Document 10467472
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Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 769537, 5 pages
http://dx.doi.org/10.1155/2013/769537
Research Article
Univalence of a New General Integral Operator Associated with
the π-Hypergeometric Function
Huda Aldweby and Maslina Darus
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
Correspondence should be addressed to Maslina Darus; maslina@ukm.my
Received 11 December 2012; Revised 3 February 2013; Accepted 17 February 2013
Academic Editor: Shyam Kalla
Copyright © 2013 H. Aldweby and M. Darus. 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.
Motivated by the familiar π-hypergeometric functions, we introduce a new family of integral operators and obtain new sufficient
conditions of univalence criteria. Several corollaries and consequences of the main results are also pointed out.
(π = π + 1; π, π ∈ N0 = N ∪ {0}; π§ ∈ U), where N denotes
the set of positive integers and (π, π)π is the π-shifted factorial
defined by
1. Introduction
Let A denote the class of functions of the form
∞
π (π§) = π§ + ∑ ππ π§π ,
π=2
ππ ≥ 0,
(1)
which are analytic in the open unit disk U = {π§ ∈ C : |π§| < 1},
and S the class of functions π ∈ A which are univalent in U.
Let π, π ∈ A, where π is defined by (1) and π is given by
∞
π (π§) = π§ + ∑ ππ π§π ,
π=2
ππ ≥ 0.
(2)
Then the Hadamard product (or convolution) π ∗ π of the
functions π and π is defined by
∞
(π ∗ π) (π§) = π§ + ∑ ππ ππ π§π .
(π, π)π = {
By using the ratio test, we should note that, if |π| < 1, the
series (4) converges absolutely for |π§| < 1 if π = π + 1. For
more mathematical background of these functions, one may
refer to [1].
Corresponding to the function defined by (4), consider
π Gπ
π Φπ
∞
(ππ ; ππ ; π, π§) = ∑
(π1 , π)π ⋅ ⋅ ⋅ (ππ , π)π
(3)
π=0 (π, π)π (π1 , π)π ⋅ ⋅ ⋅ (ππ , π)π
π§π
(4)
(ππ ; ππ ; π, π§) = π§
π Φπ
(ππ ; ππ ; π, π§) .
(6)
Recently, the authors [2] defined the linear operator
M(ππ , ππ ; π)π : A → A by
M (ππ , ππ ; π) π (π§) =
π=2
For complex parameters ππ , ππ , and π (π = 1, . . . , π, π =
1, . . . , π , ππ ∈ C \ {0, −1, −2, . . .}, |π| < 1), we define the πhypergeometric function π Φπ (π1 , . . . , ππ ; π1 , . . . , ππ ; π, π§) by
1,
π = 0;
(1 − π) (1 − ππ) (1 − ππ2 ) ⋅ ⋅ ⋅ (1 − πππ−1 ) , π ∈ N.
(5)
π Gπ
(ππ ; ππ ; π, π§) ∗ π (π§)
(7)
∞
= π§ + ∑ Υπ ππ π§π ,
π=2
where
Υπ =
(π1 , π)π−1 ⋅ ⋅ ⋅ (ππ , π)π−1
(π, π)π−1 (π1 , π)π−1 ⋅ ⋅ ⋅ (ππ , π)π−1
,
σ΅¨ σ΅¨
(σ΅¨σ΅¨σ΅¨πσ΅¨σ΅¨σ΅¨ < 1) .
(8)
2
International Journal of Mathematics and Mathematical Sciences
It should be remarked that the linear operator (7) is a
generalization of many operators considered earlier. For ππ =
ππΌπ , ππ = ππ½π , πΌπ , π½π ∈ C, π½π =ΜΈ 0, −1, −2, . . . , (π = 1, . . . , π, π =
1, . . . , π ), and π → 1, we obtain the Dziok-Srivastava linear
operator [3] (for π = π + 1), so that it includes (as its special
cases) various other linear operators introduced and studied
by Ruscheweyh [4], Carlson and Shaffer [5] and the BernardiLibera-Livingston operators [6–8].
The π-difference operator is defined by
ππ β (π§) =
β (ππ§) − β (π§)
,
(π − 1) π§
(9)
where βσΈ (π§) is the ordinary derivative. For more properties of
ππ see [9, 10].
Lemma 1 (see [2]). Let π ∈ A; then
(i) for π = 1, π = 0, and π1 = π, one has M(π, −; π)π(π§) =
π(π§).
(ii) For π = 1, π = 0, and π1 = π2 , one
has M(π2 , −; π)π(π§) = π§ππ π(π§) and limπ → 1
M(π2 , −; π)π(π§) = π§πσΈ (π§), where ππ is the π-derivative
defined by (9).
Definition 2. A function π ∈ A is said to be in the class
Bππ (ππ , ππ ; π; π) if it is satisfying the condition
(π§ ∈ U; 0 ≤ π < 1) ,
(10)
(12)
ππ‘)
1/(1+π(πΌ−1))
Note that B10 (π, −; π; π) = B(π), where the class B(π) of
analytic and univalent functions was introduced and studied
by Frasin and Darus [11].
Using the operator M(ππ , ππ ; π)π(π§)π, we now introduce
the following new general integral operator.
For π ∈ N ∪ {0}, πΎ1 , πΎ2 , . . . , πΎπ , πΏ ∈ C \ {0, −1, −2, . . .},
and |π| < 1, we define the integral operator πΌπΎπ ,πΏ (ππ , ππ ; π; π§) :
Aπ → Aπ by
where for convenience π»π π (πΌ1 , π½1 )π := π»(πΌ1 , . . . , πΌπ ; π½1 ,
. . . , π½π ; π§)π(π§), and π»π π (πΌ1 , π½1 )π(π§) = π§ + ∑∞
π=2 ((πΌ1 )π−1
⋅ ⋅ ⋅ (πΌπ )π−1 /(π½1 )π−1 ⋅ ⋅ ⋅ (π½π )π−1 (π − 1)!)ππ π§π is the Dziok-Srivastava operator [3].
(2) For π = 1, π = 0, π1 = π, πΎπ = 1/(πΌ − 1), and πΏ =
1 + π(πΌ − 1), we obtain the integral operator
πΉπ,πΌ (π§) = (1 + π (πΌ − 1)
π§
πΌ−1
× ∫ (π1 (π‘))
0
πΌ−1
⋅ ⋅ ⋅ (ππ (π‘))
ππ‘)
1/(1+π(πΌ−1))
(13)
studied recently by Breaz et al. [13].
(3) For π = 1, π = 0, π1 = π, πΎπ = 1/πΌπ , and πΏ = 1, we
obtain the integral operator
π§ π
π (π‘) πΌπ
(π‘) πΌ1
(14)
πΉπΌ (π§) = ∫ ( 1 ) ⋅ ⋅ ⋅ ( π ) ππ‘
π‘
π‘
0
introduced and studied by D. Breaz and N. Breaz [14].
(4) For π = 1, π = 0, π1 = π2 , πΎπ = 1/(πΌ − 1), and πΏ =
1 + π(πΌ − 1), we obtain the integral operator
π§
× ∫ π‘π(πΌ−1) (π1σΈ (π‘))
πΌ−1
(15)
0
πΌ−1
⋅ ⋅ ⋅ (ππσΈ (π‘))
ππ‘)
1/(1+π(πΌ−1))
introduced by Selvaraj and Karthikeyan [12].
(5) For π = 1, π = 0, π1 = π2 , πΎπ = 1/πΌ, and πΏ = 1, we
obtain the integral operator
π§
πΌ
πΌ
πΊπΌ (π§) = ∫ (π1σΈ (π‘)) ⋅ ⋅ ⋅ (ππσΈ (π‘)) ππ‘,
πΌπΎπ ,πΏ (ππ , ππ ; π; π§)
0
πΏ−1
= (πΏ ∫ π‘
π
∏(
π=1
M (ππ , ππ ; π) π (π§) ππ (π‘)
π‘
1/πΏ
1/πΎπ
)
,
πΊπΌ (π§) = (1 + π (πΌ − 1)
where M(ππ , ππ ; π)π is the operator defined by (7).
0
πΌ−1
0
πΌ−1
π→1
π§
π§
= (1 + π (πΌ − 1) ∫ (π»π π (πΌ1 , π½1 ) π1 (π‘))
⋅ ⋅ ⋅ (π»π π (πΌ1 , π½1 ) ππ (π‘))
π =ΜΈ 1, π§ =ΜΈ 0,
lim ππ β (π§) = βσΈ (π§) ,
σ΅¨σ΅¨ 2
σ΅¨σ΅¨
σΈ
σ΅¨σ΅¨ π§ (M (ππ , ππ ; π) π (π§))
σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨
− 1σ΅¨σ΅¨σ΅¨ < 1 − π
σ΅¨σ΅¨
2
σ΅¨σ΅¨ [M (π , π ; π) π (π§)]
σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨
π π
(1) for π = π + 1, ππ = ππΌπ , ππ = ππ½π , π = 1, . . . , π, π =
1, . . . , π , π → 1, πΎπ = 1/(πΌ − 1), and πΏ = 1 + π(πΌ − 1),
where πΌ ∈ C and R(πΌ) > 0, we obtain the following integral
operator introduced and studied by Selvaraj and Karthikeyan
[12]:
πΉπΌ (πΌ1 , π½1 ; π§)
ππ‘)
,
(11)
where ππ ∈ A.
Remark 3. It is interesting to note that the integral operator
πΌπΎπ ,πΏ (ππ , ππ ; π; π§) generalizes many operators introduced and
studied by several authors, for example,
(16)
recently introduced and studied by Breaz and GuΜney [15].
(6) For π = 1, π = 0, π1 = π, π1 = ⋅ ⋅ ⋅ = ππ = π ∈ A, πΎπ =
1/(πΌ − 1), and πΏ = πΌ, where πΌ ∈ C and R(πΌ) > 0, we obtain
the integral operator
π§
πΊπΌ (π§) = (πΌ ∫ (π (π‘))
0
πΌ−1
1/πΌ
)
ππ‘,
(17)
introduced and studied by Pescar [16].
In order to derive our main results, we have to recall the
following univalence criteria.
International Journal of Mathematics and Mathematical Sciences
Lemma 4 (see [17, 18]). Let πΏ ∈ C with Re(πΏ) > 0. If π ∈ A
satisfies
2 Re(πΏ)
1 − |π§|
Re (πΏ)
σ΅¨σ΅¨ σΈ σΈ
σ΅¨
σ΅¨σ΅¨ π§π (π§) σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨ σΈ
σ΅¨σ΅¨ ≤ 1,
σ΅¨σ΅¨σ΅¨ π (π§) σ΅¨σ΅¨σ΅¨
3
and for π§ = 0, we have
(
(18)
M (ππ , ππ ; π) π (π§) π1 (π§)
π§
for all π§ ∈ U, then the integral operator
π§
πΏ−1 σΈ
πΉπΏ (π§) = {πΏ ∫ π‘
π (π‘) ππ‘}
0
⋅⋅⋅(
1/πΏ
π§
(27)
1/πΎπ
= 1.
)
(19)
We define the function β(π§) by the form
Lemma 5 (see [16]). Let πΏ ∈ C with Re(πΏ) > 0, π ∈ C, with
|π| ≤ 1, π =ΜΈ − 1. If π ∈ A satisfies
σ΅¨
σ΅¨σ΅¨
π§πσΈ σΈ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨ 2πΏ
σ΅¨σ΅¨ ≤ 1,
σ΅¨σ΅¨π|π§| + (1 − |π§|2πΏ ) σΈ
σ΅¨σ΅¨
πΏπ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨
(20)
π§ π
M (ππ , ππ ; π) π (π§) ππ (π‘)
0 π=1
π‘
β (π§) = ∫ ∏(
π
β (π§) = ∏(
for all π§ ∈ U then the integral operator
M (ππ , ππ ; π) π (π§) ππ (π§)
π§
π=1
1/πΏ
(21)
0
1/πΎπ
ππ‘.
)
(28)
Therefore
σΈ
π§
)
M (ππ , ππ ; π) π (π§) ππ (π§)
is in the class S.
πΉπΏ (π§) = {πΏ ∫ π‘πΏ−1 πσΈ (π‘) ππ‘}
1/πΎ1
1/πΎπ
.
)
(29)
Differentiating logarithmically and multiplying by π§ on both
sides of (29)
is in the class S.
σΈ
π
π§βσΈ σΈ (π§)
1 π§(M (ππ , ππ ; π) π (π§) ππ (π§))
=
− 1) .
(
∑
σΈ
β (π§)
πΎ
M (ππ , ππ ; π) π (π§) ππ (π§)
π=1 π
(30)
Lemma 6 (Generalized Schwarz Lemma, see [19]). (Generalized Schwarz Lemma) Let the function π be analytic in the disk
Uπ
= {π§ : |π§| < π
}, with |π(π§)| < π for fixed π. If π(π§) has
one zero with multiplicity order bigger that π for π§ = 0, then
σ΅¨ π π
σ΅¨σ΅¨
σ΅¨σ΅¨π (π§)σ΅¨σ΅¨σ΅¨ ≤ π |π§| ,
π
Thus we have
(π§ ∈ Uπ
) .
(22)
σ΅¨σ΅¨
σΈ
σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨ σΈ σΈ
σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨ π§β (π§) σ΅¨σ΅¨σ΅¨ π 1 σ΅¨σ΅¨σ΅¨σ΅¨ π§(M (ππ , ππ ; π) π (π§) ππ (π§))
σ΅¨
σ΅¨σ΅¨ σΈ
σ΅¨
σ΅¨σ΅¨ β (π§) σ΅¨σ΅¨σ΅¨ ≤ ∑ σ΅¨σ΅¨σ΅¨πΎ σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨σ΅¨σ΅¨ M (π , π ; π) π (π§) π (π§) − 1σ΅¨σ΅¨σ΅¨σ΅¨ .
σ΅¨
σ΅¨ π=1 σ΅¨ π σ΅¨ σ΅¨σ΅¨
π π
π
σ΅¨σ΅¨
(31)
Equality can hold only if
π (π§) = πππ (
π π
)π§ ,
π
π
(23)
So
where π is constant.
1 − |π§|2 Re(πΏ)
Re (πΏ)
2. Univalence Conditions for πΌπΎπ ,πΏ (ππ ,ππ ;π;π§)
Theorem 7. Let ππ ∈ A for all π = 1, . . . , π, πΎπ ∈ C, and
π ≥ 1 with
1 π [(2 − ππ ) π + 1]
≤ 1.
∑
σ΅¨σ΅¨ σ΅¨σ΅¨
Re (πΏ) π=1
σ΅¨σ΅¨πΎπ σ΅¨σ΅¨
≤
(π§ ∈ U)
(25)
then the integral operator πΌπΎπ ,πΏ (ππ , ππ ; π; π§) defined by (11) is
analytic and univalent in U.
Proof. From the definition of the operator M(ππ , ππ ; π)π(π§)π
it can be observed that
M (ππ , ππ ; π) π (π§)
π§
=ΜΈ 0,
(π§ ∈ U) ,
(26)
1 − |π§|2 Re(πΏ)
Re (πΏ)
σ΅¨σ΅¨
σΈ σ΅¨
σ΅¨σ΅¨ π§(M (ππ , ππ ; π) π (π§) ππ (π§)) σ΅¨σ΅¨σ΅¨
1
σ΅¨
σ΅¨σ΅¨
× [ ∑ σ΅¨σ΅¨ σ΅¨σ΅¨ (σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨ + 1)]
σ΅¨
πΎ
σ΅¨
σ΅¨
σ΅¨σ΅¨ M (ππ , ππ ; π) π (π§) ππ (π§) σ΅¨σ΅¨σ΅¨
π=1 σ΅¨ π σ΅¨
σ΅¨
σ΅¨
]
[
π
(24)
If for all π = 1, . . . , π, ππ ∈ Bππ (ππ , ππ , π, ππ ), 0 ≤ ππ < 1, and
σ΅¨σ΅¨
σ΅¨
σ΅¨σ΅¨M (ππ , ππ ; π) π (π§) ππ (π§)σ΅¨σ΅¨σ΅¨ ≤ π,
σ΅¨
σ΅¨
σ΅¨σ΅¨ σΈ σΈ
σ΅¨
σ΅¨σ΅¨ π§β (π§) σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨ σΈ
σ΅¨
σ΅¨σ΅¨ β (π§) σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨
≤
1 − |π§|2 Re(πΏ)
Re (πΏ)
σ΅¨σ΅¨ 2
σΈ σ΅¨
π
σ΅¨σ΅¨ π§ (M (ππ , ππ ; π) π (π§) ππ (π§)) σ΅¨σ΅¨σ΅¨
1
σ΅¨σ΅¨
σ΅¨σ΅¨
[
× ∑ σ΅¨σ΅¨ σ΅¨σ΅¨ (σ΅¨σ΅¨
2 σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨πΎ σ΅¨ σ΅¨
σ΅¨
[π=1 σ΅¨ π σ΅¨ σ΅¨σ΅¨ [M (ππ , ππ ; π) π (π§) ππ (π§)] σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨
σ΅¨σ΅¨ M (ππ , ππ ; π) π (π§) ππ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨σ΅¨ + 1)] .
× σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨
π§
σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨
σ΅¨
]
(32)
4
International Journal of Mathematics and Mathematical Sciences
Since |M(ππ , ππ ; π)π(π§)ππ (π§)| ≤ π, (π§ ∈ U, π = 1, . . . , π),
and ππ ∈ Bππ (ππ , ππ , π, ππ ) for all π = 1, . . . , π, then from the
Schwarz Lemma and (10), we obtain
σ΅¨
σ΅¨
1 − |π§|2 Re(πΏ) σ΅¨σ΅¨σ΅¨ π§βσΈ σΈ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨ σΈ
σ΅¨σ΅¨
Re (πΏ) σ΅¨σ΅¨σ΅¨ β (π§) σ΅¨σ΅¨σ΅¨
≤
2 Re(πΏ)
1 − |π§|
Re (πΏ)
π
1
× [( ∑ σ΅¨σ΅¨ σ΅¨σ΅¨
πΎ
π=1 σ΅¨σ΅¨ π σ΅¨σ΅¨
|π| ≤ 1 −
1 π [(2 − ππ ) π + 1]
,
∑
σ΅¨σ΅¨ σ΅¨σ΅¨
πΏ π=1
σ΅¨σ΅¨πΎπ σ΅¨σ΅¨
σ΅¨σ΅¨ 2
σΈ σ΅¨
σ΅¨σ΅¨ π§ (M (ππ , ππ ; π) π (π§) ππ (π§)) σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨π
σ΅¨σ΅¨ [M (π , π ; π) π (π§) π (π§)]2 σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨
π π
π
(39)
(π§ ∈ U) ,
(40)
then the integral operator πΌπΎπ ,πΏ (ππ , ππ ; π) defined by (11) is
analytic and univalent in U.
Proof. From the proof of Theorem 7, we have
σΈ
1 π 1
[(2 − ππ ) π + 1] ,
∑
Re (πΏ) π=1 σ΅¨σ΅¨σ΅¨σ΅¨πΎπ σ΅¨σ΅¨σ΅¨σ΅¨
π
π§βσΈ σΈ (π§)
1 π§(M (ππ , ππ ; π) π (π§) ππ (π§))
=
(
− 1) .
∑
σΈ
β (π§)
πΎ
M (ππ , ππ ; π) π (π§) ππ (π§)
π=1 π
(41)
(π§ ∈ U)
(33)
which, in the light of the hypothesis (24), yields
σ΅¨
σ΅¨
1 − |π§|2 Re(πΏ) σ΅¨σ΅¨σ΅¨ π§βσΈ σΈ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨ σΈ
σ΅¨σ΅¨ ≤ 1, (π§ ∈ U) .
(34)
Re (πΏ) σ΅¨σ΅¨σ΅¨ β (π§) σ΅¨σ΅¨σ΅¨
Applying Lemma (1) for the function β(π§) we obtain that
πΌπΎπ ,πΏ (ππ , ππ ; π; π§) is univalent.
Taking ππ = 0 (for all π = 1, . . . , π), π = 1, ππ =
π , ππ = ππ½π , π → 1, and πΎπ = 1/(πΌ − 1), πΏ = 1 + π(πΌ − 1) in
Theorem 7, we have the following.
πΌπ
Corollary 8 (see [12]). Let ππ ∈ A for all π = 1, . . . , π and
πΌ ∈ C with
Re (πΌ)
(35)
.
|πΌ − 1| ≤
3π
If
σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨σ΅¨ π§2 (π»π (πΌ , π½ ) π (π§))σΈ
σ΅¨σ΅¨
σ΅¨σ΅¨
1 1
π
π
−
1
(36)
σ΅¨σ΅¨ < 1, (π§ ∈ U)
σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨ (π»π (πΌ , π½ ) π (π§))2
σ΅¨
σ΅¨
1 1
π
π
and for all π = 1, . . . , π, then the integral operator πΉπΌ (πΌ1 , π½1 ; π§)
defined by (12) is analytic and univalent in U.
Taking ππ = 0 (for all π = 1, . . . , π), π = 1, π = 1, π =
0, π1 = π, and πΎπ = 1/(πΌ − 1), πΏ = 1 + π(πΌ − 1) in Theorem 7,
we have the following.
Corollary 9. Let ππ ∈ A for all π = 1, . . . , π and πΌ ∈ C with
|πΌ − 1| ≤
π ∈ C.
If for all π = 1, . . . , π, ππ ∈ Bππ (ππ , ππ , π, ππ ), 0 ≤ ππ < 1, and
σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨M (ππ , ππ ; π) π (π§) ππ (π§)σ΅¨σ΅¨σ΅¨ ≤ π,
σ΅¨
σ΅¨
+π + 1)]
]
≤
Theorem 10. Let ππ ∈ A for all π = 1, . . . , π, πΏ, πΎπ ∈ C, and
π ≥ 1 with
Re (πΌ)
.
3π
(37)
If
σ΅¨σ΅¨
σ΅¨σ΅¨ 2 σΈ
σ΅¨σ΅¨
σ΅¨σ΅¨ π§ ππ (π§)
σ΅¨σ΅¨ < 1, (π§ ∈ U)
σ΅¨σ΅¨
−
1
(38)
σ΅¨σ΅¨
σ΅¨σ΅¨
2
σ΅¨σ΅¨
σ΅¨σ΅¨ (ππ (π§))
and for all π = 1, . . . , π, then the integral operator πΉπ,πΌ (π§)
defined by (13) is analytic and univalent in U.
Thus we have
σ΅¨σ΅¨
σ΅¨
σ΅¨
σ΅¨σ΅¨
π§βσΈ σΈ (π§) σ΅¨σ΅¨σ΅¨
π§βσΈ σΈ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨σ΅¨ 2πΏ
σ΅¨σ΅¨ ≤ |π| + σ΅¨σ΅¨σ΅¨(1 − |π§|2πΏ ) σΈ
σ΅¨σ΅¨ .
σ΅¨σ΅¨π|π§| + (1 − |π§|2πΏ ) σΈ
σ΅¨σ΅¨
πΏβ (π§) σ΅¨σ΅¨σ΅¨
πΏβ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨σ΅¨
σ΅¨
(42)
From this result and using the proof of Theorem 7 we obtain
σ΅¨σ΅¨
σ΅¨
π§βσΈ σΈ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨ 2πΏ
σ΅¨σ΅¨π|π§| + (1 − |π§|2πΏ ) σΈ
σ΅¨σ΅¨
σ΅¨σ΅¨
πΏβ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨
(43)
1 π 1
≤ |π| + ∑ σ΅¨σ΅¨ σ΅¨σ΅¨ [(2 − ππ ) π + 1] .
πΏ π=1 σ΅¨σ΅¨πΎπ σ΅¨σ΅¨
Since |π| ≤ 1 − (1/πΏ) ∑π
π=1 (1/πΎπ )[(2 − ππ )π + 1], then we have
σ΅¨σ΅¨
σΈ σΈ
σ΅¨σ΅¨σ΅¨ 2πΏ
σ΅¨σ΅¨π|π§| + (1 − |π§|2πΏ ) π§β (π§) σ΅¨σ΅¨σ΅¨ ≤ 1, (π§ ∈ U) .
(44)
σ΅¨σ΅¨
σ΅¨
πΏβσΈ (π§) σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨
Applying Lemma (4) for the function β(π§) we obtain that
πΌπΎπ ,πΏ (ππ , ππ ; π; π§) is univalent.
Taking ππ = 0 (for all π = 1, . . . , π), π = 1, π = 0, π1 = π,
and πΎπ = 1/(πΌ − 1), πΏ = 1 + π(πΌ − 1)(πΌ ∈ R) in Theorem 10,
we have the following.
Corollary 11. Let ππ ∈ A for all π = 1, . . . , π; π ∈ C, πΌ ∈ R,
and π ≥ 1 with
|π| ≤ 1 + (
1−πΌ
) (2π + 1) π,
1 + π (πΌ − 1)
2ππ + 1
πΌ ∈ [1,
].
2ππ
(45)
If for all π = 1, . . . , π
σ΅¨σ΅¨
σ΅¨σ΅¨ π§2 πσΈ (π§)
σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨ < 1, (π§ ∈ U) ,
σ΅¨σ΅¨ 2π
−
1
σ΅¨σ΅¨
σ΅¨σ΅¨ π (π§)
σ΅¨
σ΅¨ π
σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨ππ (π§)σ΅¨σ΅¨ ≤ π, (π§ ∈ U; π = 1, . . . , π) ,
(46)
then the integral operator πΉπ,πΌ (π§) defined by (13) is analytic
and univalent in U.
International Journal of Mathematics and Mathematical Sciences
Letting π = 1, π = 1, and π1 = π in Corollary 11, we
have the following.
Corollary 12. Let π ∈ A, π ∈ C and πΌ ∈ R with
|π| ≤
3 − 2πΌ
,
πΌ
(π =ΜΈ − 1) ,
3
πΌ ∈ [1, ] .
2
(47)
If
σ΅¨σ΅¨ 2 σΈ
σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨ π§ π (π§)
σ΅¨σ΅¨ < 1, (π§ ∈ U) ,
σ΅¨σ΅¨ 2
−
1
σ΅¨σ΅¨
σ΅¨σ΅¨ π (π§)
σ΅¨σ΅¨
σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨π (π§)σ΅¨σ΅¨ ≤ 1, (π§ ∈ U) ,
(48)
then the integral operator πΊπΌ (π§) defined by (17) is analytic and
univalent in U.
Remark 13. Many other interesting corollaries and results can
be obtained by specializing the parameters in Theorem 10; for
example, see [13, 20, 21].
Acknowledgments
The work presented here was partially supported by GUP2012-023 and UKM-DLP-2011-050.
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