Volume 2007, Article ID 51079,10pages doi:10.1155/2007/51079
Research Article
Inclusion Properties for Certain Subclasses of Analytic Functions Associated with the Dziok-Srivastava Operator
Oh Sang Kwon and Nak Eun Cho
Received 14 February 2007; Accepted 21 August 2007 Recommended by Andrea Laforgia
The purpose of the present paper is to introduce several new classes of analytic functions defined by using the Choi-Saigo-Srivastava operator associated with the Dziok-Srivastava operator and to investigate various inclusion properties of these classes. Some interesting applications involving classes of integral operators are also considered.
Copyright © 2007 O. S. Kwon and N. E. Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
LetᏭdenote the class of functions of the form f(z)=z+
∞ k=2
akzk (1.1)
which are analytic in the open unit diskU= {z:z∈Cand|z|<1}. If f andgare analytic in U, we say that f is subordinate tog, written f ≺g or f(z)≺g(z), if there exists a Schwarz functionw, analytic inUwithw(0)=0 and|w(z)|<1 (z∈U), such that f(z)= g(w(z)) (z∈U). In particular, if the functiongis univalent inU, the above subordination is equivalent tof(0)=g(0) andf(U)⊂g(U). For 0≤η,β <1, we denote by∗(η),(η), andᏯ(η,β) the subclasses ofᏭconsisting of all analytic functions which are, respectively, starlike of orderη, convex of orderη, close-to-convex of orderη, and typeβin U. For various other interesting developments involving functions in the classᏭ, the reader may be referred (for example) to the work of Srivastava and Owa [1].
Letᏺ be the class of all functionsφwhich are analytic and univalent inUand for whichφ(U) is convex withφ(0)=1 and Re{φ(z)}>0 forz∈U.
Making use of the principle of subordination between analytic functions, we introduce the subclasses∗(η;φ),(η;φ), andᏯ(η,δ;φ,ψ) of the class Ꮽfor 0≤η, β <1, and φ,ψ∈ᏺ(cf. [2,3]), which are defined by
∗(η;φ) :=
f ∈Ꮽ: 1 1−η
z f(z) f(z) −η
≺φ(z) inU ,
(η;φ) :=
f ∈Ꮽ: 1 1−η
1 +z f(z) f(z) −η
≺φ(z) inU
, Ꮿ(η,β;φ,ψ) :=
f ∈Ꮽ:∃g∈∗(η;φ) s.t. 1 1−β
z f(z) g(z) −β
≺ψ(z) inU
. (1.2)
We note that the classes mentioned above are the familiar classes which have been used widely on the space of analytic and univalent functions inU, and for special choices for the functionsφandψinvolved in these definitions, we can obtain the well-known sub- classes ofᏭ. For examples, we have
∗η;1 +z 1−z
=∗(η), η;1 +z 1−z
=(η), Ꮿη,β; 1 +z
1−z,1 +z 1−z
=Ꮿ(η,β).
(1.3)
Also let the Hadamard product (or convolution) f∗gof two analytic functions f(z)=
∞ k=0
akzk, g(z)= ∞ k=0
bkzk (1.4)
be given (as usual) by
(f∗g)(z)= ∞ k=0
akbkzk. (1.5)
Making use of the Hadamard product (or convolution) given by (1.5), we now define the Dziok-Srivastava operator
Hα1,. . .,αq;β1,. . .,βs
:Ꮽ−→Ꮽ, (1.6)
which was introduced and studied in a series of recent papers by Dziok and Srivastava ([4–6]; see also [7,8]). Indeed, for complex parameters
α1,. . .,αq, β1,. . .,βs
βj∈C\Z−0;Z−0 =0,−1,−2,. . .;j=1,. . .,s, (1.7) the generalized hypergeometric functionqFs(α1,. . .,αq;β1,. . .,βs;z) is given by
qFs
α1,. . .,αq;β1,. . .,βs;z:= ∞ n=0
α1
n···
αq
n
β1
n···
βs
n
zn n!
q≤s+ 1;q,s∈N0:=N∪ {0};z∈U ,
(1.8)
where (ν)kis the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma function) by
(ν)k:=Γ(ν+k) Γ(ν) =
⎧⎨
⎩
1 ifk=0,ν∈C\{0},
ν(ν+ 1)···(ν+k−1) ifk∈N,ν∈C. (1.9) Corresponding to a functionᏲ(α1,. . .,αq;β1,. . .,βs;z), defined by
Ᏺα1,. . .,αq;β1,. . .,βs;z:=zqFs
α1,. . .,αq;β1,. . .,βs;z, (1.10) Dziok and Srivastava [5] considered a linear operator defined by the following Hadamard product (or convolution):
Hα1,. . .,αq;β1,. . .,βs
f(z) :=Ᏺα1,. . .,αq;β1,. . .,βs;z∗f(z). (1.11) We note that the linear operatorH(α1,. . .,αq;β1,. . .,βs) includes various other linear operators which were introduced and studied by Carlson and Shaffer [9], Hohlov [10], Ruscheweyh [11], and so on [12,13].
Corresponding to the functionᏲ(α1,. . .,αq;β1,. . .,βs;z), defined by (1.10), we intro- duce a functionᏲλ(α1,. . .,αq;β1,. . .,βs;z) given by
Ᏺα1,. . .,αq;β1,. . .,βs;z∗Ᏺλ
α1,. . .,αq;β1,. . .,βs;z= z
(1−z)λ (λ >0). (1.12) Analogous to H(α1,. . .,αq;β1,. . .,βs), we now define the linear operator Hλ(α1,. . .,αq; β1,. . .,βs) onᏭas follows:
Hλ
α1,. . .,αq;β1,. . .,βs
f(z)=Ᏺλ
α1,. . .,αq;β1,. . .,βs;z∗f(z)
αi,βj∈C\Z−0;i=1,. . .,q; j=1,. . .,s;λ >0;z∈U; f ∈Ꮽ. (1.13) For convenience, we write
Hλ,q,s α1
:=Hλ
α1,. . .,αq;β1,. . .,βs
. (1.14)
It is easily verified from the definition (1.13) that zHλ,q,sα1+ 1f(z)=α1Hλ,q,sα1
f(z)−
α1−1Hλ,q,sα1+ 1f(z), (1.15) zHλ,q,s
α1
f(z)=λHλ+1,q,s
α1
f(z)−(λ−1)Hλ,q,s
α1
f(z). (1.16)
In particular, the operatorHλ(γ+ 1, 1; 1)(λ >0;γ >−1) was introduced by Choi et al. [2], who investigated (among other things) several inclusion properties involving various subclasses of analytic and univalent functions. Forγ=n(n∈N∪0;N= {1, 2,. . .}) and λ=2, we also note that the Choi-Sago-Srivastava operatorHλ,2,1(γ+ 1, 1; 1)f is the Noor integral operator ofnth order of f studied by Liu [14] and K. I. Noor and M. A. Noor [15,16].
Next, by using the operatorHλ,q,s(α1), we introduce the following classes of analytic functions forφ,ψ∈ᏺ, and 0≤η,β <1:
λ,α1(q,s;η;φ) :=
f ∈Ꮽ:Hλ,q,s
α1
f ∈∗(η;φ),
λ,α1(q,s;η;φ) :=
f ∈Ꮽ:Hλ,q,sα1
f ∈(η;φ), Ꮿλ,α1(q,s;η,β;φ,ψ) :=
f ∈Ꮽ:Hλ,q,sα1
f ∈Ꮿ(η,β;φ,ψ).
(1.17)
We also note that
f(z)∈λ,α1(q,s;η;φ)⇐⇒z f(z)∈λ,α1(q,s;η;φ). (1.18) In particular, we set
λ,α1
q,s;η;1 +Az 1 +Bz
=:λ,α1(q,s;η;A,B) (−1≤B < A≤1),
λ,α1
q,s;η;1 +Az 1 +Bz
=:λ,α1(q,s;η;A,B) (−1≤B < A≤1).
(1.19)
In this paper, we investgate several inclusion properties of the classesλ,α1(q,s;η;φ),
λ,α1(q,s;η;φ), andᏯλ,α1(q,s;η,β;φ,ψ) associated with the operatorHλ,q,s(α1). Some ap- plications involving integral operators are also considered.
2. Inclusion Properties Involving the OperatorHλ,q,s(α1) The following results will be required in our investigation.
Lemma 2.1 [17]. Letφbe convex univalent in Uwithφ(0)=1 and Re{κφ(z) +ν}>0 (κ,ν∈C). Ifpis analytic inUwithp(0)=1, then
p(z) + z p(z)
κp(z) +ν≺φ(z) (z∈U) (2.1)
implies
p(z)≺φ(z) (z∈U). (2.2)
Lemma 2.2 [18]. Letφbe convex univalent inUand letωbe analytic inUwith Re{ω(z)} ≥ 0. Ifpis analytic inUandp(0)=φ(0), then
p(z) +ω(z)z p(z)≺φ(z) (z∈U) (2.3) implies
p(z)≺φ(z) (z∈U). (2.4)
Theorem 2.3. Letα1,λ >1 andφ∈ᏺ. Then,
λ+1,α1(q,s;η;φ)⊂λ,α1(q,s;η;φ)⊂λ,α1+1(q,s;η;φ). (2.5)
Proof. First of all, we will show that
λ+1,α1(q,s;η;φ)⊂λ,α1(q,s;η;φ). (2.6)
Let f ∈λ+1,α1(q,s;η;φ) and set p(z)= 1
1−η
zHλ,q,s α1
f(z) Hλ,q,s
α1
f(z) −η
, (2.7)
wherepis analytic inUwithp(0)=1. Using (1.16) and (2.7), we have 1
1−η
zHλ+1,q,s α1
f(z) Hλ+1,q,s
α1
f(z) −η
=p(z) + z p(z)
(1−η)p(z) +λ−1 +η (z∈U). (2.8) Sinceλ >1 andφ∈ᏺ, we see that
Re(1−η)φ(z) +λ−1 +η>0 (z∈U). (2.9) ApplyingLemma 2.1to (2.8), it follows thatp≺φ, that is, f ∈λ,α1(q,s;η;φ).
To prove the second part, let f ∈λ,α1(q,s;η;φ) and put s(z)= 1
1−η
zHλ,q,sα1+ 1f(z) Hλ,q,sα1+ 1f(z) −η
, (2.10)
wheresis analytic function withs(0)=1. Then, by using the arguments similar to those detailed above with (1.15), it follows thats≺φinU, which implies that f ∈λ,α1+1(q,s;
η;φ). Therefore, we complete the proof ofTheorem 2.3.
Theorem 2.4. Letα1,λ >1 andφ∈ᏺ. Then,
λ+1,α1(q,s;η;φ)⊂λ,α1(q,s;η;φ)⊂λ,α1+1(q,s;η;φ). (2.11) Proof. Applying (1.18) andTheorem 2.3, we observe that
f(z)∈λ+1,α1(q,s;η;φ)⇐⇒Hλ+1,q,s α1
f(z)∈(η;φ)
⇐⇒Hλ+1,q,s
α1
z f(z)∈(η;φ)
⇐⇒z f(z)∈λ+1,α1(q,s;η;φ)
=⇒z f(z)∈λ,α1(q,s;η;φ)
⇐⇒zHλ,q,s α1
f(z)∈(η;φ)
⇐⇒ f(z)∈λ,α1(q,s;η;φ), f(z)∈λ,α1(q,s;η;φ)⇐⇒z f(z)∈λ,α1(q,s;η;φ)
=⇒z f(z)∈λ,α1+1(q,s;η;φ)
⇐⇒ f(z)∈λ,α1+1(q,s;η;φ),
(2.12)
which evidently provesTheorem 2.4.
Taking
φ(z)=1 +Az
1 +Bz (−1≤B < A≤1;z∈U) (2.13) in Theorems2.3and2.4, we have the following.
Corollary 2.5. Letα1,λ >1. Then,
λ+1,α1(q,s;η;A,B)⊂λ,α1(q,s;η;A,B)⊂λ,α1+1(q,s;η;A,B),
λ+1,α1(q,s;η;A,B)⊂λ,α1(q,s;η;A,B)⊂λ,α1+1(q,s;η;A,B). (2.14) Next, by using Lemma 2.2, we obtain the following inclusion relation for the class Ꮿλ,α1(q,s;η,β;φ,ψ).
Theorem 2.6. Letα1,λ >1 andφ,ψ∈ᏺ. Then,
Ꮿλ+1,α1(q,s;η,β;φ,ψ)⊂Ꮿλ,α1(q,s;η,β;φ,ψ)⊂Ꮿλ,α1+1(q,s;η,β;φ,ψ). (2.15) Proof. We begin by proving that
Ꮿλ+1,α1(q,s;η,β;φ,ψ)⊂Ꮿλ,α1(q,s;η,β;φ,ψ). (2.16) Let f ∈Ꮿλ+1,α1(q,s;η,β;φ,ψ). Then, from the definition ofᏯλ+1,α1(q,s;η,β;φ,ψ), there exists a functionr∈∗(η;φ) such that
1 1−β
zHλ+1,q,s
α1
f(z)
r(z) −β
≺ψ(z) (z∈U). (2.17)
Choose the functiongsuch thatHλ+1,q,s(α1)g(z)=r(z). Then,g∈λ+1,α1(q,s;η;φ) and 1
1−β
zHλ+1,q,sα1
f(z) Hλ+1,q,sα1
g(z) −β
≺ψ(z) (z∈U). (2.18)
Now let
p(z)= 1 1−β
zHλ,q,sα1
f(z) Hλ,q,sα1
g(z) −β
, (2.19)
wherepis analytic inUwithp(0)=1. Using (1.16), we have (1−β)z p(z)Hλ,q,s
α1
g(z) +(1−β)p(z) +βzHλ,q,s α1
g(z)
=λzHλ+1,q,sα1
f(z)−(λ−1)zHλ,q,sα1
f(z). (2.20)
Sinceg∈λ+1,α1(q,s;η;φ), byTheorem 2.3, we know thatg∈λ,α1(q,s;η;φ). Let q(z)= 1
1−η
zHλ,q,s
α1
g(z) Hλ,q,s
α1
g(z) −η
. (2.21)
Then, using (1.16) once again, we have λHλ+1,q,s
α1
g(z) Hλ,q,s
α1
g(z) =(1−η)q(z) +λ−1 +η. (2.22) From (2.20) and (2.22), we obtain
1 1−β
zHλ+1,q,sα1
f(z) Hλ+1,q,sα1
g(z) −β
=p(z) + z p(z)
(1−η)q(z) +λ−1 +η. (2.23) Sinceλ >1 andq≺φinU,
Re(1−η)q(z) +λ−1 +η>0 (z∈U). (2.24) Hence, applyingLemma 2.2, we can show thatp≺ψ, so that f ∈Ꮿλ,α1(q,s;η,β;φ,ψ).
For the second part, by using the arguments similar to those detailed above with (1.15), we obtain
Ꮿλ,α1(q,s;η,β;φ,ψ)⊂Ꮿλ,α1+1(q,s;η,β;φ,ψ). (2.25)
Therefore, we complete the proof ofTheorem 2.6.
3. Inclusion Properties Involving the Integral OperatorFc
In this section, we consider the generalized Libera integral operatorFc[13] (cf. [2,12]) defined by
Fc(f) :=Fc(f)(z)=c+ 1 zc
z
0tc−1f(t)dt (f ∈Ꮽ;c >−1). (3.1) We first prove the following.
Theorem 3.1. If f ∈λ,α1(q,s;η;φ), thenFc(f)∈λ,α1(q,s;η;φ) (c≥0).
Proof. Let f ∈λ,α1(q,s;η;φ) and set p(z)= 1
1−η
zHλ,q,sα1
Fc(f)(z) Hλ,q,sα1
Fc(f)(z) −η
, (3.2)
wherepis analytic inUwithp(0)=1. From (3.1), we have zHλ,q,sα1
Fc(f)(z)=(c+ 1)Hλ,q,sα1
f(z)−cHλ,q,sα1
Fc(f)(z). (3.3) Then, by using (3.2) and (3.3), we obtain
(c+ 1) Hλ,q,s α1
f(z) Hλ,q,s
α1
Fc(f)(z)=(1−η)p(z) +c+η. (3.4)
Taking the logarithmic differentiation on both sides of (3.4) and multiplying byz, we have
p(z) + z p(z)
(1−η)p(z) +c+η= 1 1−η
zHλ,q,s
α1
f(z) Hλ,q,s
α1
f(z) −η
(z∈U). (3.5) Hence, by virtue ofLemma 2.1, we conclude thatp≺φinU, which implies thatFc(f)∈
λ,α1(q,s;η;φ).
Next, we derive an inclusion property involvingFc, which is given by the following.
Theorem 3.2. If f ∈λ,α1(q,s;η;φ), thenFc(f)∈λ,α1(q,s;η;φ) (c≥0).
Proof. By applyingTheorem 3.1, it follows that
f(z)∈λ,α1(q,s;η;φ)⇐⇒z f(z)∈λ,α1(q,s;η;φ)
=⇒Fc
z f(z)∈λ,α1(q,s;η;φ)
⇐⇒zFc(f)(z)∈λ,α1(q,s;η;φ)
⇐⇒Fc(f)(z)∈λ,α1(q,s;η;φ),
(3.6)
which provesTheorem 3.2.
From Theorems3.1and3.2, we have the following.
Corollary 3.3. If f belongs to the class λ,α1(q,s;η;A,B) (orλ,α1(q,s;η;A,B)), then Fc(f) belongs to the classλ,α1(q,s;η;A,B) (orλ,α1(q,s;η;A,B)) (c≥0).
Finally, we prove.
Theorem 3.4. If f ∈Ꮿλ,α1(q,s;η,β;φ,ψ), thenFc(f)∈Ꮿλ,α1(q,s;η,β;φ,ψ) (c≥0).
Proof. Letf ∈Ꮿλ,α1(q,s;η,β;φ,ψ). Then, in view of the definition of the classᏯλ,α1(q,s;η, β;φ,ψ), there exists a functiong∈λ,α1(q,s;η;φ) such that
1 1−β
zHλ,q,s α1
f(z) Hλ,q,s
α1
g(z) −β
≺ψ(z) (z∈U). (3.7)
Thus, we set
p(z)= 1 1−β
zHλ,q,sα1
Fc(f)(z) Hλ,q,sα1
Fc(g)(z) −β
, (3.8)
wherepis analytic inUwithp(0)=1. Sinceg∈λ,α1(q,s;η;φ), we see fromTheorem 3.1 thatFc(g)∈λ,α1(q,s;η;φ). Using (3.3), we have
(1−β)p(z) +βHλ,q,s
α1
Fc(g)(z) +cHλ,q,s
α1
Fc(f)(z)=(c+ 1)Hλ,q,s
α1
f(z).
(3.9)
Then, by a simple calculation, we get (c+ 1)zHλ,q,s
α1 f(z) Hλ,q,s
α1
Fc(g)(z) =
(1−β)p(z) +β(1−η)q(z) +c+η+ (1−β)z p(z), (3.10) where
q(z)= 1 1−η
zHλ,q,s
α1
Fc(g)(z) Hλ,q,s
α1
Fc(g)(z) −η
. (3.11)
Hence, we have 1 1−β
zHλ,q,s
α1
f(z) Hλ,q,s
α1
g(z) −β
=p(z) + z p(z)
(1−η)q(z) +c+η. (3.12) The remaining part of the proof inTheorem 3.4is similar to that ofTheorem 2.6and so
we omit it.
Acknowledgment
This research was supported by Kyungsung University grants in 2007.
References
[1] S. Owa and H. M. Srivastava, “Analytic solution of a class of Briot-Bouquet differential equa- tions,” in Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa, Eds., pp.
252–259, World Scientific, River Edge, NJ, USA, 1992.
[2] J. H. Choi, M. Saigo, and H. M. Srivastava, “Some inclusion properties of a certain family of integral operators,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 432–
445, 2002.
[3] W. Ma and D. Minda, “An internal geometric characterization of strongly starlike functions,”
Annales Universitatis Mariae Curie-Skłodowska, Sectio A, vol. 45, pp. 89–97, 1991.
[4] J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hy- pergeometric function,” Applied Mathematics and Computation, vol. 103, no. 1, pp. 1–13, 1999.
[5] J. Dziok and H. M. Srivastava, “Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function,” Advanced Studies in Con- temporary Mathematics, vol. 5, no. 2, pp. 115–125, 2002.
[6] J. Dziok and H. M. Srivastava, “Certain subclasses of analytic functions associated with the gen- eralized hypergeometric function,” Integral Transforms and Special Functions, vol. 14, no. 1, pp.
7–18, 2003.
[7] J.-L. Liu and H. M. Srivastava, “Certain properties of the Dziok-Srivastava operator,” Applied Mathematics and Computation, vol. 159, no. 2, pp. 485–493, 2004.
[8] J.-L. Liu and H. M. Srivastava, “Classes of meromorphically multivalent functions associated with the generalized hypergeometric function,” Mathematical and Computer Modelling, vol. 39, no. 1, pp. 21–34, 2004.
[9] B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,” SIAM Jour- nal on Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984.
[10] Yu. E. Hohlov, “Operators and operations in the class of univalent functions,” Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, vol. 10, pp. 83–89, 1978.
[11] S. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109–115, 1975.
[12] R. M. Goel and N. S. Sohi, “A new criterion forp-valent functions,” Proceedings of the American Mathematical Society, vol. 78, no. 3, pp. 353–357, 1980.
[13] S. Owa and H. M. Srivastava, “Some applications of the generalized Libera integral operator,”
Proceedings of the Japan Academy, Series A, vol. 62, no. 4, pp. 125–128, 1986.
[14] J.-L. Liu, “The Noor integral and strongly starlike functions,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 441–447, 2001.
[15] K. I. Noor, “On new classes of integral operators,” Journal of Natural Geometry, vol. 16, no. 1-2, pp. 71–80, 1999.
[16] K. I. Noor and M. A. Noor, “On integral operators,” Journal of Mathematical Analysis and Appli- cations, vol. 238, no. 2, pp. 341–352, 1999.
[17] P. Eenigenburg, S. S. Miller, P. T. Mocanu, and M. O. Reade, “On a Briot-Bouquet differential subordination,” in General Inequalities, 3 (Oberwolfach, 1981), vol. 64 of Internationale Schriften- reihe zur Numerischen Mathematik, pp. 339–348, Birkh¨auser, Basel, Switzerland, 1983.
[18] S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michi- gan Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981.
Oh Sang Kwon: Department of Mathematics, Kyungsung University, Pusan 608-736, Korea Email address:[email protected]
Nak Eun Cho: Department of Applied Mathematics, Pukyong National University, Pusan 608-737, Korea
Email address:[email protected]
