SOME SUBORDINATION CRITERIA CONCERNING
THE SǍLǍGEAN OPERATOR
Subordination Criteria
KAZUO KUROKI AND SHIGEYOSHI OWA
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502, Japan
EMail: freedom@sakai.zaq.ne.jp
owa@math.kindai.ac.jp
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
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Received:
14 September, 2008
Accepted:
11 January, 2009
Communicated by:
H.M. Srivastava
2000 AMS Sub. Class.:
26D15.
Key words:
Janowski function, Starlike, Convex, Univalent, Subordination.
Abstract:
Applying Sǎlǎgean operator, for the class A of analytic functions f (z) in the
open unit disk U which are normalized by f (0) = f 0 (0) − 1 = 0, the generalization of an analytic function to discuss the starlikeness is considered. Furthermore, from the subordination criteria for Janowski functions generalized by
some complex parameters, some interesting subordination criteria for f (z) ∈ A
are given.
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Contents
1
Introduction, Definition and Preliminaries
3
2
Subordinations for the Class Defined by the Sǎlǎgean Operator
11
3
Subordination Criteria for Other Analytic Functions
17
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
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1.
Introduction, Definition and Preliminaries
Let A denote the class of functions f (z) of the form:
(1.1)
f (z) = z +
∞
X
an z n
n=2
Subordination Criteria
which are analytic in the open unit disk
Kazuo Kuroki and Shigeyoshi Owa
U = {z : z ∈ C
|z| < 1}.
and
vol. 10, iss. 2, art. 36, 2009
Also, let P denote the class of functions p(z) of the form:
(1.2)
p(z) = 1 +
∞
X
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pn z n
Contents
n=1
which are analytic in U. If p(z) ∈ P satisfies Re p(z) > 0 (z ∈ U), then we say
that p(z) is the Carathéodory function (cf. [1]).
By the familiar principle of differential subordination between analytic functions
f (z) and g(z) in U, we say that f (z) is subordinate to g(z) in U if there exists an
analytic function w(z) satisfying the following conditions:
w(0) = 0 and |w(z)| < 1
(z ∈ U),
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such that
f (z) = g w(z)
(z ∈ U).
We denote this subordination by
f (z) ≺ g(z)
(z ∈ U).
In particular, if g(z) is univalent in U, then it is known that
f (z) ≺ g(z)
(z ∈ U)
⇐⇒
f (0) = g(0)
and f (U) ⊂ g(U).
For the function p(z) ∈ P, we introduce the following function
p(z) =
(1.3)
1 + Az
1 + Bz
(−1 5 B < A 5 1)
Subordination Criteria
which has been investigated by Janowski [3]. Thus, the function p(z) given by (1.3)
is said to be the Janowski function. And, as a generalization of the Janowski function, Kuroki, Owa and Srivastava [2] have discussed the function
p(z) =
1 + Az
1 + Bz
(ii) |A| 5 1, |B| = 1, A 6= B, and 1 − AB > 0.
Here, for some complex numbers A and B which satisfy condition (i), the function
p(z) is analytic and univalent in U and p(z) maps the open unit disk U onto the open
disk given by
1 − AB |A − B|
(1.4)
p(z) − 1 − |B|2 < 1 − |B|2 .
Thus, it is clear that
(1.5)
Re(1 − AB) − |A − B|
Re p(z) >
=0
1 − |B|2
vol. 10, iss. 2, art. 36, 2009
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for some complex parameters A and B which satisfy one of following conditions

 (i) |A| 5 1, |B| < 1, A 6= B, and Re 1 − AB = |A − B|

Kazuo Kuroki and Shigeyoshi Owa
(z ∈ U).
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Also, for some complex numbers A and B which satisfy condition (ii), the function p(z) is analytic and univalent in U and the domain p(U) is the right half-plane
satisfying
(1.6)
1 − |A|2
Re p(z) >
= 0.
2(1 − AB)
Hence, we see that the generalized Janowski function maps the open unit disk U
onto some domain which is on the right half-plane.
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
Remark 1. For the function
1 + Az
1 + Bz
defined with the condition (i), the inequalities (1.4) and (1.5) give us that
p(z) =
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p(z) 6= 0 namely,
1 + Az 6= 0
(z ∈ U).
Since, after a simple calculation, we see the condition |A| 5 1, we can omit the
condition |A| 5 1 in (i).
Hence, the condition (i) is newly defined by the following conditions
(1.7)
|B| < 1, A 6= B, and Re 1 − AB = |A − B|.
A function f (z) ∈ A is said to be starlike of order α in U if it satisfies
0 zf (z)
(1.8)
Re
>α
(z ∈ U)
f (z)
for some α (0 5 α < 1). We denote by S ∗ (α) the subclass of A consisting of all
functions f (z) which are starlike of order α in U.
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Similarly, if f (z) ∈ A satisfies the following inequality
zf 00 (z)
>α
(z ∈ U)
(1.9)
Re 1 + 0
f (z)
for some α (0 5 α < 1), then f (z) is said to be convex of order α in U. We denote
by K(α) the subclass of A consisting of all functions f (z) which are convex of order
α in U.
As usual, in the present investigation, we write
∗
S (0) ≡ S
∗
and K(0) ≡ K.
D0 f (z) = f (z) = z +
(1.10)
an z n ,
n=2
(1.11)
1
0
D f (z) = Df (z) = zf (z) = z +
∞
X
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nan z n ,
n=2
(1.12)
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
The classes S ∗ (α) and K(α) were introduced by Robertson [7].
We define the following differential operator due to Sǎlǎgean [8].
For a function f (z) and j = 1, 2, 3, . . . ,
∞
X
Subordination Criteria
∞
X
j
j−1
D f (z) = D D f (z) = z +
nj an z n .
n=2
Also, we consider the following differential operator
Z z
∞
X
f (ζ)
−1
dζ = z +
n−1 an z n ,
(1.13)
D f (z) =
ζ
0
n=2
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(1.14)
∞
X
D−j f (z) = D−1 D−(j−1) f (z) = z +
n−j an z n
n=2
for any negative integers.
Then, for f (z) ∈ A given by (1.1), we know that
(1.15)
Dj f (z) = z +
∞
X
Subordination Criteria
n j an z n
(j = 0, ±1, ±2, . . . ).
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
n=2
We consider the subclass Sjk (α) as follows:
k
D f (z)
k
>α
Sj (α) = f (z) ∈ A : Re
Dj f (z)
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(z ∈ U ; 0 5 α < 1) .
Sjj+1 (α)
In particular, putting k = j + 1, we also define
j+1
D f (z)
j+1
Sj (α) = f (z) ∈ A : Re
>α
Dj f (z)
by
(z ∈ U ; 0 5 α < 1) .
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0
z zf 0 (z)
D2 f (z)
zf 00 (z)
=
=
1
+
,
D1 f (z)
zf 0 (z)
f 0 (z)
we see that
S01 (α) ≡ S ∗ (α),
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Remark 2. Noting
D1 f (z)
zf 0 (z)
=
,
D0 f (z)
f (z)
Contents
S12 (α) ≡ K(α)
(0 5 α < 1).
Close
Furthermore, by applying subordination, we consider the following subclass
Dk f (z)
1 + Az
k
Pj (A, B) = f (z) ∈ A : j
≺
(z ∈ U ; A 6= B, |B| 5 1) .
D f (z)
1 + Bz
In particular, putting k = j + 1, we also define
Dj+1 f (z)
1 + Az
j+1
Pj (A, B) = f (z) ∈ A :
≺
Dj f (z)
1 + Bz
(z ∈ U; A 6= B, |B| 5 1) .
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
Remark 3. Noting
vol. 10, iss. 2, art. 36, 2009
Dk f (z)
1 + (1 − 2α)z
≺
Dj f (z)
1−z
⇐⇒
Re
Dk f (z)
Dj f (z)
>α
(z ∈ U; 0 5 α < 1),
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we see that
P01 (1 − 2α, −1) ≡ S ∗ (α),
P12 (1 − 2α, −1) ≡ K(α)
(0 5 α < 1).
In our investigation here, we need the following lemma concerning the differential subordination given by Miller and Mocanu [5] (see also [6, p. 132]).
Lemma 1.1. Let the function q(z) be analytic and univalent in U. Also let φ(ω) and
ψ(ω) be analytic in a domain C containing q(U), with
ψ(ω) 6= 0 ω ∈ q(U) ⊂ C .
Set
Q(z) = zq 0 (z)ψ q(z)
and h(z) = φ q(z) + Q(z),
and suppose that
(i)
Q(z) is starlike and univalent in U;
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and
(ii)
Re
zh0 (z)
Q(z)
= Re
!
φ0 q(z)
zQ0 (z)
+
>0
Q(z)
ψ q(z)
(z ∈ U).
If p(z) is analytic in U, with
p(0) = q(0)
and p(U) ⊂ C,
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
and
vol. 10, iss. 2, art. 36, 2009
φ p(z) + zp0 (z)ψ p(z) ≺ φ q(z) + zq 0 (z)ψ q(z) =: h(z)
(z ∈ U),
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then
p(z) ≺ q(z)
(z ∈ U)
Contents
and q(z) is the best dominant of this subordination.
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By making use of Lemma 1.1, Kuroki, Owa and Srivastava [2] have investigated
some subordination criteria for the generalized Janowski functions and deduced the
following lemma.
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Lemma 1.2. Let the function f (z) ∈ A be chosen so that f (z)
6= 0 (z ∈ U).
z
Also, let α (α 6= 0), β (−1 5 β 5 1), and some complex parameters A and B satisfy
one of following conditions:
(i) |B| < 1, A 6= B, and Re(1 − AB) = |A − B| be such that
β(1 − α) (1 + β) Re 1 − AB − |A − B|
1−β
1+β
+
+
+
−1 = 0,
2
α
1 − |B|
1 + |A| 1 + |B|
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(ii) |B| = 1, |A| 5 1, A 6= B, and 1 − AB > 0 be such that
β(1 − α) (1 + β)(1 − |A|2 ) (1 − β)(1 − |A|)
+
+
= 0.
α
2(1 + |A|)
2(1 − AB)
If
(1.16)
zf 0 (z)
f (z)
β zf 00 (z)
1+α 0
f (z)
≺ h(z)
(z ∈ U),
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
where
h(z) =
Subordination Criteria
1 + Az
1 + Bz
β−1 1 + Az α(1 + Az)2 + α(A − B)z
(1 − α)
+
,
1 + Bz
(1 + Bz)2
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then
zf 0 (z)
1 + Az
≺
f (z)
1 + Bz
(z ∈ U).
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2.
Subordinations for the Class Defined by the Sǎlǎgean
Operator
First of all, by applying the Sǎlǎgean operator for f (z) ∈ A, we consider the following subordination criterion in the class Pjk (A, B) for some complex parameters
A and B.
j
Theorem 2.1. Let the function f (z) ∈ A be chosen so that D fz (z) 6= 0 (z ∈ U).
Also, let α (α 6= 0), β (−1 5 β 5 1), and some complex parameters A and B satisfy
one of following conditions:
(i) |B| < 1, A 6= B, and Re(1 − AB) = |A − B| be so that
1−β
1+β
β(1 − α) (1 + β) Re 1 − AB − |A − B|
+
+
−1 = 0,
+
1 − |B|2
1 + |A| 1 + |B|
α
(ii) |B| = 1, |A| 5 1, A 6= B, and 1 − AB > 0 be so that
β(1 − α) (1 + β)(1 − |A|2 ) (1 − β)(1 − |A|)
+
+
= 0.
α
2(1 + |A|)
2(1 − AB)
If
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
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k
β k
D f (z)
D f (z) Dk+1 f (z) Dj+1 f (z)
(2.1)
(1−α)+α
+ k
− j
≺ h(z),
Dj f (z)
Dj f (z)
D f (z)
D f (z)
where
h(z) =
1 + Az
1 + Bz
β−1 1 + Az α(1 + Az)2 + α(A − B)z
(1 − α)
+
,
1 + Bz
(1 + Bz)2
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then
Dk f (z)
1 + Az
≺
Dj f (z)
1 + Bz
(z ∈ U).
Proof. If we define the function p(z) by
Dk f (z)
p(z) = j
D f (z)
(z ∈ U),
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
then p(z) is analytic in U with p(0) = 1. Further, since
k
k+1
D f (z) Dj+1 f (z)
D f (z)
0
zp (z) =
−
,
Dj f (z)
Dk f (z)
Dj f (z)
the condition (2.1) can be written as follows:
β β−1
p(z)
(1 − α) + αp(z) + αzp0 (z) p(z)
≺ h(z)
vol. 10, iss. 2, art. 36, 2009
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(z ∈ U).
We also set
q(z) =
1 + Az
,
1 + Bz
φ(z) = z β (1 − α + αz),
and ψ(z) = αz β−1
for z ∈ U. Then, it is clear that the function q(z) is analytic and univalent in U and
has a positive real part in U for the conditions (i) and (ii).
Therefore, φ and ψ are analytic in a domain C containing q(U), with
ψ(ω) 6= 0
ω ∈ q(U) ⊂ C .
Also, for the function Q(z) given by
α(A − B)z(1 + Az)β−1
,
Q(z) = zq 0 (z)ψ q(z) =
(1 + Bz)β+1
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we obtain
1−β
1+β
zQ0 (z)
=
+
− 1.
Q(z)
1 + Az 1 + Bz
(2.2)
Furthermore, we have
h(z) = φ q(z) + Q(z)
β 1 + Az
1 + Az
α(A − B)z(1 + Az)β−1
=
1−α+α
+
1 + Bz
1 + Bz
(1 + Bz)β+1
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
and
zh0 (z)
β(1 − α)
zQ0 (z)
=
+ (1 + β)q(z) +
.
Q(z)
α
Q(z)
(2.3)
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Hence,
(i) For the complex numbers A and B such that
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|B| < 1, A 6= B,
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and
Re(1 − AB) = |A − B|,
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it follows from (2.2) and (2.3) that
0 zQ (z)
1−β
1+β
Re
>
+
− 1 = 0,
Q(z)
1 + |A| 1 + |B|
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Re
zh0 (z)
Q(z)
β(1 − α) (1 + β) Re 1 − AB − |A − B|
>
+
α
1 − |B|2
1+β
1−β
+
+
−1=0
1 + |A| 1 + |B|
(z ∈ U).
(ii) For the complex numbers A and B such that
|B| = 1, |A| 5 1, A 6= B,
and
1 − AB > 0,
from (2.2) and (2.3), we get
0 zQ (z)
1−β
1
(1 − β)(1 − |A|)
Re
>
+ (1 + β) − 1 =
= 0,
Q(z)
1 + |A| 2
2(1 + |A|)
Subordination Criteria
and
0 zh (z)
β(1 − α) (1 + β)(1 − |A|2 ) (1 − β)(1 − |A|)
Re
>
+
+
=0
Q(z)
α
2(1 + |A|)
2(1 − AB)
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
(z ∈ U).
Since all the conditions of Lemma 1.1 are satisfied, we conclude that
Dk f (z)
1 + Az
≺
j
D f (z)
1 + Bz
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(z ∈ U),
which completes the proof of Theorem 2.1.
Remark 4. We know that a function f (z) satisfying the conditions in Theorem 2.1
belongs to the class Pjk (A, B).
Letting k = j + 1 in Theorem 2.1, we obtain the following theorem.
j
Theorem 2.2. Let the function f (z) ∈ A be chosen so that D fz (z) 6= 0 (z ∈ U).
Also, let α (α 6= 0), β (−1 5 β 5 1), and some complex parameters A and B satisfy
one of following conditions
(i) |B| < 1, A 6= B, and Re(1 − AB) = |A − B| be so that
1−β
1+β
β(1 − α) (1 + β) Re 1 − AB − |A − B|
+
+
− 1 = 0,
+
2
1 − |B|
1 + |A| 1 + |B|
α
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(ii) |B| = 1, |A| 5 1, A 6= B, and 1 − AB > 0 be so that
β(1 − α) (1 + β)(1 − |A|2 ) (1 − β)(1 − |A|)
+
= 0.
+
α
2(1 + |A|)
2(1 − AB)
If
(2.4)
Dj+1 f (z)
Dj f (z)
β Dj+2 f (z)
1 − α + α j+1
D f (z)
≺ h(z),
where
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
h(z) =
then
Subordination Criteria
β−1 1 + Az α(1 + Az)2 + α(A − B)z
1 + Az
(1 − α)
+
,
1 + Bz
1 + Bz
(1 + Bz)2
Dj+1 f (z)
1 + Az
≺
j
D f (z)
1 + Bz
Contents
(z ∈ U).
Remark 5. A function f (z) satisfying the conditions in Theorem 2.2 belongs to the
class Pjj+1 (A, B). Setting j = 0 in Theorem 2.2, we obtain Lemma 1.2 proven by
Kuroki, Owa and Srivastava [2].
Also, if we assume that
1 − µ iθ
α = 1, β = A = 0, and B =
e
(0 5 µ < 1, 0 5 θ < 2π),
1+µ
Theorem 2.2 becomes the following corollary.
j
Corollary 2.3. If f (z) ∈ A D fz (z) 6= 0 in U satisfies
Dj+2 f (z)
1 + µ − (1 − µ)eiθ z
≺
Dj+1 f (z)
1 + µ + (1 − µ)eiθ z
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(z ∈ U; 0 5 θ < 2π)
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for some µ (0 5 µ < 1), then
Dj+1 f (z)
1+µ
≺
j
D f (z)
1 + µ + (1 − µ)eiθ z
(z ∈ U).
From the above corollary, we have
j+2
j+1
D f (z)
D f (z)
1+µ
> µ =⇒ Re
>
Re
j+1
j
D f (z)
D f (z)
2
In particular, making j = 0, we get
0 zf 00 (z)
zf (z)
1+µ
> µ =⇒ Re
>
Re 1 + 0
f (z)
f (z)
2
(z ∈ U; 0 5 µ < 1).
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
(z ∈ U; 0 5 µ < 1),
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namely
f (z) ∈ K(µ)
=⇒
f (z) ∈ S
∗
1+µ
2
(z ∈ U; 0 5 µ < 1).
And, taking µ = 0, we find that every convex function is starlike of order 21 . This
fact is well-known as the Marx-Strohhäcker theorem in Univalent Function Theory
(cf. [4, 9]).
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3.
Subordination Criteria for Other Analytic Functions
In this section, by making use of Lemma 1.1, we consider some subordination critej
ria concerning the analytic function D fz (z) for f (z) ∈ A.
Theorem 3.1. Let α (α 6= 0), β (−1 5 β 5 1), and some complex parameters A
and B which satisfy one of following conditions
Subordination Criteria
(i) |B| < 1, A 6= B, and Re(1 − AB) = |A − B| be so that
β
1−β
1+β
+
+
− 1 = 0,
α 1 + |A| 1 + |B|
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
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(ii) |B| = 1, |A| 5 1, A 6= B, and 1 − AB > 0 be so that
β (1 − β)(1 − |A|)
+
= 0.
α
2(1 + |A|)
If f (z) ∈ A satisfies
β j
Dj+1 f (z)
D f (z)
1−α+α j
≺ h(z),
(3.1)
z
D f (z)
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where
h(z) =
then
1 + Az
1 + Bz
β
+
α(A − B)z(1 + Az)β−1
,
(1 + Bz)β+1
Dj f (z)
1 + Az
≺
z
1 + Bz
(z ∈ U).
Close
Proof. If we define the function p(z) by
p(z) =
Dj f (z)
z
(z ∈ U),
then p(z) is analytic in U with p(0) = 1 and the condition (3.1) can be written as
follows:
β
β−1
p(z) + αzp0 (z) p(z)
≺ h(z)
(z ∈ U).
We also set
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
1 + Az
q(z) =
,
1 + Bz
vol. 10, iss. 2, art. 36, 2009
β
φ(z) = z ,
and
ψ(z) = αz
β−1
for z ∈ U. Then, the function q(z) is analytic and univalent in U and satisfies
Re q(z) > 0
(z ∈ U)
for the condition (i) and (ii).
Thus, the functions φ and ψ satisfy the conditions required by Lemma 1.1.
Further, for the functions Q(z) and h(z) given by
Q(z) = zq 0 (z)ψ q(z) and h(z) = φ q(z) + Q(z),
we have
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1−β
1+β
zQ0 (z)
=
+
− 1 and
Q(z)
1 + Az 1 + Bz
zh0 (z)
β zQ0 (z)
= +
.
Q(z)
α
Q(z)
Then, similarly to the proof of Theorem 2.1, we see that
0 0 zQ (z)
zh (z)
Re
> 0 and Re
>0
Q(z)
Q(z)
(z ∈ U)
Close
for the conditions (i) and (ii).
Thus, by applying Lemma 1.1, we conclude that p(z) ≺ q(z)
The proof of the theorem is completed.
(z ∈ U).
Letting j = 0 in Theorem 3.1, we obtain the following theorem.
Theorem 3.2. Let α (α 6= 0), β (−1 5 β 5 1), and some complex parameters A
and B satisfy one of following conditions:
(i) |B| < 1, A 6= B, and Re(1 − AB) = |A − B| be so that
β
1−β
1+β
+
+
− 1 = 0,
α 1 + |A| 1 + |B|
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
Title Page
(ii) |B| = 1, |A| 5 1, A 6= B, and 1 − AB > 0 be so that
β (1 − β)(1 − |A|)
+
= 0.
α
2(1 + |A|)
If f (z) ∈ A satisfies
(3.2)
then
f (z)
z
β−1 Contents
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f (z)
0
(1 − α)
+ αf (z)
z
β
1 + Az
α(A − B)z(1 + Az)β−1
≺
+
,
1 + Bz
(1 + Bz)β+1
f (z)
1 + Az
≺
z
1 + Bz
(z ∈ U).
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Also, taking
α = 1, β = A = 0,
1 − ν iθ
e
and B =
ν
1
5 ν < 1, 0 5 θ < 2π
2
in Theorem 3.2, we have
Corollary 3.3. If f (z) ∈ A satisfies
for some ν
1
2
zf 0 (z)
ν
≺
f (z)
ν + (1 − ν)eiθ z
5 ν < 1 , then
Subordination Criteria
(z ∈ U; 0 5 θ < 2π)
f (z)
ν
≺
z
ν + (1 − ν)eiθ z
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
Title Page
(z ∈ U).
Contents
Further, making
α = β = 1, A = 0,
and B =
1 − ν iθ
e
ν
1
5 ν < 1, 0 5 θ < 2π
2
in Theorem 3.2, we get
Corollary 3.4. If f (z) ∈ A satisfies
2
ν
0
f (z) ≺
ν + (1 − ν)eiθ z
for some ν 21 5 ν < 1 , then
ν
f (z)
≺
z
ν + (1 − ν)eiθ z
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(z ∈ U ; 0 5 θ < 2π)
(z ∈ U).
Close
The above corollaries give:
0 zf (z)
(3.3) Re
> ν =⇒
f (z)
Re
f (z)
z
>ν
1
z ∈ U; 5 ν < 1 ,
2
and
(3.4)
Re
p
f 0 (z)
>ν
=⇒
Re
f (z)
z
>ν
1
z ∈ U; 5 ν < 1 .
2
1
,
2
Here, taking ν =
we find some results that are known as the Marx-Strohhäcker
theorem in Univalent Function Theory (cf. [4], [9]).
Setting j = 1 in Theorem 3.1, we obtain the following theorem.
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
Title Page
Theorem 3.5. Let α (α 6= 0), β (−1 5 β 5 1), and some complex parameters A
and B satisfy one of following conditions
(i) |B| < 1, A 6= B, and Re(1 − AB) = |A − B| be so that
1−β
1+β
β
+
+
− 1 = 0,
α 1 + |A| 1 + |B|
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(ii) |B| = 1, |A| 5 1, A 6= B, and 1 − AB > 0 be so that
β (1 − β)(1 − |A|)
+
= 0.
α
2(1 + |A|)
If f (z) ∈ A satisfies
β
β
1 + Az
α(A − B)z(1 + Az)β−1
zf 00 (z)
0
(3.5) f (z)
1+α 0
≺
+
,
f (z)
1 + Bz
(1 + Bz)β+1
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then
f 0 (z) ≺
1 + Az
1 + Bz
(z ∈ U).
Here, making
α = 1, β = A = 0,
1 − ν iθ
and B =
e
ν
1
5 ν < 1, 0 5 θ < 2π
2
in Theorem 3.5, we have:
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
Corollary 3.6. If f (z) ∈ A satisfies
zf 00 (z)
ν
1+ 0
≺
f (z)
ν + (1 − ν)eiθ z
for some ν 12 5 ν < 1 , then
f 0 (z) ≺
ν
ν + (1 − ν)eiθ z
(z ∈ U ; 0 5 θ < 2π)
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(z ∈ U).
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Also, from Corollary 3.6 we have:
zf 00 (z)
(3.6) Re 1 + 0
> ν =⇒ Re (f 0 (z)) > ν
f (z)
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1
z ∈ U; 5 ν < 1 .
2
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References
[1] P.L. DUREM, Univalent Functions, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983.
[2] K. KUROKI, S. OWA AND H.M. SRIVASTAVA, Some subordination criteria
for analytic functions, Bull. Soc. Sci. Lett. Lodz, Vol., 52 (2007), 27–36.
[3] W. JANOWSKI, Extremal problem for a family of functions with positive real
part and for some related families, Ann. Polon. Math., 23 (1970), 159–177.
[4] A. MARX, Untersuchungen uber schlichte Abbildungen, Math. Ann., 107
(1932/33), 40–67.
Subordination Criteria
Kazuo Kuroki and Shigeyoshi Owa
vol. 10, iss. 2, art. 36, 2009
Title Page
[5] S.S. MILLER AND P.T. MOCANU, On some classes of first-order differential
subordinations, Michigan Math. J., 32 (1985), 185–195.
[6] S.S. MILLER AND P.T. MOCANU, Differential Subordinations, Pure and Applied Mathematics 225, Marcel Dekker, 2000.
[7] M.S. ROBERTSON, On the theory of univalent functions, Ann. Math., 37
(1936), 374–408.
[8] G.S. SǍLǍGEAN, Subclass of univalent functions, Complex Analysis-Fifth
Romanian-Finnish Seminar, Part 1(Bucharest, 1981), Lecture Notes in Math.,
vol. 1013, Springer, Berlin, 1983, pp. 362–372.
[9] E. STROHHÄCKER, Beitrage zur Theorie der schlichten Funktionen, Math.
Z., 37 (1933), 356–380.
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