Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 845724, 11 pages
doi:10.1155/2008/845724
Research Article
Differential Subordinations Associated with
Multiplier Transformations
Adriana Cătaş, Georgia Irina Oros, and Gheorghe Oros
Department of Mathematics, Faculty of Science, University of Oradea, 1 University Street,
410087 Oradea, Romania
Correspondence should be addressed to Adriana Cătaş, acatas@gmail.com
Received 26 February 2008; Accepted 15 May 2008
Recommended by Ferhan Atici
The authors introduce new classes of analytic functions in the open unit disc which are defined by
using multiplier transformations. The properties of these classes will be studied by using techniques
involving the Briot-Bouquet differential subordinations. Also an integral transform is established.
Copyright q 2008 Adriana Cătaş et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction and definitions
Let H be the class of analytic functions in the open unit disc
U z ∈ C : |z| < 1
1.1
and let Ha, n be the subclass of H consisting of functions of the form fz a an zn an1 zn1 · · · . Let Ap, n denote the class of functions fz normalized by
fz zp ∞
ak zk
p, n ∈ N : {1, 2, 3, . . .}
1.2
kpn
which are analytic in the open unit disc. In particular, we set
Ap, 1 : Ap ,
A1, n : An,
A1, 1 : A A1 .
1.3
If a function fz belongs to the class An, it has the form
fz z ∞
ak zk
kn1
n ∈ N : {1, 2, 3, . . .} .
1.4
2
Abstract and Applied Analysis
For two functions fz given by 1.4 and for gz given by
∞
gz z bk z k
n ∈ N,
1.5
kn1
the Hadamard product or convolution f∗gz is defined, as usual, by
f∗gz : z ∞
ak bk zk : g∗fz.
1.6
kn1
If f and g are analytic in U, we say that f is subordinate to g, written symbolically as
f ≺g
or fz ≺ gz z ∈ U
1.7
if there exists a Schwarz function wz in U which is analytic in U with w0 0 and |wz| < 1
such that fz gwz, z ∈ U.
We consider the following multiplier transformations.
Definition 1.1 see 1. Let f ∈ Ap, n. For δ, λ ∈ R, λ ≥ 0, δ ≥ 0, l ≥ 0, define the multiplier
transformations Ip δ, λ, l on Ap, n by the following infinite series:
Ip δ, λ, lfz : zp ∞ p λk − p l δ
ak zk .
p
l
kpn
1.8
It follows from 1.8 that
Ip 0, λ, lfz fz,
p lIp 2, λ, lfz p1 − λ l Ip 1, λ, lfz λz Ip 1, λ, lfz ,
Ip δ1 , λ, l Ip δ2 , λ, l fz Ip δ2 , λ, l Ip δ1 , λ, l fz .
1.9
Remark 1.2 see 1. For p 1, l 0, λ ≥ 0, δ m, m ∈ N0 , N0 N ∪ {0}, the operator Dλm :
I1 m, λ, 0 was introduced and studied by Al-Oboudi 2 which is reduced to the Sălăgean
differential operator 3 for λ 1. The operator Ilm : I1 m, 1, l was studied recently by Cho
and Srivastava 4 and by Cho and Kim 5. The operator Im : I1 m, 1, 1 was studied by
Uralegaddi and Somanatha 6, the operator Dλδ : I1 δ, λ, 0 was introduced by Acu and Owa
7 and the operator Ip m, l : Ip m, 1, l was investigated recently by Sivaprasad Kumar et al.
8.
If f is given by 1.2, then we have
Ip δ, λ, lfz f∗ϕδp,λ,l z,
where
ϕδp,λ,l z zp ∞ p λk − p l δ k
z .
pl
kpn
1.10
1.11
In particular, we set
I1 δ, λ, lfz : Iδ, λ, lfz.
In order to prove our main results, we will make use of the following lemmas.
1.12
Adriana Cătaş et al.
3
Lemma 1.3 see 9. For real or complex numbers a, b, and c c/
∈Z−0 : {0, −1, −2, . . .}, the following
hold:
1
ΓbΓc − b
· 2 F1 a, b; c; z
tb−1 1 − tc−b−1 1 − tz−a dt Re c > Re b > 0,
1.13
Γc
0
z
−a
,
1.14
2 F 1 a, b; c; z 1 − z ·2 F1 a, c − b; c;
z−1
2 F 1 a, b; c; z
2 F1 b, a; c; z,
b 1· 2 F1 1, b; b 1; z b 1 bz·2 F1 1, b 1; b 2; z,
√
π·Γ a b 1/2
ab1 1
;
.
2 F 1 a, b;
2
2
Γ a 1/2 Γ b 1/2
1.15
1.16
1.17
Lemma 1.4 see 10. Let β, γ ∈ C, β / 0 and let h be convex in U, with
Re βhz γ > 0
z ∈ U.
1.18
If the function p ∈ Hh0, n, then
pz zp z
≺ hz ⇒ pz ≺ hz.
βpz γ
1.19
Lemma 1.5 see 11. Let μ be a positive measure on the unit interval I 0, 1. Let gt, z be a
function analytic in 0, 1 × U, for each t ∈ I and integrable in t, for each z ∈ U and for almost all t ∈ I.
Suppose also that
Re gt, z > 0 z ∈ U; t ∈ I,
1.20
gt, −r is real for real r and
Re
1
1
≥
gt, z gt, −r
|z| ≤ r < 1, t ∈ I .
1.21
If
gz gt, zdμt,
1.22
I
then
Re
1
gz
≥
1
g−r
|z| ≤ r < 1 .
1.23
Lemma 1.6 see 12. Let ψz be univalent in the unit disc U and let v and ϕ be analytic in a domain
D ⊃ ψU with ϕw / 0, when w ∈ ψU. Set
Qz : zψ zϕ ψz ,
hz : v ψz Qz.
1.24
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Abstract and Applied Analysis
Suppose that
1 Qz is starlike in U and
2 Re zh z/Qz > 0 for z ∈ U.
If qz is analytic in U, with q0 ψ0, qU ⊂ D, and
v qz zq zϕ qz ≺ v ψz zψ zϕ ψz ,
1.25
then qz ≺ ψz and ψz is the best dominant.
Lemma 1.7 see 12, Theorem 3.3d. Let β, γ, A ∈ C, with Re β γ > 0 and let B ∈ −1, 0 satisfy
either
Re β1 AB γ1 B 2 ≥ βA βB Bγ γ,
1.26
when B ∈ −1, 0, or
Re β1 − A 2γ ≥ 0,
β1 A > 0,
1.27
when B −1. If p ∈ H1, n satisfies
pz zp z
1 Az
≺
,
βpz γ
1 Bz
1.28
then
pz ≺ qz qn z ≺
1 Az
,
1 Bz
1.29
where qn is the univalent solution of the differential equation
qz nzq z
1 Az
.
βqz γ
1 Bz
1.30
In addition, the function qn , is the best 1, n-dominant and the function qn is given by
γ
β γ kz β/n γ zK z
1
− ,
qz − β
Kz
β
Kz
βgz β
where
kz z exp
z
ht − 1t−1 dt,
0
βγ
Kz nzγ/n
1.31
z
k
β/n
γ/n−1
tt
n/β
dt
,
1.32
0
and the univalent function g is given by
1
gz n
1
0
ktz
kz
β/n
tγ/n−1 dt.
1.33
Now we define new classes of analytic functions by using the multiplier transformations
Im, λ, l defined by 1.8 as follows.
Adriana Cătaş et al.
5
2. Main results
Definition 2.1. Let −1 ≤ B < A ≤ 1, λ > 0, l ≥ 0, m ∈ N ∪ {0}. A function f ∈ An is said to be in
the class Sm, λ, l; A, B if it satisfies the following subordination:
l 1 Im 1, λ, lfz 1 − λ l 1 Az
·
−
≺
λ
Im, λ, lfz
l1
1 Bz
z ∈ U.
2.1
Remark 2.2. We note that
S0, 1, 0; 1 − 2α, −1 ≡ S∗ α,
S1, 1, 0; 1 − 2α, −1 ≡ Kα,
2.2
where S∗ α and Kα 0 ≤ α < 1 denote the subclasses of functions in A which are,
respectively, starlike of order α and convex of order α in U. Also we have the class
Sm, λ, 0; A, B ≡ Sm
λ A, B
2.3
studied by Patel 13.
Let φz be analytic in U and φ0 1. We introduce the following definition.
Definition 2.3. A function f ∈ An is said to be in the class Am, λ, l, n; φ if it satisfies the
following subordination:
Im 1, λ, lfz
≺ φz,
Im, λ, lfz
z ∈ U.
2.4
Remark 2.4. We note that the classes Am, 1, l, n; φ were investigated recently by Sivaprasad
Kumar et al. 8.
Theorem 2.5. Let −1 ≤ B < A ≤ 1, l ≥ 0, λ > 0, and
1 − B1 − λ l λ1 − A > 0.
2.5
Sm 1, λ, l; A, B ⊂ Sm, λ, l; A, B.
2.6
i Then
Further, for f ∈ Sm 1, λ, l; A, B, the following hold:
l 1 Im 1, λ, lfz 1 − λ l
1 Az
·
−
≺ qz ≺
,
λ
Im, λ, lfz
l1
1 Bz
2.7
6
Abstract and Applied Analysis
where
qz ⎧
1−λl
1
⎪
⎪
−
⎪
⎪
λ
⎨ 1/n 1 sl1/λn−1 1 Bzs/1 Bz1/nA/B−1 ds
if B /
0,
⎪
⎪
⎪
⎪
⎩
if B 0,
0
1−λl
1
−
λ
l1/λn−1
1/n 0 s
· exp Azs − 1/n ds
1
2.8
and q is the best dominant of 2.7.
ii Furthermore, in addition to 2.5, one consider the inequality
B l 1 λn − 1
A≤−
,
λ
2.9
Sm 1, λ, l; A, B ⊂ Sm, λ, l; 1 − 2η, −1,
2.10
where −1 ≤ B < 0, then
where
η
−1
A l1
B
1
1
−
,
1,
1,
F
−
1
−
λ
l
/λ.
2 1
n
B
λn
B−1
2.11
The result is the best possible.
Proof. Setting
z Im, λ, lfz
χz :
Im, λ, lfz
z ∈ U,
2.12
we note that χz is analytic in U and
χz 1 an zn an1 zn1 · · · .
2.13
l 1Im 1, λ, lfz 1 − λ lIm, λ, lfz λz Im, λ, lfz
2.14
Using the identity
in definition of χz and carrying out logarithmic differentiation in the resulting equation, one
obtains
z Im 1, λ, lfz
zχ z
χz .
Im 1, λ, lfz
χz 1 − λ l/λ
2.15
Since f ∈ Sm 1, λ, l; A, B, we get
χz zχ z
1 Az
≺
.
χz 1 − λ l/λ 1 Bz
2.16
Adriana Cătaş et al.
7
By applying Lemma 1.4, we obtain that
1 Az
.
2.17
1 Bz
Hence we have shown the inclusion 2.6. Also, making use of Lemma 1.7 with β 1
and γ 1 − λ l/λ, q is the best dominant of 2.7 and q is defined by 2.8. This proves part
i of Theorem 2.5.
To establish 2.10, we need to show that
inf Re qz q−1.
2.18
χz ≺
|z|<1
The proof of the assertion 2.18 will be deduced on the same lines as in 14 making use of
Lemma 1.5. If we set a 1/n1 − A/B, b l 1/λn, c l 1/λn 1, B < 0, then by
using 1.13, 1.14, and 1.15, we find from 2.8 that
qz 1
1−λl
−
,
1/nQz
λ
where
Qz 1 Bza
1
sb−1 1 Bsz−a ds
0
Bz
Γb
.
·2 F1 1, a; c;
Γc
Bz 1
By using 1.13, the above equality yields
Qz 2.19
1
gs, zdμs,
2.20
2.21
0
where
gs, z dμs 1 Bz
,
1 1 − sBz
Γc
sa−1 1 − sc−a−1 ds
ΓaΓc − a
2.22
is a positive measure on the closed interval 0, 1.
For −1 ≤ B < 1, we note that Re gs, z > 0, gs, −r is real for 0 ≤ r < 1 and s ∈ 0, 1 and
1
1 − 1 − sBr
1
Re
≥
, |z| ≤ r < 1.
2.23
gs, z
1 − Br
gs, −r
Therefore, by using Lemma 1.5, one obtains
1
1
Re
≥
,
Qz
Q−r
which, upon letting r→1−1 , yields
Re
1
Qz
≥
|z| ≤ r < 1
1
.
Q−1
2.24
2.25
Now, the assertion 2.18 follows by using Lemma 1.5. The result is the best possible and q is
the best dominant of 2.7. This completes the proof of Theorem 2.5.
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Abstract and Applied Analysis
Taking m 0, n 1, λ 1, l 0, A 1 − 2α, B −1 in Theorem 2.5, we get the following
result due to MacGregor 15.
Corollary 2.6. For 0 ≤ α < 1, one obtains
Kα ⊂ S∗ η1 ,
2.26
where
η1 2 F1
1
1, 21 − α, 2,
2
−1
⎧
⎪
⎪
⎨
1
1 − 2α
,
, α /
2α−1
2
1−2
⎪
1
1
⎪
⎩
,
α .
2 ln 2
2
221−α
2.27
The result is the best possible.
Theorem 2.7. Let ψz be univalent in U with ψ0 1, Re ψz > 0, and let zψ z/ψz be starlike
in U. Let φz be defined by
zψ z
l1
λ
ψz .
φz :
l1
λ
ψz
2.28
Am 1, λ, l, n; φ ⊂ Am, λ, l, n; ψ.
2.29
Then
Proof. Setting
qz Im 1, λ, lfz
,
Im, λ, lfz
2.30
we note that qz is analytic in U.
By a simple computation, we observe from 2.30 that
z Im, λ, lfz
zq z z Im 1, λ, lfz
−
.
qz
Im 1, λ, lfz
Im, λ, lfz
2.31
Making use of the identity 2.14, one obtains from 2.31
Im 2, λ, lfz
zq z
l1
λ
qz .
Im 1, λ, lfz l 1
λ
qz
2.32
By the hypothesis of Theorem 2.7 that f belongs to the class Am 1, λ, l, n; φ and in
view of 2.32, we have
zq z
zψ z
l1
l1
λ
λ
qz ≺
ψz .
2.33
l1
λ
qz
l1
λ
ψz
If we let
Qz : zψ zϕ ψz ,
2.34
Adriana Cătaş et al.
9
where
ϕ ψz λ
1
·
,
l 1 ψz
2.35
hz : ψz Qz
and since Qz is starlike, our theorem is an immediate consequence of Lemma 1.6.
Theorem 2.8. Let ψ be univalent in U, ψ0 1 and let γ be a complex number. Suppose that
1 Re λγ 1 − l 1 l 1ψz > 0 and
2 zψ z/λγ 1 − l 1 l 1ψz is starlike in U.
Let the function Fz be defined by
γ 1
zγ
z
tγ−1 ftdt
2.36
λzψ z
,
λγ 1 − l 1 l 1ψz
2.37
Fz :
0
and the function
hz : ψz then f ∈ Am, λ, l, n; h implies F ∈ Am, λ, l, n; ψ.
Proof. From the definition of Fz and
γ 1Im, λ, lfz 1−λl
l1
Im 1, λ, lFz γ −
Im, λ, lFz,
λ
λ
2.38
if we let
qz :
Im 1, λ, lFz
,
Im, λ, lFz
2.39
then we note that qz is analytic in U. Using 2.38 and 2.39, one obtains
γ 1
Im, λ, lfz
1−λl l1
γ−
qz.
Im, λ, lFz
λ
λ
2.40
Differentiating this equality, we obtain
λzq z
Im 1, λ, lfz
qz .
Im, λ, lfz
λγ 1 − l 1 l 1qz
2.41
For f ∈ Am, λ, l, n; h, we have from 2.41
qz λzψ z
λzq z
≺ ψz .
λγ 1 − l 1 l 1qz
λγ 1 − l 1 l 1ψz
2.42
10
Abstract and Applied Analysis
If we let
Qz : zψ zϕ ψz ,
2.43
where
ϕ ψz λ
,
λγ 1 − l 1 l 1ψz
hz : v ψz Qz,
v ψz : ψz
2.44
and since Qz is starlike in U, our theorem is an immediate consequence of Lemma 1.6.
Theorem 2.9. Let fz ∈ An. Then f belongs to the class Am, λ, l, n; χ if and only if Fz defined
by
z
l1
t1−λl/λ−1 ftdt
2.45
1−λl
Fz zF z l 1fz.
λ
2.46
Fz :
z1−λl/λ
0
belongs to the class Am 1, λ, l, n; χ.
Proof. From the definition of Fz, we have
By convoluting 2.46 with the function
∞ 1 λk − 1 l m k
z
um, λ, l, n; z : z l1
kn1
2.47
and using a convolution property
zf∗g z fz∗zg z,
2.48
1−λl
Im, λ, lFz z um, λ, l, n; z∗Fz ,
λ
2.49
1−λl
Im, λ, lFz z Im, λ, lFz .
λ
2.50
one obtains
l 1Im, λ, lfz that is,
l 1Im, λ, lfz By using identity 2.14, we get
Im, λ, lfz 1
Im 1, λ, lFz.
λ
2.51
Adriana Cătaş et al.
11
Also, we obtain
l 1Im 1, λ, lfz 1 − λ lIm, λ, lfz λz Im, λ, lfz
1−λl
Im 1, λ, lFz z Im 1, λ, lFz
λ
l1
Im 2, λ, lFz.
λ
2.52
From 2.51 and 2.52, we get
Im 2, λ, lFz Im 1, λ, lfz
.
Im 1, λ, lFz
Im, λ, lfz
2.53
By the hypothesis of Theorem 2.9 that
Im 1, λ, lfz
≺ χz
Im, λ, lfz
2.54
and using 2.53, the desired result follows at once.
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