0) = p, which are convex univalent in U
and for which
Re h(z) > 0 (z ∈ U ).
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
Lp (a + 1, c)f (z)
p h(z)
≺1− +
,
Lp (a, c)f (z)
a
a
(1.9)
where
(1.10)
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
(A − B)z
(−1 ≤ B < A ≤ 1),
1 + Bz
Yang [12] introduced and studied the class
(1.11)
h(z) = p +
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).
Lemma 1.1 (Ruscheweyh [9]). Let α < 1, f (z) ∈ S ∗ (α) and g(z) ∈ R(α). Then, for any
analytic function F (z) in U ,
g ∗ (f F )
(U ) ⊂ co(F (U )),
g∗f
where co(F (U )) denotes the closed convex hull of F (U ).
Lemma 1.2 (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 ).
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).
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 50, 10 pp.
http://jipam.vu.edu.au/
4
D ING -G ONG YANG , N- ENG X U ,
AND
S HIGEYOSHI OWA
2. M AIN R ESULTS
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)
then
Qp (a2 , c; h) ⊂ Qp (a1 , c; h).
Proof. Define
g(z) = z +
∞
X
(a1 )k
k=1
(a2 )k
z k+1 (z ∈ U ; 0 < a1 < a2 ).
Then
ϕp (a1 , a2 ; z)
= g(z) ∈ A,
z p−1
where ϕp (a1 , a2 ; z) is defined as in (1.5), and
z
z
(2.4)
∗
g(z)
=
.
a
(1 − z) 2
(1 − z)a1
(2.3)
From (2.4) we have
z
a1 a2 ∗
∗
∗ g(z) ∈ S 1 −
⊂S 1−
(1 − z)a2
2
2
for 0 < a1 < a2 , which implies that
a2 g(z) ∈ R 1 −
.
2
(2.5)
Since
(2.6)
Lp (a1 , c)f (z) = ϕp (a1 , a2 ; z) ∗ Lp (a2 , c)f (z) (f (z) ∈ Ap ),
we deduce from (1.7) and (2.6) that
a1 Lp (a1 + 1, c)f (z) = z (Lp (a1 , c)f (z))0 + (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))
(2.7)
= ϕp (a1 , a2 ; z) ∗ (a2 Lp (a2 + 1, c)f (z) + (a1 − a2 )Lp (a2 , c)f (z)).
By using (2.3), (2.6) and (2.7), we find that
a2
a2
p−1
L
(a
+
1,
c)f
(z)
+
1
−
L
(a
,
c)f
(z)
(z
g(z))
∗
p 2
a1 p 2
a1
Lp (a1 + 1, c)f (z)
=
Lp (a1 , c)f (z)
(z p−1 g(z)) ∗ Lp (a2 , c)f (z)
(z)
,c)f (z)
g(z) ∗ aa12 Lp (a2z+1,c)f
+ 1 − aa21 Lp (az2p−1
p−1
=
,c)f (z)
g(z) ∗ Lp (az2p−1
g(z) ∗ (q(z)F (z))
(2.8)
=
(f (z) ∈ Ap ),
g(z) ∗ q(z)
where
q(z) =
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 50, 10 pp.
Lp (a2 , c)f (z)
∈A
z p−1
http://jipam.vu.edu.au/
A NALYTIC AND M ULTIVALENT F UNCTIONS D EFINED BY L INEAR O PERATORS
and
F (z) =
5
a2 Lp (a2 + 1, c)f (z)
a2
+1− .
a1 Lp (a2 , c)f (z)
a1
Let f (z) ∈ Qp (a2 , c; h). Then
p
h(z)
a2
1−
+
+1−
a2
a2
a1
h(z)
p
(2.9)
=1−
+
= h1 (z) (say),
a1
a1
where h1 (z) is convex univalent in U , and, by (1.7),
a2
F (z) ≺
a1
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)
≺ 1 − p + h(z).
(2.10)
By using (2.1), (2.2) and (2.10), we get
zq 0 (z)
a2
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 1.1 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.
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
γ
(z ∈ U ; 0 < γ ≤ 1; −1 ≤ B < A ≤ 1).
If
0 < a1 < a2 and a2 ≥ 2 1 −
1−A
1−B
γ ,
then
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 and
γ
1−A
Re h(z) > β = h(−1) = p − 1 +
≥ 0 (z ∈ U ).
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
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 50, 10 pp.
http://jipam.vu.edu.au/
6
D ING -G ONG YANG , N- ENG X U ,
(i) a ≥ 1 − 2
or
1−A
1−B
(ii) a > 0 ≥ 1 − 2
then
AND
S HIGEYOSHI OWA
>0
1−A
1−B
,
Qp (a + 1, c; h) ⊂ Qp (a, c; h).
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−A
≥1−2
(−1 ≤ B < A ≤ 1)
1−B
1−B
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 +
∞ X
γ+1
γ+k
k=1
δ k z k (z ∈ U ; 0 < δ ≤ 1; γ ≥ 0).
If
0 < a1 < a2 and a2 ≥ 2
∞
X
k+1
(−1)
k=1
γ+1
γ+k
δk ,
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.
Corollary 2.5. Let
(2.14)
2
h(z) = p + 2
π
log
√ 2
1 + γz
√
1 − γz
(z ∈ U ; 0 < γ ≤ 1).
If
0 < a1 < a2 and a2 ≥
16
√
(arctan γ)2 ,
2
π
then
Qp (a2 , c; h) ⊂ Qp (a1 , c; h).
Proof. The function h(z) defined by (2.14) belongs to the class P (cf. [7]) and satisfies h(z) =
h(z). Thus
8
1
√
Re h(z) > β = h(−1) = p − 2 (arctan γ)2 ≥ p − > 0 (z ∈ U ).
π
2
Hence an application of Theorem 2.1 yields the desired result.
For γ = 1, Corollary 2.5 leads to
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 50, 10 pp.
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A NALYTIC AND M ULTIVALENT F UNCTIONS D EFINED BY L INEAR O PERATORS
Corollary 2.6. Let
2
h(z) = p + 2
π
log
√ 2
1+ z
√
1− z
7
(z ∈ U ).
Then, for a > 0,
Qp (a + 1, c; h) ⊂ Qp (a, c; h).
Theorem 2.7. Let h(z) ∈ P and
Re h(z) > p − 1 + α (z ∈ U ; α < 1).
(2.15)
If f (z) ∈ Qp (a, c; h),
g(z)
∈ R(α) (α < 1),
z p−1
g(z) ∈ Ap and
(2.16)
then
(f ∗ g)(z) ∈ Qp (a, c; h).
Proof. Let f (z) ∈ Qp (a, c; h) and suppose that
q(z) =
(2.17)
Lp (a, c)f (z)
.
z p−1
Then
F (z) =
(2.18)
p h(z)
Lp (a + 1, c)f (z)
≺1− +
,
Lp (a, c)f (z)
a
a
q(z) ∈ A and
zq 0 (z)
≺ 1 − p + h(z)
q(z)
(2.19)
(see (2.10) used in the proof of Theorem 2.1). By (2.15) and (2.19), we see that
q(z) ∈ S ∗ (α).
(2.20)
For g(z) ∈ Ap , it follows from (2.17) and (2.18) that
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 1.1 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 = 21 , Theorem 2.7 reduces to
Corollary 2.8. Let h(z) ∈ P and g(z) ∈ Ap satisfy either
(i)
g(z)
z p−1
is convex univalent in U and
Re h(z) > p − 1 (z ∈ U )
or
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 50, 10 pp.
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8
D ING -G ONG YANG , N- ENG X U ,
(ii)
g(z)
z p−1
AND
S HIGEYOSHI OWA
∈ S ∗ ( 12 ) and
Re h(z) > p −
1
(z ∈ U ).
2
If f (z) ∈ Qp (a, c; h), then
(f ∗ g)(z) ∈ Qp (a, c; h).
Theorem 2.9. Let h(z) ∈ P and
Re h(z) > − Re λ (z ∈ U ),
(2.22)
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
t f (t)dt
(2.23)
g(z) =
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
= aLp (a + 1, c)g(z) + (λ + p − a)Lp (a, c)g(z).
(2.24)
If we let
q(z) =
(2.25)
Lp (a + 1, c)g(z)
,
Lp (a, c)g(z)
then (2.24) and (2.25) lead to
(2.26)
aq(z) + λ + p − a = (λ + p)
Lp (a, c)f (z)
.
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)
Let f (z) ∈ Qp (a, c; h). Then it follows from (2.27) that
(2.28)
zq 0 (z)
p h(z)
q(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) + Reλ > 0 (z ∈ U ).
a
a
Therefore, it follows from (2.28), (2.29) and Lemma 1.2 that
q(z) ≺ 1 −
p h(z)
+
.
a
a
This proves that g(z) ∈ Qp (a, c; h).
From Theorem 2.9 we have the following corollaries.
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A NALYTIC AND M ULTIVALENT F UNCTIONS D EFINED BY L INEAR O PERATORS
9
Corollary 2.10. Let h(z) be defined as in Corollary 2.2. If f (z) ∈ Qp (a, c; h) and
γ
1−A
Reλ ≥ 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
Reλ ≥
(−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).
Corollary 2.12. Let h(z) be defined as in Corollary 2.5. If f (z) ∈ Qp (a, c; h) and
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).
Reλ ≥
Theorem 2.13. Let h(z) ∈ P and
Re h(z) > −
(2.30)
Reλ
(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
(2.31)
Lp (a, c)g(z) =
t
(Lp (a, c)f (t)) dt
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
q(z) =
(2.34)
Lp (a + 1, c)g(z)
.
Lp (a, c)g(z)
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) = β Re h(z) + Reλ > 0 (z ∈ U ).
a
a
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 50, 10 pp.
http://jipam.vu.edu.au/
10
D ING -G ONG YANG , N- ENG X U , AND S HIGEYOSHI OWA
Now, in view of (2.34), (2.35) and (2.36), an application of Lemma 1.2 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
Reλ ≥ β 1 − p −
(0 < γ ≤ 1; −1 ≤ B < A ≤ 1; β > 0),
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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