Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 520698, 11 pages
doi:10.1155/2008/520698
Research Article
On Integral Operator Defined by Convolution
Involving Hybergeometric Functions
K. Al-Shaqsi and M. Darus
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
43600 Bangi, Selangor Darul Ehsan, Malaysia
Correspondence should be addressed to M. Darus, maslina@pkrisc.cc.ukm.my
Received 16 September 2007; Accepted 9 January 2008
Recommended by Brigitte Forster-Heinlein
μ
For λ > −1 and μ ≥ 0, we consider a liner operator Iλ on the class A of analytic functions in the unit
disk defined by the convolution fμ −1 ∗fz, where fμ 1 − μz2 F1 a, b, c; z μzz2 F1 a, b, c; z ,
and introduce a certain new subclass of A using this operator. Several interesting properties of these
classes are obtained.
Copyright q 2008 K. Al-Shaqsi 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.
1. Introduction
Let A denote the class of functions of the form
fz z ∞
ak zk ,
1.1
k2
which are analytic in the unit disk U {z : |z| < 1}.
If fz ∈ A satisfies
arg zf z − α < π β
2
fz
z ∈ U, 0 ≤ α < 1, 0 < β ≤ 1,
1.2
then fz is said to be strongly starlike of order β and type α in U, and denoted by S∗ α, β.
If fz ∈ A satisfies
arg 1 zf z − α < π β
2
f z
z ∈ U, 0 ≤ α < 1, 0 < β ≤ 1,
1.3
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International Journal of Mathematics and Mathematical Sciences
then fz is said to be strongly convex of order β and type α in U, and denoted by Cα, β.
It is obvious that fz ∈ A belongs to Cα, β if and if zf z ∈ S∗ α, β. Further, we note
that S∗ α, 1 ≡ S∗ α and Cα, 1 ≡ Cα which are, respectively, starlike and convex univalent
functions of order α.
Let P denote the class of functions of the form pz 1 p1 z · · · analytic in U which
satisfy the condition Re{P z} > 0.
∞
k
For functions f given by 1.1 and gz z k2 bk z , let f∗gz denote the
Hadamard product or convolution of fz and gz, defined by
f∗gz fz∗gz z ∞
ak bk zk .
1.4
k2
If f and g are analytic in U, we say that f is subordinate to g, written f ≺ g or fz ≺ gz, if
there exists a Schwarz function w in U such that fz gwz 1.
Let f ∈ A. Denote by Dλ : A → A the operator defined by
Dλ z
1 − zλ1
∗fz λ > −1.
1.5
It is obvious that D0 fz fz, D1 fz zf z, and
Dδ fz δ
z zδ−1 fz
δ!
δ ∈ N0 N ∪ {0} .
1.6
The operator Dδ f is called the δth-order Ruscheweyh derivative of f. Recently, K. I. Noor 2
and K. I. Noor and M. A. Noor 3 defined and studied an integral operator In : A → A,
analogous to Dδ fz as follows.
−1
Let fn z/1 − zn1 , n ∈ N0 , and fn z be defined such that
−1
fn z∗fn
z z
1 − z2
.
1.7
∗fz f ∈ A.
1.8
Then
In fz −1
fn z∗fz
z
−1
1 − zn1
We note that I0 fz zf z, I1 fz fz. The operator In is called the Noor integral of
nth order of f see 4, 5, which is an important tool in defining several classes of analytic
functions. In recent years, it has been shown that Noor integral operator has fundamental and
significant applications in the geometric function theory.
For real or complex numbers a, b, c other than 0, −1, −2, . . . , the hypergeometric series is
defined by
2 F1 a, b; c; z
∞
ak bk
ck 1k
k0
zk ,
1.9
K. Al-Shaqsi and M. Darus
3
where xk is Pochhammer symbol defined by
xk Γx k
xx 1 · · · x k − 1
Γx
for k 1, 2, 3, . . . , x ∈ C, x0 1.
1.10
We note that the series 1.9 converges absolutely for all z ∈ U so that it represents an
analytic function in U. Also an incomplete beta function φa, c; z is related to Gauss hypergeometric function z2 F1 a, b; c; z as
1.11
φa, c; z z2 F1 1, a; c; z,
and we note that φa, 1; z z/1 − za , where φ2, 1; z is Koebe function. Using φa, c; z, a
convolution operator 6, was defined by Carlson and Shaferr. Furthermore, Hohlov 7 introduced a convolution operator using 2 F1 a, b; c; z.
N. Shukla and P. Shukla 8 studied the mapping properties of a function fμ to be as
given in
fμ a, b, cz 1 − μz2 F1 a, b, c; z μz z2 F1 a, b, c; z
μ ≥ 0,
1.12
and investigated the geometric properties of an integral operator of the form
Iz z
fμ t
dt.
t
0
1.13
Kim and Shon 9 considered linear operator Lμ : A → A defined by Lμ a, b, cfz fμ a, b, cz∗fz.
We now introduce a function fμ −1 given by
−1
fμ a, b, cz∗ fμ a, b, cz
z
1 − zλ1
μ ≥ 0, λ > −1,
1.14
and obtain the following linear operator:
−1
∗fz.
Iμλ a, b, cfz fμ a, b, cz
1.15
The operator Iμλ is known as the generalized integral operator. For μ 0 in 1.14, Iλ a, b;
cfz : Iμλ a, b; cfz, which was introduced by K. I. Noor 10.
Now we find the explicit form of the function fμ −1 . It is well known that λ > −1
z
1 − zλ1
∞
λ 1k
k0
k!
zk1
z ∈ U.
1.16
Putting 1.9 and 1.16 in 1.14, we get
∞
μk 1ak bk
k0
ck 1k
∞
−1 λ 1k k1
z .
zk1 ∗ fμ
k!
k0
1.17
4
International Journal of Mathematics and Mathematical Sciences
Therefore, the function fμ −1 has the following form:
−1
fμ a, b, cz
∞
λ 1k ck
zk1
μk
1a
b
k
k
k0
z ∈ U.
1.18
Now we note that
Iμλ a, b, cfz z ∞
λ 1k ck
ak1 zk1 .
μk
1a
b
k
k
k1
1.19
From 1.19, we note that
I0λ a, λ 1, afz fz,
I01 a, 1, afz zf z.
Also it can easily be verified that
z Iμλ a, b, cfz λ 1Iμλ1 a, b, cfz − λIμλ a, b, cfz,
z Iμλ a 1, b, cfz aIμλ a, b, cfz − a − 1Iμλ a 1, b, cfz.
1.20
1.21
1.22
Now we introduce the following classes in term of the new operator Iμλ a, b, c. For λ > −1,
μ ≥ 0, 0 ≤ α < 1, and 0 < β ≤ 1, let Sμλ a, b, c; α, β be the class of functions f ∈ A satisfying
λ
π
z Iμ a, b, cfz
−
α
< β
arg
2
Iμλ a, b, cfz
z ∈ U.
1.23
Observe that Iμλ a, b, cfz ∈ S∗ α, β and zIμλ a, b, cfz /Iμλ a, b, cfz /
α. Also, for λ >
λ
−1, μ ≥ 0, 0 ≤ α < 1, and 0 < β ≤ 1, let Cμ a, b, c; α, β be the class of functions f ∈ A satisfying
π
z Iμλ a, b, cfz
−
α
< β
arg 1 λ
2
Iμ a, b, cfz
z ∈ U.
1.24
Observe that Iμλ a, b, cfz ∈ Cα, β and 1 zIμλ a, b, cfz /Iμλ a, b, cfz / α.
λ
λ
Clearly, f ∈ Cμ a, b, c; α, β if and only if zf z ∈ Sμ a, b, c; α, β.
Note that S0λ a, λ 1, a; α, β ≡ S∗ α, β, S0λ a, λ 1, a; α, 1 ≡ S∗ α, C0λ a, λ 1, a; α, β ≡
Cα, β, and C0λ a, λ 1, a; α, 1 ≡ Cα.
Finally, let Kλμ a, b, c; α, β, γ; A, B be the class of functions f ∈ A satisfying
λ
π
z Iμ a, b, cfz
−
α
< β
arg
λ
2
Iμ a, b, cgz
z ∈ U
1.25
for λ > −1, μ ≥ 0, 0 ≤ α < 1, 0 < β ≤ 1, and g ∈ Qλμ a, b, c; γ; A, B, where
Qλμ a, b, c; γ; A, B
λ
z Iμ a, b, cgz
1
1 Az
g∈A:
−γ ≺
z ∈ U; 0 ≤ γ < 1, − 1 ≤ B < A ≤ 1.
1−γ
1 Bz
Iμλ a, b, cgz
1.26
K. Al-Shaqsi and M. Darus
5
We also note that Kλ0 a, λ 1, a; α, 1, γ; 1, −1 and Kλ0 a, 1, a; α, 1, γ; 1, −1 are the classes of quasiconvex and close-to-convex functions of order α and type γ, respectively, introduced and
studied by Noor and Alkhorasani 11 and Silverman 12.
2. Main results
In order to give our results, we need the following lemmas.
Lemma 2.1 see 13. Let β, γ be complex numbers. Let φz be convex univalent in U with φ0 1
and Reβφz γ > 0, z ∈ U and q ∈ A with qz ≺ φz, z ∈ U. If p ∈ P is analytic in U, then
pz zp z
≺ φz z ∈ U
βqz γ
2.1
implies
2.2
pz ≺ φz z ∈ U.
Lemma 2.2 see 14. Let δ, η be complex numbers. Let φz be convex univalent in U with φ0 1
and Reδφz η > 0, z ∈ U. If p ∈ P is analytic in U, then
pz zp z
≺ φz
δpz η
2.3
z ∈ U
implies
2.4
pz ≺ φz z ∈ U.
Lemma 2.3 see 15. Let φz be convex univalent in U and let E ≥ 0. Suppose Bz is analytic in
U with Re Bz ≥ E, If g ∈ P is analytic in U, then
Ez2 g z Bzzg z gz ≺ φz
z ∈ U
2.5
implies
gz ≺ φz
2.6
z ∈ U.
Theorem 2.4. Let φz be convex univalent in U with φ0 1 and Re φz ≥ 0. If fz ∈ A satisfies
the condition
λ1
z Iμ a, b, cfz
1
2.7
− γ ≺ φz z ∈ U,
1−γ
Iμλ1 a, b, cfz
then
1
1−γ
λ
z Iμ a, b, cfz
for λ > −1, μ ≥ 0, and 0 ≤ γ < 1.
Iμλ a, b, cfz
−γ
≺ φz
z ∈ U
2.8
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International Journal of Mathematics and Mathematical Sciences
Proof. Let
1
pz 1−γ
λ
z Iμ a, b, cfz
Iμλ a, b, cfz
−γ
,
2.9
where p ∈ P . By using 1.21 in 2.9 and then differentiating, we get
1
1−γ
λ1
z Iμ a, b, cfz
Iμλ1 a, b, cfz
−γ
pz zp z
,
λ 1qz
2.10
where qz Iμλ1 a, b, cfz/Iμλ a, b, cfz and qz ≺ φz. Hence by applying Lemma 2.1,
we obtain the required result.
Theorem 2.5. Let φz be convex univalent in U with φ0 1 and Re φz ≥ 0. If fz ∈ A satisfies
the condition
λ
z Iμ a, b, cfz
1
2.11
− γ ≺ φz z ∈ U,
1−γ
Iμλ a, b, cfz
then
1
1−γ
λ
z Iμ a 1, b, cfz
Iμλ a 1, b, cfz
−γ
≺ φz
z ∈ U
2.12
for λ > −1, μ ≥ 0, and 0 ≤ γ < 1.
Proof. By using the same technique in the proof of Theorem 2.4 and using 1.22 and applying
Lemma 2.2, we obtain the required result.
Taking φz 1 Az/1 Bz −1 ≤ B < A ≤ 1 in Theorem 2.4 and in Theorem 2.5,
we have the following.
Corollary 2.6. It holds that
λ
Qλ1
μ a, b, c; γ; A, B ⊂ Qμ a, b, c; γ; A, B,
Qλμ a, b, c; γ; A, B ⊂ Qμλ a 1, b, c; γ; A, B
2.13
for λ > −1, μ ≥ 0, 0 ≤ γ < 1, and Re a > 1 − γ.
Also, by taking φz 1 z/1 − zβ 0 < β ≤ 1 in Theorem 2.4 and in Theorem 2.5,
we have the following.
Corollary 2.7. It holds that
Sμλ1 a, b, c; γ, β ⊂ Sμλ a, b, c; γ, β,
Sμλ a, b, c; γ, β ⊂ Sμλ a 1, b, c; γ, β
for λ > −1, μ ≥ 0, 0 ≤ γ < 1, and Re a > 1 − β.
2.14
K. Al-Shaqsi and M. Darus
7
Corollary 2.8. For λ > −1, μ ≥ 0, 0 ≤ γ < 1, and Re a > 1 − β, one has
λ
Cλ1
μ a, b, c; γ, β ⊂ Cμ a, b, c; γ, β,
2.15
Cλμ a, b, c; γ, β ⊂ Cλμ a 1, b, c; γ, β.
Proof. We will proof the first relation and by the same method we can proof the second relation
λ1
fz ∈ Cλ1
μ a, b, c; γ, β ⇐⇒ zf z ∈ Sμ a, b, c; γ, β
⇐⇒ zf z ∈ Sμλ a, b, c; γ, β
⇐⇒ Iμλ a, b, c zf z ∈ S∗ γ, β
⇐⇒ z Iμλ a, b, cfz ∈ S∗ γ, β
2.16
⇐⇒ Iμλ a, b, cfz ∈ Cγ, β
⇐⇒ fz ∈ Cλμ a, b, c; γ, β.
Theorem 2.9. Let φz be convex univalent in U with φ0 1 and Re φz ≥ 0. If fz ∈ A satisfies
the condition
λ
z Iμ a, b, cfz
1
2.17
− γ ≺ φz 0 ≤ γ < 1; z ∈ U,
1−γ
Iμλ a, b, cfz
then
1
1−γ
λ
z Iμ a, b, cFz
Iμλ a, b, cFz
−γ
0 ≤ γ < 1; z ∈ U,
≺ φz
2.18
where F be the integral operator defined by
Fz c1
zc
z
tc−1 ftdt
2.19
c > −1.
0
Proof. From 2.19, we have
z Iμλ a, b, cFz c 1Iμλ a, b, cfz − cIμλ a, b, cFz.
2.20
Now, let
1
pz 1−γ
λ
z Iμ a, b, cFz
Iμλ a, b, cFz
−γ
2.21
,
where p ∈ P . Then by using 2.20, we get
1 − γpz c γ c 1Iμλ a, b, cfz
Iμλ a, b, cFz
.
2.22
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International Journal of Mathematics and Mathematical Sciences
Differentiating both sides of 2.22 logarithmically, we obtain
zp z
1
pz c γ 1 − γpz 1 − γ
λ
z Iμ a, b, cfz
Iμλ a, b, cfz
−γ
.
2.23
Then, by Lemma 2.2, we obtain that
1
1−γ
λ
z Iμ a, b, cfz
Iμλ a, b, cfz
−γ
≺ φz
0 ≤ γ < 1; z ∈ U.
2.24
Now, by letting φz 1 Az/1 Bz −1 ≤ B < A ≤ 1 in Theorem 2.9, we have the
following.
Corollary 2.10. For λ > −1, μ ≥ 0, c > −γ, and 0 ≤ γ < 1. If f ∈ Qλμ a, b, c; γ; A, B, then F ∈
Qλμ a, b, c; γ; A, B, where F given by 2.19.
Also, by taking φz 1 z/1 − zβ 0 < β ≤ 1 in Theorem 2.9, we have the
following.
Corollary 2.11. For λ > −1, μ ≥ 0, c > −β, 0 < β ≤ 1, and 0 ≤ γ < 1. If f ∈ Sμλ a, b, c; γ, β, then
F ∈ Sμλ a, b, c; γ, β.
Corollary 2.12. For λ > −1, μ ≥ 0, c > −β, 0 < β ≤ 1, and 0 ≤ γ < 1. If f ∈ Cλμ a, b, c; γ, β, then
F ∈ Cλμ a, b, c; γ, β.
Proof. It holds that
fz ∈ Cλμ a, b, c; γ, β ⇐⇒ F zf z ∈ Sμλ a, b, c; γ, β
⇐⇒ z Fz ∈ Sμλ a, b, c; γ, β
2.25
⇐⇒ Fz ∈ Cλμ a, b, c; γ, β.
Theorem 2.13. Let f ∈ A. Then
λ
Kλ1
μ a, b, c; α, β, γ; A, B ⊂ Kμ a, b, c; α, β, γ; A, B
2.26
for Re a > 1 − β, 0 < β ≤ 1, 0 ≤ α < 1, 0 ≤ γ < 1, and −1 ≤ B < A ≤ 1.
Proof. Let f ∈ Kλ1
μ a, b, c; α, β, γ; A, B, then by the definition, we can write
1
1−α
λ1
z Iμ a, b, cfz
Iμλ1 a, b, cgz
for some g ∈ Qμλ1 a, b, c; γ; A, B.
−α
≺
1z
1−z
β
z ∈ U
2.27
K. Al-Shaqsi and M. Darus
9
Letting hz zIμλ a, b, cfz /Iμλ a, b, cgz and Hz zIμλ a, b, cgz Iμλ a, b,
cgz, we observe that hz, Hz ∈ P z. Now by Corollary 2.6, g ∈ Qμλ a, b, c; γ; A, B and so
Re Hz > γ. Also, note that
z Iμλ a, b, cfz Iμλ a, b, cgz hz.
2.28
Differentiating both sides in 2.28 yields
z Iμλ a, b, cfz
Iμλ a, b, cgz
z Iμλ a, b, cgz
Iμλ1 a, b, cgz
hz zh z Hzhz zh z.
2.29
Now by using the identity 1.21, we obtain
z Iμλ1 a, b, cfz
Iμλ1 a, b, cgz
Iμλ1 a, b, c zf z
Iμλ1 a, b, cgz
z Iμλ a, b, c zf z λIμλ a, b, c zf z
z Iμλ a, b, cgz λIμλ a, b, cgz
z Iμλ a, b, c zf z /Iμλ a, b, cgz λ Iμλ a, b, c zf z /Iμλ a, b, cgz
z Iμλ a, b, cgz /Iμλ a, b, cgz λ
2.30
Hzhz zh z λhz
Hz λ
hz zh z
.
Hz λ
From 2.27, 2.28, and 2.30, we conclude that
zh z
1
1z β
hz .
−α ≺
1−α
Hz λ
1−z
2.31
Letting E 0 and Bz 1/1 − α1/Hz λ, we obtain
Re Bz 1
1
Re Hz λ > 0.
1 − α Hz λ2
2.32
The above inequality satisfies the conditions required by Lemma 2.3. Hence φz ≺ 1 z/
1 − zβ and so the proof is complete.
Theorem 2.14. Let f ∈ A. Then
Kλμ a, b, c; α, β, γ; A, B ⊂ Kλμ a 1, b, c; α, β, γ; A, B
for Re a > 1 − β, 0 < β ≤ 1, 0 ≤ α < 1, 0 ≤ γ < 1, and −1 ≤ B < A ≤ 1.
2.33
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International Journal of Mathematics and Mathematical Sciences
Proof. By using the same technique as in the proof of Theorem 2.13, we get
1
zh z
1z β
hz .
−α ≺
1−α
Hz α − 1
1−z
2.34
By letting E 0 and Bz 1/1 − α1/Hz a − 1, we obtain
Re Bz 1
1
2 Re Hz a − 1 > 0.
1 − α Hz a − 1
2.35
Then, by applying Lemma 2.3, we obtain the required result.
Theorem 2.15. Let c > −β, 0 < β ≤ 1, 0 ≤ α < 1, 0 ≤ γ < 1, and −1 ≤ B < A ≤ 1. If f ∈
Kλμ a, b, c; α, β, γ; A, B, then F ∈ Kλμ a, b, c; α, β, γ; A, B, where F is given by 2.19.
Proof. Also, by using the same technique as in the proof of Theorem 2.13, we get
zh z
1z β
1
hz − α ≺
.
1 − α Hz c
1−z
2.36
By letting E 0 and Bz 1/1 − α1/Hz c, we obtain
Re Bz 1
1
2 Re Hz c > 0.
1 − α Hz c
2.37
Then, applying Lemma 2.3, we obtain the required result.
Acknowledgment
The work presented here was supported by Fundamental Research Grant Scheme UKM-ST-01FRGS0055-2006.
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