Volume 2009, Article ID 158408,13pages doi:10.1155/2009/158408
Research Article
A New General Integral Operator Defined by
Al-Oboudi Differential Operator
Serap Bulut
Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 ˙Izmit-Kocaeli, Turkey
Correspondence should be addressed to Serap Bulut,[email protected]
Received 8 December 2008; Accepted 22 January 2009
Recommended by Narendra Kumar Govil
We define a new general integral operator using Al-Oboudi differential operator. Also we introduce new subclasses of analytic functions. Our results generalize the results of Breaz, G ¨uney, and S˘al˘agean.
Copyrightq2009 Serap Bulut. 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
LetAdenote the class of functions of the form
fz z∞
n2
anzn 1.1
which are analytic in the open unit diskU{z∈C:|z|<1}, andS:{f∈ A:fis univalent inU}.
Forf∈ A, Al-Oboudi1introduced the following operator:
D0fz fz, 1.2
D1fz 1−λfz λzfz D
λfz, λ≥0, 1.3
Dkfz D
Iffis given by1.1, then from1.3and1.4we see that
Dkfz z∞
n2
1 n−1λkanzn, k∈N0:N∪ {0}
, 1.5
withDkf0 0.
Remark 1.1. Whenλ1, we get S˘al˘agean’s differential operator2.
Now we introduce new classesSkδ, b, λandKkδ, b, λas follows.
A functionf ∈ Ais in the classesSkδ, b, λ, whereδ∈0,1,b∈C−{0},λ≥0,k∈N0,
if and only if
Re
11
b
Dk1fz
Dkfz −1
> δ 1.6
or equivalently
Re
1λ
b
zDkfz
Dkfz −1
> δ 1.7
for allz∈U.
A functionf∈ Ais in the classsKkδ, b, λ, whereδ∈0,1,b∈C− {0},λ≥0,k∈N0,
if and only if
Re
1λ
b
zDkfz
Dkfz
> δ 1.8
for allz∈U.
We note thatf∈ Kkδ, b, λif and only ifzf ∈ Skδ, b, λ.
Remark 1.2. iFork0 andλ1, we have the classes
S0δ, b,1≡ S∗δb, K0δ, b,1≡ Cδb 1.9
introduced by Frasin3.
iiForb1 andλ1, we have the class
Skδ,1,1≡ Skδ 1.10
ofk-starlike functions of orderδdefined by S˘al˘agean2.
iiiIn particular, the classes
are the classes of starlike functions of orderδ and convex functions of orderδ in U, resp-ectively.
ivFurthermore, the classes
S00,1,1≡ S∗, K00,1,1≡ K 1.12
are familiar classes of starlike and convex functions inU, respectively.
vForλ1, we get
Kkδ, b,1≡ Sk1δ, b,1. 1.13
Let us introduce the new subclassesUSkα, δ, b, λ,UKkα, δ, b, λandSHkα, b, λ,
KHkα, b, λas follows.
A functionf ∈ Ais in the classUSkα, δ, b, λif and only iffsatisfies
Re
11
b
Dk1fz
Dkfz −1
> α 1b
Dk1fz
Dkfz −1
δ z∈U 1.14
or equivalently
Re
1λ
b
zDkfz
Dkfz −1
> α λb
zDkfz
Dkfz −1
δ, 1.15
whereα≥0,δ∈−1,1,αδ≥0,b∈C− {0},λ≥0,k∈N0.
A functionf ∈ Ais in the classUKkα, δ, b, λif and only iffsatisfies
Re
1λ
b
zDkfz
Dkfz
> α λbz
Dkfz
Dkfz
δ, 1.16
whereα≥0,δ∈−1,1,αδ≥0,b∈C− {0},λ≥0,k∈N0.
We note thatf∈ UKkα, δ, b, λif and only ifzf∈ USkα, δ, b, λ.
Remark 1.3. iForα0, we have
USk0, δ, b, λ≡ Skδ, b, λ, UKk0, δ, b, λ≡ Kkδ, b, λ. 1.17
iiForb1 andλ1, we have the class
USkα, δ,1,1≡ USkα, δ. 1.18
ofk-uniform starlike functions of orderδand typeα,4.
iiiForλ1, we have
ivForb1 andλ1, we have
UKkα, δ,1,1≡ USk1α, δ. 1.20
Geometric Interpretation
f ∈ USkα, δ, b, λandf ∈ UKkα, δ, b, λif and only if 1 λ/bzDkfz/Dkfz−1
and 1 λ/bzDkfz/Dkfz, respectively, take all the values in the conic domain
Rα,δ which is included in the right-half plane such that
Rα,δ
uiv:u > α
u−12v2δ. 1.21
From elementary computations we see that ∂Rα,δ represents the conic sections
symmetric about the real axis. ThusRα,δ is an elliptic domain forα >1, a parabolic domain
forα1, a hyperbolic domain for 0< α <1 and a right-half planeu > δforα0. A functionf ∈ Ais in the classSHkα, b, λif and only iffsatisfies
1 1
b
Dk1fz
Dkfz −1
−2α√2−1 <Re
√ 2
11
b
Dk1fz
Dkfz −1
2α√2−1 z∈U,
1.22
whereα >0,b∈C− {0},λ≥0,k∈N0.
A functionf ∈ Ais in the classKHkα, b, λif and only iffsatisfies
1λ
b
zDkfz
Dkfz −2α
√
2−1 <Re
√ 2
1λ
b
zDkfz
Dkfz
2α√2−1 z∈U,
1.23
whereα >0,b∈C− {0},λ≥0,k∈N0.
We note thatf∈ KHkα, b, λif and only ifzf∈ SHkα, b, λ.
Remark 1.4. iForb1 andλ1, we have the classes
SHkα,1,1≡ SHkα,
KHkα,1,1≡ SHk1α,1,1≡ SHk1α
1.24
defined in5.
iiForλ1, we have
D. Breaz and N. Breaz6introduced and studied the integral operator
Fnz
z
0
f1t
t
μ1 · · ·
fnt
t
μn
dt, 1.26
wherefi∈ Aandμi>0 for alli∈ {1, . . . , n}.
By using the Al-Oboudi differential operator, we introduce the following integral operator. So we generalize the integral operatorFn.
Definition 1.5. Letk ∈ N0,l l1, . . . , ln ∈Nn0,andμi > 0, 1 ≤ i ≤ n.One defines the integral
operatorIk,n,l,μ:An → A,
Ik,n,l,μf1, . . . , fnF,
DkFz
z
0
Dl1f
1t
t
μ1 · · ·
Dlnfnt
t
μn
dt,
1.27
wheref1, . . . , fn∈ AandDis the Al-Oboudi differential operator.
Remark 1.6. InDefinition 1.5, if we set
iλ1, then we have7, Definition 1.
ii λ 1, k 0 andl1 · · · ln 0, then we have the integral operator defined
by1.26.
iiik0,l1· · ·lnl∈N0, then we have8, Definition 1.1.
2. Main Results
The following lemma will be required in our investigation.
Lemma 2.1. For the integral operatorIk,n,l,μf1, . . . , fn F, defined by1.27, one has
λzDkFz
DkFz
n
i1
μiD
li1f iz
Dlifiz −
n
i1
μi. 2.1
Proof. By1.27, we get
DkFz
Dl1f
1z
z
μ1 · · ·
Dlnfnz
z
μn
. 2.2
Also, using1.3and1.4, we obtain
DkFz Dk1Fz−1−λDkFz
On the other hand, from2.2and2.3, we find
DkFzn
i1
μi
Dlifiz
z
μizDlif
iz−Dlifiz
zDlifiz
n
j1
j /i
Dljfjz
z
μj
, 2.4
DkFz Dk2Fz−2−λDk1Fz 1−λDkFz
λ2z2 . 2.5
Thus by2.2and2.4, we can write
DkFz
DkFz
n
i1
μi
zDlifiz−Dlifiz
zDlifiz
n
i1
μi
Dli1f
iz−Dlifiz
λzDlifiz
.
2.6
Finally, we obtain
λzDkFz
DkFz
n
i1
μi
Dli1f
iz
Dlifiz −1
, 2.7
which is the desired result.
Theorem 2.2. Letαi≥0,δi∈−1,1,αiδi≥01≤i≤n, andb∈C− {0},λ≥0. Also suppose
that
n
i1
μi11−δαi
i ≤1. 2.8
Iffi∈ USliαi, δi, b, λ1≤i≤n, then the integral operatorIk,n,l,μ F, defined by1.27, is in the
classKkγ, b, λ, where
γ1−
n
i1
μi11−δαi
i. 2.9
Proof. Sincefi∈ USliαi, δi, b, λ1≤i≤n, by1.14we have
Re
11
b
Dli1f
iz
Dlifiz −1
> ααiδi
for allz∈U. By2.1, we get
11
b
λzDkFz
DkFz 1
n
i1
μi1b
Dli1f
iz
Dlifiz −1
1
n
i1
μi
11
b
Dli1f
iz
Dlifiz −1
−n
i1
μi.
2.11
So,2.10and2.11give us
Re
11
b
λzDkFz
DkFz
1−
n
i1
μi
n
i1
μiRe
1 1
b
Dli1f
iz
Dlifiz −1
>1−
n
i1
μi
n
i1
μiααiδi
i1 1− n
i1
μi11−δαi i
2.12
for allz∈U. Hence, we obtainF∈ Kkγ, b, λ, whereγ 1−in1μi1−δi/1αi.
Corollary 2.3. Letαi≥0,δi∈−1,1,αiδi≥01≤i≤n, andb∈C− {0}. Also suppose that
n
i1
μi11−δαi
i ≤1. 2.13
Iffi∈ USliαi, δi, b,1 1≤i≤n, then the integral operatorIk,n,l,μF, defined by1.27, is in the classSk1γ, b,1, whereγis defined as in2.9.
Proof. InTheorem 2.2, we considerλ1.
FromCorollary 2.3, we immediately getCorollary 2.4.
Corollary 2.4. Letαi≥0,δi∈−1,1,αiδi≥0 1≤i≤n, andb∈C− {0}. Also suppose that
n
i1
μi11−δαi
i ≤1. 2.14
Iffi∈ USliαi, δi, b,1 1≤i≤n, then the integral operatorIk,n,l,μF, defined by1.27, is in the classSk10, b,1.
Remark 2.5. If we setb1 inCorollary 2.4, then we have7, Theorem 1. SoCorollary 2.4is an extension of Theorem 1.
Corollary 2.6. Letδi∈0,11≤i≤nandb∈C− {0},λ≥0. Also suppose that
n
i1
Iffi∈ Sliδi, b, λ1≤i≤n, then the integral operatorIk,n,l,μ F, defined by1.27, is in the class
Kkρ, b, λ, where
ρ1−
n
i1
μi1−δi. 2.16
Proof. InTheorem 2.2, we considerα1 α2 · · ·αn0.
Corollary 2.7. Letδi∈0,1 1≤i≤nandb∈C− {0}. Also suppose that
n
i1
μi1−δi≤1. 2.17
Iffi∈ Sliδi, b,1 1≤i≤n, then the integral operatorIk,n,l,μF, defined by1.27, is in the class
Sk1ρ, b,1, whereρis defined as in2.16.
Proof. InCorollary 2.6, we considerλ1.
Corollary 2.8readily follows fromCorollary 2.7.
Corollary 2.8. Letδi∈0,1 1≤i≤n, andb∈C− {0}. Also suppose that
n
i1
μi1−δi≤1. 2.18
Iffi∈ Sliδi, b,1 1≤i≤n, then the integral operatorIk,n,l,μF, defined by1.27, is in the class Sk10, b,1.
Remark 2.9. If we setb1 inCorollary 2.8, then we have7, Corollary 1.
Theorem 2.10. Letαi≥0,δi∈−1,1,αiδi≥0 1≤i≤nandb∈C− {0},λ≥0. Also suppose that
n
i1
μi≤1. 2.19
Iffi∈ USliαi, δi, b, λ 1≤i≤n, then the integral operatorIk,n,l,μF, defined by1.27, is in the
classKkγ, b, λ, whereγis defined as in2.9.
Proof. The proof is similar to the proof ofTheorem 2.2.
Corollary 2.11. Letαi≥0,δi∈−1,1,αiδi≥01≤i≤nandb∈C− {0}. Also suppose that
n
i1
Iffi∈ USliαi, δi, b,1 1≤i≤n, then the integral operatorIk,n,l,μF, defined by1.27, is in the classSk1γ, b,1, whereγis defined as in2.9.
Proof. InTheorem 2.10, we considerλ1.
Remark 2.12. If we setb1 inCorollary 2.11, then we have7, Theorem 2.
Corollary 2.13. Letδi∈0,11≤i≤nandb∈C− {0},λ≥0. Also suppose that
n
i1
μi≤1. 2.21
Iffi∈ Sliδi, b, λ 1≤i≤n, then the integral operatorIk,n,l,μ F, defined by1.27, is in the class
Kkρ, b, λ, whereρis defined as in2.16.
Proof. InTheorem 2.10, we considerα1α2· · ·αn0.
Corollary 2.14. Letδi∈0,11≤i≤nandb∈C− {0}. Also suppose that
n
i1
μi≤1. 2.22
Iffi∈ Sliδi, b,1 1≤i≤n, then the integral operatorIk,n,l,μF, defined by1.27, is in the class
Sk1ρ, b,1, whereρis defined as in2.16.
Proof. InCorollary 2.13, we considerλ1.
Remark 2.15. If we setb1 inCorollary 2.14, then we have7, Corollary 2.
Theorem 2.16. Letα≥0,δ∈−1,1,αδ≥0andb∈C− {0},λ≥0. Also suppose that
n
i1
μi≤1. 2.23
Iffi ∈ USliα, δ, b, λ 1≤i≤n, then the integral operatorIk,n,l,μ F, defined by1.27, is in the
classUKkα, δ, b, λ.
Proof. Sincefi∈ USliα, δ, b, λ 1≤i≤n, by1.14we have
Re
11
b
Dli1f
iz
Dlifiz −1
> α 1b
Dli1f
iz
Dlifiz −1
δ 2.24
On the other hand, from2.1, we obtain
1λ
b
zDkFz
DkFz 1
n
i1
μib1
Dli1f
iz
Dlifiz −1
1−
n
i1
μi
n
i1
μi
11
b
Dli1f
iz
Dlifiz −1
.
2.25
Considering1.16with the above equality, we find
Re
1 λ
b
zDkFz
DkFz
−α λbz
DkFz
DkFz
−δ
1−
n
i1
μi
n
i1
μiRe
11
b
Dli1f
iz
Dlifiz −1
−α n
i1
μib1
Dli1f
iz
Dlifiz −1
−δ
≥1−
n
i1
μi
n
i1
μiRe
11
b
Dli1f
iz
Dlifiz −1
−αn
i1
μi
1
b
Dli1f
iz
Dlifiz −1
−δ
>1−
n
i1
μi
n
i1
μi
α 1b
Dli1f
iz
Dlifiz −1
δ
−αn
i1
μi
1
b
Dli1f
iz
Dlifiz −1
−δ
1−δ
1−
n
i1
μi
≥0
2.26
for allz∈U. This completes proof.
Corollary 2.17. Letα≥0,δ∈−1,1,αδ≥0, andb∈C− {0}. Also suppose that
n
i1
μi≤1. 2.27
Iffi ∈ USliα, δ, b,1 1≤i≤n, then the integral operatorIk,n,l,μ F, defined by1.27, is in the
classUSk1α, δ, b,1.
Proof. InTheorem 2.16, we considerλ1.
Remark 2.18. If we setb1 inCorollary 2.17, then we have7, Theorem 3.
Theorem 2.19. Letα≥0,b∈C− {0}, andλ≥0. Also suppose that
n
i1
μi≤1. 2.28
Iffi ∈ SHliα, b, λ 1 ≤i≤ n, then the integral operatorIk,n,l,μ F, defined by1.27, is in the
Proof. Sincefi∈ SHliα, b, λ 1≤i≤n, by1.22we have Re √ 2 √ 2 b
Dli1f
iz
Dlifiz −1
2α√2−1− 1 1
b
Dli1f
iz
Dlifiz −1
−2α√2−1 >0
2.29
for allz∈U. Considering this inequality and2.1, we obtain
Re √ 2 √ 2 b
λzDkFz
DkFz
2α√2−1− 1 1
b
λzDkFz
DkFz −2α
√ 2−1
Re √ 2 √ 2 b n
i1
μi
Dli1f
iz
Dlifiz −1
2α√2−1
− 11
b
n
i1
μi
Dli1f
iz
Dlifiz −1
−2α√2−1
√2
n
i1
μiRe
√ 2
b
Dli1f
iz
Dlifiz −1
2α√2−1
− 1
n
i1
μi1b
Dli1f
iz
Dlifiz −1
−2α√2−1
√2
n
i1
μiRe
√ 2 √ 2 b
Dli1f
iz
Dlifiz −1
−√2
n
i1
μi2α
√ 2−1
− 1
n
i1
μi
11
b
Dli1f
iz
Dlifiz −1
−2α√2−1
−n
i1
μi2α√2−1
n
i1
μi−2α√2−1
√2 1− n
i1
μi
2α√2−1
n
i1
μiRe
√ 2 √ 2 b
Dli1f
iz
Dlifiz −1
− 1−2α√2−1
1−
n
i1
μi
n
i1
μi
11
b
Dli1f
iz
Dlifiz −1
−2α√2−1
≥√2
1−
n
i1
μi
2α√2−1
n
i1
μiRe
√ 2 √ 2 b
Dli1f
iz
Dlifiz −1
− 1−2α√2−1
1−
n
i1
μi
−n
i1
μi
11
b
Dli1f
iz
Dlifiz −1
−2α√2−1
2α√2−1
n
i1
μi−2α
√ 2−1
n
i1
√22α√2−1− 1−2α√2−1
1−
n
i1
μi
n
i1
μi
Re
√ 2
√ 2
b
Dli1f
iz
Dlifiz −1
2α√2−1− 11
b
Dli1f
iz
Dlifiz −1
−2α√2−1
>√22α√2−1− 1−2α√2−1
1−
n
i1
μi
>
1−
n
i1
μi
min√2−114α,√21≥0
2.30
for allz∈U. Hence by1.23, we haveF∈ KHkα, b, λ.
Corollary 2.20. Letα≥0andb∈C− {0}. Also suppose that
n
i1
μi≤1. 2.31
Iffi ∈ SHliα, b,1 1 ≤ i≤ n, then the integral operatorIk,n,l,μ F, defined by1.27, is in the
classSHk1α, b,1.
Proof. InTheorem 2.19, we considerλ1.
Remark 2.21. If we setb1 inCorollary 2.20, then we have7, Theorem 4.
Theorem 2.22. Letα≥0,b∈C− {0}andλ≥0. Also suppose that
1√2α√2−1
n
i1
μi<1. 2.32
Iffi ∈ SHliα, b, λ 1 ≤i≤ n, then the integral operatorIk,n,l,μ F, defined by1.27, is in the
classKk0, b, λ.
Proof. Sincefi∈ SHliα, b, λ 1≤i≤n, by1.22we have
Re
√
2
√ 2
b
Dli1f
iz
Dlifiz −1
2α√2−1> 11
b
Dli1f
iz
Dlifiz −1
−2α√2−1
for allz∈U. Considering this inequality and2.1, we obtain
√ 2Re
1λ
b
zDkfz
Dkfz
Re
√
2
√ 2
b
n
i1
μi
Dli1f
iz
Dlifiz −1
√2−√2
n
i1
μi
n
i1
μiRe
√ 2
√ 2
b
Dli1f
iz
Dlifiz −1
√2−√2
n
i1
μi−2α
√ 2−1
n
i1
μi
n
i1
μi
Re
√ 2
√ 2
b
Dli1f
iz
Dlifiz −1
2α√2−1
>√2
1−1√2α√2−1
n
i1
μi
>0
2.34
for allz∈U. Hence, by1.8, we haveF∈ Kk0, b, λ.
Corollary 2.23. Letα≥0andb∈C− {0}. Also suppose that
1√2α√2−1
n
i1
μi<1. 2.35
Iffi ∈ SHliα, b,1 1 ≤ i≤ n, then the integral operatorIk,n,l,μ F, defined by1.27, is in the
classSk10, b,1.
Proof. InTheorem 2.22, we considerλ1.
Remark 2.24. If we setb1 inCorollary 2.23, then we have7, Theorem 5.
References
1 F. M. Al-Oboudi, “On univalent functions defined by a generalized S˘al˘agean operator,”International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 27, pp. 1429–1436, 2004.
2 G. S¸. S˘al˘agean, “Subclasses of univalent functions,” inComplex Analysis-Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, Germany, 1983.
3 B. A. Frasin, “Family of analytic functions of complex order,”Acta Mathematica. Academiae Paedagogicae Ny´ıregyh´aziensis, vol. 22, no. 2, pp. 179–191, 2006.
4 I. Magdas¸, Doctoral thesis, University “Babes¸-Bolyai”, Cluj-Napoca, Romania, 1999.
5 M. Acu, “Subclasses of convex functions associated with some hyperbola,”Acta Universitatis Apulensis, no. 12, pp. 3–12, 2006.
6 D. Breaz and N. Breaz, “Two integral operators,”Studia Universitatis Babes¸-Bolyai. Mathematica, vol. 47, no. 3, pp. 13–19, 2002.
7 D. Breaz, H. ¨O. G ¨uney, and G. S¸. S˘al˘agean, “A new general integral operator,”Tamsui Oxford Journal of Mathematical Sciences. Accepted.
