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Volume 2007, Article ID 51079, 10 pagesdoi:10.1155/2007/51079 Research Article Inclusion Properties for Certain Subclasses of Analytic Functions Associated with the Dziok-Srivastava Oper

Volume 2007, Article ID 51079, 10 pages

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

LetᏭ denote the class of functions of the form

f (z) = z +

∞



k =2

which are analytic in the open unit diskU = { z : z ∈ Cand| z | < 1 } If f and g are analytic

in U, we say that f is subordinate to g, 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 functiong is univalent inU, the above subordination

is equivalent tof (0) = g(0) and f (U)⊂ g(U) For 0≤ η, β < 1, we denote by᏿∗(η), ᏷(η),

andᏯ(η,β) the subclasses of Ꮽ consisting of all analytic functions which are, respectively,

starlike of orderη, convex of order η, close-to-convex of order η, and type β inU 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 functions φ which are analytic and univalent in Uand for whichφ(U) is convex withφ(0) =1 and Re{ φ(z) } > 0 for z ∈ U

Making use of the principle of subordination between analytic functions, we introduce the subclasses᏿∗(η; φ), ᏷(η;φ), and Ꮿ(η,δ;φ,ψ) of the class Ꮽ for 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 functionsφ and ψ involved 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 ∗ g of two analytic functions

f (z) =

∞



k =0

a k z k, g(z) =

∞



k =0

be given (as usual) by

(f ∗ g)(z) =

∞



k =0

Making use of the Hadamard product (or convolution) given by (1.5), we now define the Dziok-Srivastava operator

H

α1, , α q;β1, , β s



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, , α q, β1, , β s



β j ∈ C\Z −

0;Z−

0 =0,−1,−2, .; j =1, , s

the generalized hypergeometric functionq F s(α1, , α q;β1, , β s;z) is given by

q F s



α1, , α q;β1, , β s;z

:=

∞



n =0



α1



n ···α q



n



β1



n ···β s



n

z n 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) =

⎧

⎨

⎩

ν(ν + 1) ···(ν + k −1) ifk ∈ N,ν ∈ C (1.9)

Corresponding to a functionᏲ(α1, , α q;β1, , β s;z), defined by

Ᏺα1, , α q;β1, , β s;z

:= z q F s



α1, , α q;β1, , β s;z

Dziok and Srivastava [5] considered a linear operator defined by the following Hadamard product (or convolution):

H

α1, , α q;β1, , β s

f (z) : =Ᏺα1, , α q;β1, , β s;z

We note that the linear operatorH(α1, , α q;β1, , β s) 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, , α q;β1, , β s;z), defined by (1.10), we intro-duce a functionᏲλ(α1, , α q;β1, , β s;z) given by

Ᏺα1, , α q;β1, , β s;z

∗Ᏺλ



α1, , α q;β1, , β s;z

(1− z) λ (λ > 0). (1.12) Analogous to H(α1, , α q;β1, , β s), we now define the linear operator H λ(α1, , α q;

β1, , β s) onᏭ as follows:

H λ

α1, , α q;β1, , β s

f (z) =Ᏺλ

α1, , α q;β1, , β s;z

∗ f (z)



α i,β j ∈ C\Z −

0; =1, , q; j =1, , s; λ > 0; z ∈ U; f ∈Ꮽ. (1.13)

For convenience, we write

H λ,q,s

α1



:= H λ

α1, , α q;β1, , β s

It is easily verified from the definition (1.13) that

z

H λ,q,s

α1+ 1

f (z)

= α1H λ,q,s

α1



f (z) −α1−1

H λ,q,s

α1+ 1

f (z), (1.15)

z

H λ,q,s



α1



f (z)

= λH λ+1,q,s



α1



f (z) −(λ −1)H λ,q,s



α1



In particular, the operatorH λ(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 operatorH λ,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 operatorH λ,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 : H λ,q,s



α1



f ∈᏿∗(η; φ) ,

᏷λ,α1(q, s; η; φ) : =f ∈ Ꮽ : H λ,q,s



α1



f ∈ ᏷(η;φ) ,

Ꮿλ,α1(q, s; η, β; φ, ψ) : =f ∈ Ꮽ : H λ,q,s



α1



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 classes᏿λ,α1(q, s; η; φ),

᏷λ,α1(q, s; η; φ), andᏯλ,α1(q, s; η, β; φ, ψ) associated with the operator H λ,q,s(α1) Some ap-plications involving integral operators are also considered

2 Inclusion Properties Involving the OperatorH λ,q,s(α1)

The following results will be required in our investigation

Lemma 2.1 [17] Let φ be convex univalent in Uwith φ(0) = 1 and Re { κφ(z) + ν } > 0

(κ, ν ∈ C ) If p is analytic inUwith p(0) = 1, then

p(z) + z p

(z)

implies

Lemma 2.2 [18] Let φ be convex univalent inUand let ω be analytic inUwith Re { ω(z) } ≥

0 If p is analytic inUand p(0) = φ(0), then

implies

Theorem 2.3 Let α1,λ > 1 and φ ∈ ᏺ Then,

᏿λ+1,α(q, s; η; φ) ⊂᏿λ,α(q, s; η; φ) ⊂᏿λ,α+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



z

H λ,q,s

α1 

f (z)

H λ,q,s

α1 

f (z) − η



wherep is analytic inUwithp(0) =1 Using (1.16) and (2.7), we have

1

1− η



z

H λ+1,q,s

α1



f (z)

H λ+1,q,s

α1



f (z) − η



= p(z) + z p (z)

(1− η)p(z) + λ −1 +η ( ∈ U). (2.8) Sinceλ > 1 and φ ∈ᏺ, we see that

Re

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



z

H λ,q,s

α1+ 1

f (z)

H λ,q,s

α1+ 1

f (z) − η



wheres is analytic function with s(0) =1 Then, by using the arguments similar to those detailed above with (1.15), it follows thats ≺ φ inU, which implies that f ∈᏿λ,α1 +1(q, s;

Theorem 2.4 Let α1,λ > 1 and φ ∈ 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) ⇐⇒ H λ+1,q,s

α1 

f (z) ∈ ᏷(η;φ)

⇐⇒ H λ+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)

⇐⇒ z

H λ,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)

φ(z) =1 +Az

in Theorems2.3and2.4, we have the following

Corollary 2.5 Let α1,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 Let α1,λ > 1 and φ, 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



z

H λ+1,q,s



α1



f (z)



≺ ψ(z) ( ∈ U). (2.17) Choose the functiong such that H λ+1,q,s(α1)g(z) = r(z) Then, g ∈᏿λ+1,α1(q, s; η; φ) and

1

1− β



z

H λ+1,q,s

α1



f (z)

H λ+1,q,s

α1



g(z) − β



≺ ψ(z) ( ∈ U). (2.18) Now let

p(z) = 1

1− β



z

H λ,q,s

α1



f (z)

H λ,q,s

α1



g(z) − β



wherep is analytic inUwithp(0) =1 Using (1.16), we have

(1− β)z p (z)H λ,q,s

α1 

g(z) +

(1− β)p(z) + β

z

H λ,q,s

α1 

g(z)

= λz

H λ+1,q,s

α1



f (z)

−(λ −1)z

H λ,q,s

α1



f (z)

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



z

H λ,q,s



α1



g(z)

H λ,q,s



α1



g(z) − η



Then, using (1.16) once again, we have

λ H λ+1,q,s



α1



g(z)

H λ,q,s



α1



From (2.20) and (2.22), we obtain

1

1− β



z

H λ+1,q,s

α1



f (z)

H λ+1,q,s

α1



g(z) − β



= p(z) + z p (z)

(1− η)q(z) + λ −1 +η . (2.23)

Sinceλ > 1 and q ≺ φ inU,

Re

(1− η)q(z) + λ −1 +η > 0 ( ∈ 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 OperatorF c

In this section, we consider the generalized Libera integral operatorF c[13] (cf [2,12]) defined by

F c(f ) : = F c(f )(z) = c + 1 z c

z

0t c −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), then F c(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



z

H λ,q,s

α1



F c(f )(z)

H λ,q,s

α1



F c(f )(z) − η



wherep is analytic inUwithp(0) =1 From (3.1), we have

z

H λ,q,s

α1



F c(f )(z)

=(c + 1)H λ,q,s

α1



f (z) − cH λ,q,s

α1



F c(f )(z). (3.3) Then, by using (3.2) and (3.3), we obtain

(c + 1) H λ,q,s



α1 

f (z)

H λ,q,s

α1 

F c(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

(z)

(1− η)p(z) + c + η = 1

1− η



z

H λ,q,s



α1



f (z)

H λ,q,s



α1



f (z) − η



( ∈ U). (3.5)

Hence, by virtue ofLemma 2.1, we conclude thatp ≺ φ inU, which implies thatF c(f ) ∈

Next, we derive an inclusion property involvingF c, which is given by the following

Theorem 3.2 If f ∈᏷λ,α1(q, s; η; φ), then F c(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)

=⇒ F c

z f (z)

∈᏿λ,α1(q, s; η; φ)

⇐⇒ z

F c(f )(z)

∈᏿λ,α1(q, s; η; φ)

⇐⇒ F c(f )(z) ∈᏷λ,α1(q, s; η; φ),

(3.6)

From Theorems3.1and3.2, we have the following

Corollary 3.3 If f belongs to the class ᏿λ,α1(q, s; η; A, B) (or ᏷λ,α1(q, s; η; A, B)), then

F c(f ) belongs to the class᏿λ,α1(q, s; η; A, B) (or᏷λ,α1(q, s; η; A, B)) (c ≥0).

Finally, we prove

Theorem 3.4 If f ∈Ꮿλ,α1(q, s; η, β; φ, ψ), then F c(f ) ∈Ꮿλ,α1(q, s; η, β; φ, ψ) (c ≥ 0).

Proof Let f ∈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 function g ∈᏿λ,α1(q, s; η; φ) such that

1

1− β



z

H λ,q,s

α1



f (z)

H λ,q,s

α1



g(z) − β



≺ ψ(z) ( ∈ U). (3.7) Thus, we set

p(z) = 1

1− β



z

H λ,q,s

α1



F c(f )(z)

H λ,q,s

α1



F c(g)(z) − β



wherep is analytic inUwithp(0) =1 Sinceg ∈᏿λ,α1(q, s; η; φ), we see fromTheorem 3.1 thatF c(g) ∈᏿λ,α1(q, s; η; φ) Using (3.3), we have



(1− β)p(z) + β

H λ,q,s



α1



F c(g)(z) + cH λ,q,s



α1



F c(f )(z) =(c + 1)H λ,q,s



α1



f (z).

(3.9)

Then, by a simple calculation, we get

(c + 1) z



H λ,q,s

α1 

f (z)

H λ,q,s

α1 

F c(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



z

H λ,q,s



α1



F c(g)(z)

H λ,q,s



α1



F c(g)(z) − η



Hence, we have

1

1− β



z

H λ,q,s



α1



f (z)

H λ,q,s



α1



g(z) − β



(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

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 AJour-nalysis, 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:oskwon@ks.ac.kr

Nak Eun Cho: Department of Applied Mathematics, Pukyong National University,

Pusan 608-737, Korea

Email address:necho@pknu.ac.kr

3 Inclusion Properties Involving the Integral OperatorF c

In this section, we consider the generalized... “Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function,” Advanced Studies in

Con-temporary Mathematics,... H M Srivastava, ? ?Certain subclasses of analytic functions associated with the

gen-eralized hypergeometric function,” Integral Transforms and Special Functions, vol 14,