Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 783084, 10 pages
doi:10.1155/2011/783084
Research Article
On Some Integral Operators for Certain Classes of
p-Valent Functions
Lokesh Vijayvargy,1 Pranay Goswami,2 and Bushra Malik3
1
Faculty of Management, Jaipuria Institute of Management, Jaipur 302033, India
Departement of Mathematics, Amity University Rajasthan, Jaipur 302002, India
3
Mathematics Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan
2
Correspondence should be addressed to Pranay Goswami, pranaygoswami83@gmail.com
Received 3 December 2010; Revised 9 February 2011; Accepted 3 March 2011
Academic Editor: H. S. V. De Snoo
Copyright q 2011 Lokesh Vijayvargy 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.
We study some generalized integral operators for the classes of p-valent functions with bounded
radius and boundary rotation. Our work generalizes many previously known results. Many of our
results are best possible.
1. Introduction
Let Ap denote the class of functions of the form
fz zp ∞
an zn ,
p ∈ N {1, 2, . . .},
1.1
np1
which are analytic in the open unit disc U {z : |z| < 1}.
Let f and g be analytic functions in U we say that f is subordinate to g, written as
f ≺ g;
1.2
if there exists a Schwarz function wz in U, with w0 0 and |wz| < 1 z ∈ U, such that
fz gwz.
1.3
2
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In particular, when g is univalent, then the above subordination is equivalent to
f0 0,
fU ⊆ gU.
1.4
For functions f, g ∈ Ap , given by
fz zp ∞
an zn ,
gz zp np1
∞
bn zn ,
z ∈ U,
1.5
np1
we define the Hadamard product or convolution of f and g by
∞
f ∗ g z zp an bn zn ,
z ∈ U.
1.6
np1
Janowski 1 defined the class P A, B as follows.
Let h be a function, analytic in U, with h0 1. Then h is said to belong to the class
P A, B, −1 ≤ B < A ≤ 1, if and only if, for z ∈ U,
hz 1 Awz
,
1 Bwz
1.7
where wz is a Schwarz function.
Or equivalently, we can say that h ∈ P A, B, −1 ≤ B < A ≤ 1, if and only if,
hz ≺
1 Az
,
1 Bz
z ∈ U.
1.8
Geometrically, hz is in the class P A, B, if and only if, h0 1 and the image of hU lies
inside the open disc centered on the real axis with diameter end points,
D1 h−1 1−A
,
1−B
D2 h1 1A
,
1B
0 < D1 < 1 < D2 .
1.9
Clearly P A, B ⊂ P 1 − A/1 − B.
In the recent paper, Noor 2 introduced the class Pk α. We define it as follows. Let
Pk α, 0 ≤ α < p, be the class of functions pz with p0 1 and satisfying the property
pz 1
2
2π
0
1 1 − 2αze−it
dμt,
1 − ze−it
where μt is a real-valued function of bounded variation on 0, 2π and
2π
|dμt| ≤ k.
0
1.10
2π
0
dμt 2 and
International Journal of Mathematics and Mathematical Sciences
3
The classes Vk α and Rk α are related to the class Pk α and can be defined as
zf z
∈ Pk α,
iff
pf z
f ∈ Vk α,
f ∈ Rk α,
zf z
∈ Pk α,
iff
pfz
z ∈ U,
1.11
z ∈ U.
We define a class Pk A, B as follows.
Let Pk A, B, k ≥ 2, −1 ≤ B < A ≤ 1, denote the class of p-valent analytic functions hz
that are represented by
hz k 1
k 1
−
h1 z −
h2 z,
4 2
4 2
z ∈ U,
1.12
where h1 , h2 ∈ P A, B. For A 1 − 2α 0 ≤ α < p and B −1, it reduces to the class Pk α
and P2 α P α is the class of p-valent analytic functions hz with Re hz > α, z ∈ U.
Taking A 1, B −1, and p 1, we have Pk 1, −1 Pk see 3, and P2 1, −1 P is the
class of functions with positive real part.
Definition 1.1. A function f, analytic in U, and given by 1.1 is said to be in the class Rk A, B;
−1 ≤ B < A ≤ 1, k ≥ 2, if and only if,
zf z
∈ Pk A, B,
pfz
z ∈ U.
1.13
For p 1, Rk A, B is introduced and studied by Noor 4. We note that
Rk A, B ⊂ Rk
1−A
1−B
1.14
⊂ Rk ,
where Rk is the class of functions with bounded radius rotation see 5. For k 2, we have
R2 A, B ≡ S∗p A, B ⊂ S∗p
1−A
1−B
⊂ S∗p ,
1.15
where S∗p is the class of p-valent starlike functions. Similarly, we can define the class Vk A, B
as follows.
Definition 1.2. A function f, analytic in U, and given by 1.1 is said to be in the class Vk A, B;
−1 ≤ B < A ≤ 1, k ≥ 2, if and only if,
zf z
∈ Pk A, B,
pf z
z ∈ U.
1.16
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International Journal of Mathematics and Mathematical Sciences
It is clear that
f ∈ Vk A, B,
iff
zf z
∈ Rk A, B,
p
z ∈ U.
1.17
For p 1, Vk A, B is the class introduced and studied by Noor 4. It is easy to see that,
Vk A, B ⊂ Vk
1−A
1−B
⊂ Vk ,
1.18
where Vk is the class of functions with bounded boundary rotation see 5. Also
V2 A, B ≡ Cp A, B ⊂ Cp
1−A
1−B
⊂ Cp ,
1.19
where Cp is the class of p-valent convex functions.
Very recently, Frasin 6, introduced the following general integral operators for pvalent functions,
z
fn t αn
f1 t α1
·
·
·
dt,
tp
tp
0
α1
αn
z
f1 t
fn t
p−1
pt
···
dt, where αi ∈ C, z ∈ U.
Gp z ptp−1
ptp−1
0
Fp z ptp−1
1.20
1.21
Clearly, we may see that for p 1, these operators become the general integral operators
F1 z Fn z,
G1 z Fα1 ,α2 ,...,αn z,
1.22
introduced and studied by Breaz and Breaz 7 and Breaz et al. 8, see also 9,10.
z
For p n 1, α1 α ∈ 0, 1 in 1.20, we obtain the integral operator 0 ft/tα dt
studied in 11 and for p n 1, α1 δ ∈ C, |δ| < 1/4 in 1.21, we obtain the integral
z
operator 0 f tδ dt, studied in 12.
2. Main Results
Lemma 2.1. Let β > 0, β γ > 0, α ∈ α0 , 1, with α0 max{β − γ − 1/2β, −γ/β}. If
pz zp z
1 1 − 2αz
≺
,
βpz γ
1−z
2.1
then
pz ≺ Qz ≺
1 1 − 2αz
,
1−z
2.2
International Journal of Mathematics and Mathematical Sciences
5
where Qz 1/βGz − γ/β,
Gz 1 0
1−z
1 − tz
2β1−α
ρ ρ α, β, γ tβγ−1 dt 2 F1 2β1 − α, 1, β γ 1;
βγ
γ
− ,
β
β2 F1 2β1 − α, 1, β γ 1; 1/2
z
,
1−z
2.3
2.4
2 F1 denotes the Gauss hypergeometric function. From 2.2, we can deduce the sharp result p ∈
P ρ, where ρ is defined in 2.4. This result is a special case of one given in [11].
Proof. To prove this Lemma we use Theorem 3.2j of 11, page 97. Take hz 1 1 −
2αz/1 − z, 0 ≤ α < 1 and
Hz βhz γ a bz
,
1−z
2.5
where a β γ and b β1 − 2α γ.
Since H is convex to apply Theorem 3.2j of 11, page 97 we only need to determine
condition Re Hz > 0.
The range of |z| ≤ 1 under Hz is a half plane. In order to satisfy the required
condition this half plane needs to lie in the right half plane. This requirement will be satisfied
if Re H−1 Re Hi and Re H0 > Re H−1 ≥ 0. Or we can write it as
β1 − α > 0,
βα γ ≥ 0.
2.6
When β > 0, β γ > 0, these conditions imply that α ∈ −γ/β, 1, and if β γ > 1, then
α ∈ β − γ − 1/2β, 1. Hence all the conditions of Theorem 3.2j of 11, page 97 are satisfied
for α ∈ α0 , 1, with α0 max{β − γ − 1/2β, −γ/β}, thus we have the required result.
To show that the solution Qz can be represented in terms of hypergeometric
functions we take A 1 − 2α, B −1, n 1 in Theorem 3.3d of 11, page 109.
Lemma 2.2. Let f ∈ Vk α, 0 ≤ α < p, k ≥ 2. Then f ∈ Rk ρ in U, where
ρ ρ α, p 1
.
2 F1 2p1 − α, 1, p 1; 1/2
2.7
k 1
k 1
−
h1 z −
h2 z,
4 2
4 2
2.8
This result is sharp.
Proof. Let for k ≥ 2, z ∈ U, we have
zf z
hz pfz
where h, hi are analytic in U with h0 1, hi 0 1, i 1, 2.
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International Journal of Mathematics and Mathematical Sciences
We define
φp z z ∞
1
zn ,
p1
− 1
n
n1
z ∈ U.
2.9
By using 2.8, with convolution technique, see 13, we have
φp z
∗ hz z
k 1
4 2
φp z
∗ h1 z
z
−
k 1
−
4 2
φp z
∗ h2 z .
z
2.10
This implies that,
zh z
hz phz
k 1
4 2
zh z
h1 z 1
ph1 z
−
k 1
−
4 2
zh z
h2 z 2
ph2 z
.
2.11
Logarithmic differentiation of 2.8 yields,
zh1 z
zh2 z
zf z
k 1
zh z
k 1
h1 z −
h2 z hz −
.
pf z
phz
4 2
ph1 z
4 2
ph2 z
2.12
Since zf z /pf z ∈ Pk α, 0 ≤ α < p, thus
hi z zhi z
∈ P α,
phi z
i 1, 2.
2.13
By using Lemma 2.1 for β p and γ 0, we deduce that hi ∈ P ρ, where ρ is given in
2.7. This estimate is best possible because of the best dominant property of function Qz,
where
Qz 1
,
F
2p1
−
α,
1,
p 1; z/1 − z
2 1
z ∈ U.
2.14
For p 1, we have the sharp result proved in 14.
We begin with the following theorem.
Theorem 2.3. i Let αi > 0, fi ∈ Rk A, B for all i 1, 2, . . . , n, and, ni1 αi 1. Then the integral
operator Fp ∈ Vk A, B in U, where −1 ≤ B < A ≤ 1, k ≥ 2.
ii Let fi ∈ Rk α, αi > 0 for all i 1, 2, . . . , n with α 1 − A/1 − B, k ≥ 2. If
n
i1 αi 1, then the integral operator Fp defined by 1.20 also belongs to the class Rk ρ in U,
where ρ ρα, p is defined by 2.7. This result is sharp.
International Journal of Mathematics and Mathematical Sciences
7
Proof (i). From 1.20, we can see that Fp ∈ Ap in U, and
Fp z
pz
p−1
f1 z
zp
α1
···
fn z
zp
αn .
2.15
Differentiating logarithmically and multiplying by z, we obtain,
zFp z
Fp z
n
p − 1 αi
i1
zfi z
−p ,
fi z
z ∈ U.
2.16
Thus, we have
1
zFp z
Fp z
n
αi
i1
zfi z
fi z
,
2.17
or
zFp z
pFp z
zfi z
pfi z
i1
n
n
k 1
k 1
−
αi pi z −
αi hi z ,
4 2
4 2
i1
i1
n
αi
2.18
where hi , pi ∈ P A, B, for all i 1, 2, . . . , n.
Since P A, B is a convex set, see 15, it follows that,
zFp z
pFp z
k 1
k 1
H1 z −
−
H2 z,
4 2
4 2
2.19
where H1 , H2 ∈ P A, B and therefore,
zFp z
pFp z
∈ Pk A, B,
z ∈ U.
2.20
This proves the result.
Substituting p 1, in Theorem 2.3i, we have the following corollary.
Corollary 2.4. Let αi > 0, fi ∈ Rk A, B for all i 1, 2, . . . , n, −1 ≤ B < A ≤ 1, k ≥ 2. Then the
integral operator Fp ∈ Vk A, B in U.
Remark 2.5. Letting α1 α, α2 β, and n 2 in Corollary 2.4, we obtain a result due to Noor
4.
8
International Journal of Mathematics and Mathematical Sciences
For n 1, α1 α 1, and f1 f in Theorem 2.3i, we have the following.
2.6. Let f ∈ Rk A, B in U, −1 ≤ B < A ≤ 1, k ≥ 2. Then the integral operator
Corollary
z
pft/tdt
∈ Vk A, B, z ∈ U.
0
Proof (ii). Taking A 1 − 2α, B −1, with α 1 − A/1 − B, we have for all i 1, 2, . . . , n,
fi ∈ Rk 1 − 2α, −1 Rk α,
2.21
using part i of Theorem 2.3, we have
Fp ∈ Vk 1 − 2α, −1 Vk α
in U.
2.22
Now using Lemma 2.2 for Fp ∈ Vk α, α 1 − A/1 − B implies that
Fp ∈ Rk ρ ,
where ρ ρ α, p is defined in 2.7.
2.23
The sharpness of the result is clear from the function Qz defined by 2.14.
For p 1, we have the following corollary.
Corollary 2.7. Let αi > 0, fi ∈ Rk α for all i 1, 2, . . . , n, with α 1−A/1−B and A 1−2α,
B −1. Then the integral operator Fp defined by 1.20 also belongs to the class Rk ρ in U, where
ρ ρα ⎧
1
2α − 1
⎪
⎪
⎨ 2 − 221−α , if α /
2
⎪
⎪
⎩ 1 ,
2 ln 2
1
if α .
2
2.24
Remark 2.8. Letting α1 μ, α2 η, and n 2 in Corollary 2.7, we have the sharp result proved
in 14.
For A 1, B −1, and p 1, we have
fi ∈ Rk 0
1
in U.
implies that Fp ∈ Vk
2
2.25
Theorem 2.9. i Let αi > 0,fi ∈ Vk A, B for all i 1, 2, . . . , n. If ni1 αi 1, then the integral
operator Gp defined by 1.21, also belongs to the class Vk A, B in U, where −1 ≤ B < A ≤ 1, k ≥ 2.
ii Let for αi > 0, ni1 αi 1 and fi ∈ Vk α, for all i 1, 2, . . . , n with 0 ≤ α < p,
α 1 − A/1 − B, k ≥ 2. Then the integral operator Gp ∈ Rk ρ in U, where ρ ρα, p is defined
by 2.7. This result is sharp.
International Journal of Mathematics and Mathematical Sciences
9
Proof (i). From definition 1.20, we have
1
zGp z
Gp z
p
n
αi
i1
zfi z
−p1
fi z
2.26
n
zfi z
,
αi
fi z
i1
or
zGp z
pGp z
n
zfi z
αi
pfi z
i1
n
n
k 1
k 1
−
αi pi z −
αi hi z ,
4 2
4 2
i1
i1
2.27
where hi , pi ∈ P A, B, for all i 1, 2, . . . , n.
Since P A, B is a convex set, see 15, it follows that,
zGp z
pGp z
k 1
k 1
H1 z −
−
H2 z,
4 2
4 2
z ∈ U,
2.28
where H1 , H2 ∈ P A, B and therefore,
zGp z
pGp z
∈ Pk A, B
2.29
in U.
This implies that Gp ∈ Vk A, B.
Letting p 1 in Theorem 2.9i, we have the following corollary.
Corollary 2.10. Let αi > 0, fi ∈ Vk A, B for all i 1, 2, . . . , n and −1 ≤ B < A ≤ 1, k ≥ 2. If
n
i1 αi 1, then Gp ∈ Vk A, B in U.
Proof (ii). Taking A 1 − 2α, B −1, we have for all i 1, 2, . . . , n
fi ∈ Vk 1 − 2α, −1 Vk α,
where α 1−A
.
1−B
2.30
Now using part i of Theorem 2.9, we have
Gp ∈ Vk 1 − 2α, −1 Vk α
in U.
2.31
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International Journal of Mathematics and Mathematical Sciences
Now using Lemma 2.2, for α 1 − A/1 − B, we have
Gp ∈ Vk α implies that Gp ∈ Rk ρ , in U,
where ρ ρ α, p is defined in 2.7.
2.32
The sharpness of the result is clear from the function Qz defined by 2.14.
For p 1, we have the following corollary.
Corollary 2.11. i Let αi > 0, fi ∈ Vk α, i 1, 2, . . . , n, with α 1 − A/1 − B and A 1 − 2α,
B −1. Then Gp ∈ Rk ρ in U, where ρ ρα and defined in 2.24.
Also for A 1, B −1, we have.
ii If fi ∈ Vk 0 for all i 1, 2, . . . , n, then Gp ∈ Rk 1/2 in U.
Acknowledgments
Authors are thankful to anonymous referees for their very constructive comments to improve
this paper. Second author P. Goswami is also grateful to Professor S. P. Goyal for his
guidance and constant encouragement.
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