Mathematical and Computer Modelling 54 (2011) 3189–3196

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Mathematical and Computer Modelling

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Subclasses of spirallike multivalent functionsNantu Sarkar a, Pranay Goswami b,∗, Teodor Bulboacă c

a Department of Mathematics, Jadavpur University, Kolkata-700032, Indiab Department of Mathematics, AMITY University Rajasthan, Jaipur-302002, Indiac Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania

a r t i c l e i n f o

Article history:Received 24 February 2011Received in revised form 5 August 2011Accepted 8 August 2011

Keywords:Multivalent functionsSubordinationHadamard productDziok–Srivastava operator

a b s t r a c t

We studied convolution properties of spirallike, starlike and convex functions, and somespecial cases of the main results are also pointed out. Further, we also obtained inclusionand convolution properties for some new subclasses on p-valent functions defined by usingthe Dziok–Srivastava operator.

Crown Copyright© 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction and preliminaries

Let Ap denote the class of functions of the form

f (z) = zp +

∞−k=p+1

akzk, (p ∈ N = {1, 2, . . .}) , (1.1)

which are analytic and p-valent in the open unit disc U = {z ∈ C : |z| < 1}.If f and g are analytic functions in U, we say that f is subordinate to g , written f (z) ≺ g(z), if there exists a Schwarz

functionw, which (by definition) is analytic in U withw(0) = 0, and |w(z)| < 1 for all z ∈ U, such that f (z) = g(w(z)), forall z ∈ U. Furthermore, if the function g is univalent in U, then we have the following equivalence:

f (z) ≺ g(z) ⇔ f (0) = g(0) and f (U) ⊂ g(U).

We will define the following two subclasses of a multivalent function:

Definition 1.1. Let λ ∈ R with |λ| < π2 , let p ∈ N, and let φ be a univalent function in the unit disc U with φ(0) = 1, such

that

Reφ(z) > 1 −1p, z ∈ U. (1.2)

We define the classes Sλp (φ) and Cλp (φ) by

Sλp (φ) =

f ∈ Ap : eiλ

zf ′(z)f (z)

≺ p (φ(z) cos λ+ i sin λ)

∗ Corresponding author.E-mail addresses: nantu.math@gmail.com (N. Sarkar), pranaygoswami83@gmail.com (P. Goswami), bulboaca@math.ubbcluj.ro (T. Bulboacă).

0895-7177/$ – see front matter Crown Copyright© 2011 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2011.08.015

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3190 N. Sarkar et al. / Mathematical and Computer Modelling 54 (2011) 3189–3196

and

Cλp (φ) =

f ∈ Ap : eiλ

[1 +

zf ′′(z)f ′(z)

]≺ p (φ(z) cos λ+ i sin λ)

.

Remarks 1.1. 1. It is easy to see that

f ∈ Cλp (φ) ⇔zf ′(z)

p∈ Sλp (φ). (1.3)

2. For φ(z) =1+Az1+Bz (−1 ≤ B < A ≤ 1), we set Sλp [A, B] := Sλp (φ) and Cλp [A, B] := Cλp (φ).

3. For λ = 0 we write

S0p (φ) =: S∗

p (φ) and C0p (φ) =: Cp(φ),

S0p [A, B] =: S∗

p [A, B] and C0p [A, B] =: Cp[A, B].

Moreover, for the special case p = 1, A = 1 and B = −1, the class C1[1,−1] represents the class of convex (univalent)and normalized functions (with the conditions f (0) = f ′(0)− 1 = 0) in U.

4. For 0 ≤ α < 1, the classes Sλp [1 − 2α,−1] and Cλp [1 − 2α,−1] reduce to the classes Sλp (α) and Cλp (α) of multivalentλ-spirallike functions of order α, and multivalent λ-spirallike convex functions of order α in U respectively.

Lemma 1.1. 1. If f ∈ Sλp (φ), then the function F(z) =f (z)zp−1 is a λ-spirallike (univalent) function in U.

2. Consequently, for all γ ∈ C∗= C \ {0}, the multivalued function

f (z)zp

γhas an analytic branch in U, with

f (z)zp

γ z=0

= 1.

Proof. Since f ∈ Ap we have F ∈ A1. Differentiating the definition formula of F we get zF ′(z)F(z) =

zf ′(z)f (z) − p+ 1, and using the

fact that f ∈ Sλp (φ) it follows

eiλzF ′(z)F(z)

≺ [pφ(z)+ 1 − p] cos λ+ i sin λ =: H(z). (1.4)

Using the assumption (1.2) we deduce

ReH(z) = cos λ · Re [pφ(z)+ 1 − p] > 0, z ∈ U,

whenever λ ∈ R with |λ| < π2 . Since the function H is univalent, the subordination (1.4) yields that Re

eiλ zF ′(z)

F(z)

> 0,

z ∈ U, and this last inequality shows that F is a λ-spirallike function, hence it is univalent in U. It follows F(z)z = 0 for all

z ∈ U, i.e. f (z)zp = 0, z ∈ U, that implies the second part of our lemma. �

Using the above lemma we will introduce the following two subclasses of Ap:

Definition 1.2. Let γ ∈ C∗ be an arbitrary number.

1. A function g ∈ Ap is said to be in the class Sλ,γp (φ), if there exists a function f ∈ Sλp (φ) such that g(z) = zp

f (z)zp

γ,

wheref (z)zp

γ z=0

= 1.

2. A function g ∈ Ap is said to be in the class Cλ,γp (φ), if G ∈ S

λ,γp (φ), where G(z) =

zg ′(z)p .

Remarks 1.2. 1. Note that for the special case γ = 1 the classes Sλ,γp (φ) and C

λ,γp (φ) coincide with Sλp (φ) and Cλp (φ)

respectively.2. Also, for p = 1 and φ(z) =

1+Az1+Bz (−1 ≤ B < A ≤ 1)we have S

λ,γ

1 [A, B] =: Sλ,γ [A, B], which was studied by Ahuja [1].

For the functions f ∈ Ap, given by (1.1) and g ∈ Ap of the form

g(z) = zp +

∞−k=p+1

bkzk,

the Hadamard (or convolution) product of f and g is defined by

(f ∗ g)(z) = zp +

∞−k=p+1

akbkzk.

For ai ∈ C (i = 1, 2, . . . , q) and bj ∈ C\Z0−(j = 1, 2, . . . , s), whereZ0

−= {0,−1,−2,−3, . . .}, we define the generalized

hypergeometric function qFs(a1, . . . , aq; b1, . . . , bs; z) (see, for example, [2]) by the following infinite series

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qFs(a1, . . . , aq; b1, . . . , bs; z) =

∞−k=0

(a1)k · · · (aq)k(b1)k · · · (bs)k

zk

k!, z ∈ U, (1.5)

where q, s ∈ N0 = N ∪ {0} with q ≤ s + 1, and (λ)r is the Pochhammer symbol defined in terms of the Gamma function by

(λ)r =Γ (λ+ r)Γ (λ)

=

1, for r = 0, λ ∈ C \ {0},λ(λ+ 1) · · · (λ+ r − 1), for r ∈ N, λ ∈ C.

Corresponding to a function hp(a1, . . . , aq; b1, . . . , bs; z) given by

hp(a1, . . . , aq; b1, . . . , bs; z) = zp ·q Fs(a1, . . . , ai, . . . , aq; b1, . . . , bs; z).

Dziok and Srivastava (see [3,4]) considered the linear operator

Hp(a1, . . . , aq; b1, . . . , bs) : Ap → Ap

defined by the following Hadamard product:

Hp(a1, . . . , aq; b1, . . . , bs)f (z) = hp(a1, . . . , aq; b1, . . . , bs; z) ∗ f (z).

Thus, for a function f of the form (1.1) we have

Hp(a1, . . . , aq; b1, . . . , bs)f (z) = zp +

∞−k=p+1

Γk−p[a1; b1]akzk,

where

Γk−p[a1; b1] :=1

(k − p)!(a1)k−p · · · (aq)k−p

(b1)k−p · · · (bs)k−p,

and for convenience, we write

Hp;q,s(a1) := Hp(a1, . . . , aq; b1, . . . , bs).

For the various properties and special cases of this operator we may refer to [2,5,3,4,6,7], and many others.

Definition 1.3. Let λ ∈ R with |λ| < π2 , let p ∈ N, and let φ be an univalent function in the unit disc U with φ(0) = 1, that

satisfies (1.2). For q, s ∈ N0, with q ≤ s + 1, we define the following two subclasses of analytic functions:

Sλ,γ

p;q,s[a1;φ] := Sλ,γp [a1, . . . , aq; b1, . . . , bs;φ]

=

f ∈ Ap : Hp;q,s(a1)f ∈ Sλ,γp (φ)

and

Cλ,γ

p;q,s[a1;φ] := Cλ,γp [a1, . . . , aq; b1, . . . , bs;φ]

=

f ∈ Ap : Hp;q,s(a1)f ∈ Cλ,γp (φ)

.

In particular, for γ = 1 we denote Sλp;q,s[a1;φ] := Sλ,1p;q,s[a1;φ] and Cλp;q,s[a1;φ] := Cλ,1p;q,s[a1;φ].

In the present paper we studied several convolution and inclusion properties of the subclasses introduced in this section,that extend some known results obtained by Ahuja [1], and Aouf and Soudy [5].

2. Convolution properties

Theorem 2.1. Let f ∈ Ap, let γ ∈ C∗= C \ {0} be an arbitrary number, and let g(z) = zp

f (z)zp

γ, where the power function

is considered to the main branch, i.e.

f (z)zp

γ z=0

= 1. Then, g ∈ Sλp (φ) if and only if

1zp

[f (z) ∗

zp − Czp+1

(1 − z)2

]= 0, z ∈ U, (2.1)

and for all

C = Cγ (x) =[1 − φ(x)] p cos λ− γ eiλ

[1 − φ(x)] p cos λ, |x| = 1, (2.2)

and also for C = 1, if γ ∈ C∗\ {1}.

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Proof. For any f ∈ Ap, we may easily deduce that

f (z) = f (z) ∗zp

1 − zand zf ′(z) = f (z) ∗

[zp+1

(1 − z)2+

pzp

1 − z

]. (2.3)

1. Suppose that g ∈ Sλp (φ), where g(z) = zpf (z)zp

γ. Since

γ eiλzf ′(z)f (z)

+ eiλp(1 − γ ) = eiλzg ′(z)g(z)

,

it follows that g ∈ Sλp (φ) if and only if

γ eiλzf ′(z)f (z)

+ eiλp(1 − γ ) ≺ p (φ(z) cos λ+ i sin λ) . (2.4)

Using the fact that the function from the right-hand side of the above subordination is univalent, from (2.4) it followsthat

γ eiλzf ′(z)f (z)

+ eiλp(1 − γ )− pφ(x) cos λ− ip sin λ = 0, ∀|x| = 1, ∀ z ∈ U. (2.5)

From the convolution relations (2.3), a simple computation shows that (2.5) is equivalent to (2.1), where C is givenby (2.2).

If γ ∈ C∗\ {1}, since g(z) = zp

f (z)zp

γis analytic in U it follows that f (z)

zp = 0 for all z ∈ U, and according to (2.3) thisis equivalent to

1zp

[f (z) ∗

zp

1 − z

]= 0, z ∈ U, (2.6)

that represents (2.1) for C = 1.2. Reversely, if γ ∈ C∗

\ {1}, since the assumption (2.1) holds for C = 1, it follows that (2.6) holds, and according to (2.3)

this shows that f (z)zp = 0 for all z ∈ U. Thus, the power function

f (z)zp

γhas uniform branches in the unit disc U, hence

the function g is well defined.Since it was shown in the first part of the proof that the assumption (2.1) is equivalent to (2.5), we obtain that

γ eiλzf ′(z)f (z)

+ eiλp(1 − γ ) = p (φ(x) cos λ+ i sin λ) , ∀|x| = 1, ∀ z ∈ U. (2.7)

Denoting

ϕ(z) = γ eiλzf ′(z)f (z)

+ eiλp(1 − γ ), ψ(z) = p (φ(z) cos λ+ i sin λ) ,

the relation (2.7) shows that ϕ(U)∩ψ(∂U) = ∅, and thus the simply-connected domain ϕ(U) is included in a connectedcomponent of C \ ψ(∂U). From here, and using the fact that ϕ(0) = ψ(0) together with the univalence of the functionψ , it follows that ϕ(z) ≺ ψ(z), which represents in fact the subordination (2.4), i.e. g ∈ Sλp (φ). �

Theorem 2.2. If g ∈ Ap, then g ∈ Sλ,γp (φ) if and only if

1zp

[g(z) ∗

zp − Dzp+1

(1 − z)2

]= 0, z ∈ U, (2.8)

for all

D = Dγ (x) =[1 − φ(x)] pγ cos λ− eiλ

[1 − φ(x)] pγ cos λ, |x| = 1. (2.9)

Proof. From Definition 1.2, g ∈ Sλ,γp (φ) if there exists a function f ∈ Sλp (φ) such that g(z) = zp

f (z)zp

γ, where

f (z)zp

γ z=0

= 1.Since

γzf ′(z)f (z)

+ p(1 − γ ) =zg ′(z)g(z)

,

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and using the fact that g ∈ Sλ,γp (φ) if and only if f ∈ Sλp (φ), it follows that g ∈ S

λ,γp (φ) is equivalent to

eiλzg ′(z)g(z)

− eiλp(1 − γ ) = γ eiλzf ′(z)f (z)

≺ pγ (φ(z) cos λ+ i sin λ) .

Using the same arguments as in the proof of Theorem 2.1, we find that the above subordination holds if and only if therelation (2.8) is satisfied whenever D has the form (2.9). �

Remark 2.1. Settingφ(z) =1+Az1+Bz (−1 ≤ B < A ≤ 1) andwriting−x as x in Theorems 2.1 and 2.2,we obtain the convolution

results obtained by Ahuja [1].

For γ = 1 we have g(z) = f (z), where g is defined like in Theorem 2.1, and C1(x) ≡ D1(x), hence both of the results ofTheorems 2.1 and 2.2 coincide with the following one:

Corollary 2.1. If f ∈ Ap, then f ∈ Sλp (φ) if and only if (2.1) holds for all

C = C1(x) =[1 − φ(x)] p cos λ− eiλ

[1 − φ(x)] p cos λ, |x| = 1. (2.10)

Theorem 2.3. If f ∈ Ap, then f ∈ Cλp (φ) if and only if

1zp

[f (z) ∗

pzp − [p + C(p + 1)− 2] zp+1+ C(p − 1)zp+2

(1 − z)3

]= 0, z ∈ U, (2.11)

for all C , where C = C1(x) is given by (2.10).

Proof. If we let G(z) =zp−Czp+1

(1−z)2, then

zG′(z) =

[pzp − [p + C(p + 1)− 2] zp+1

+ C(p − 1)zp+2

(1 − z)3

].

Using the duality relation (1.3), we have

f ∈ Cλp (φ) ⇔zf ′(z)

p∈ Sλp (φ),

and according to Corollary 2.1 we deduce that f ∈ Cλp (φ) if and only if

1zp

[zf ′(z)

p∗ G(z)

]= 0, z ∈ U, (2.12)

for all C given by (2.2). Since the identity zf ′(z) ∗ g(z) = f (z) ∗ zg ′(z) holds for all f , g ∈ Ap, in particular we havezf ′(z) ∗ G(z) = f (z) ∗ zG′(z), hence the condition (2.12) reduces to

1zp

f (z) ∗ zG′(z)

= 0, z ∈ U.

This last relation is equivalent to (2.11), which gives the required result. �

Theorem 2.4. If f ∈ Ap, then f ∈ Cλ,γp (φ) if and only if

1zp

[f (z) ∗

pzp − [p + D(p + 1)− 2] zp+1+ D(p − 1)zp+2

(1 − z)3

]= 0, z ∈ U, (2.13)

for all D, where D = Dγ (x) is given by (2.9).

Proof. Letting φ(z) =zp−Dzp+1

(1−z)2, we have

zH ′(z) =

[pzp − [p + D(p + 1)− 2] zp+1

+ D(p − 1)zp+2

(1 − z)3

].

From Definition 1.2 we have that f ∈ Cλ,γp (φ) if and only if zf ′(z)

p ∈ Sλ,γp (φ), and according to Theorem 2.2 we deduce

that f ∈ Cλ,γp (φ) if and only if

1zp

[zf ′(z)

p∗ φ(z)

]= 0, z ∈ U,

for all D given by (2.9). Now, applying exactly the same method used in the proof of Theorem 2.3 we easily obtain thedesired result. �

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Remark 2.2. For γ = 1, since C1(x) ≡ D1(x), the results of Theorems 2.3 and 2.4 coincide.

Theorem 2.5. If f ∈ Ap is of the form (1.1), then f ∈ Sλ,γ

p;q,s[a1;φ] if and only if

1 +

∞−k=p+1

[[1 − φ(x)] pγ cos λ+ (k − p)eiλ

[1 − φ(x)] pγ cos λ

]Γk−p[a1; b1]akzk−p

= 0, z ∈ U, (2.14)

for all |x| = 1.

Proof. From Theorem 2.2 we find that f ∈ Sλ,γ

p;q,s[a1;φ] if and only if

1zp

[Hp;q,s(a1)f (z) ∗

zp − Dzp+1

(1 − z)2

]= 0, z ∈ U, (2.15)

for all |x| = 1, where D is given by (2.9). Since

zp

(1 − z)2= zp +

∞−k=p+1

(k − p + 1)zk,zp+1

(1 − z)2=

∞−k=p+1

(k − p)zk, (2.16)

after some simple computations in Eq. (2.15), with the help of (2.16) we get

1zp

zp +

∞−k=p+1

[1 + (1 − D)(k − p)]Γk−p[a1; b1]akzk

= 0, z ∈ U,

for all |x| = 1, which simplifies to (2.14), and the proof of the theorem is complete. �

Theorem 2.6. If f ∈ Ap is of the form (1.1), then f ∈ Cλ,γ

p;q,s[a1;φ] if and only if

1 +

∞−k=p+1

kp

[[1 − φ(x)] pγ cos λ+ (k − p)eiλ

[1 − φ(x)] pγ cos λ

]Γk−p[a1; b1]akzk−p

= 0, z ∈ U, (2.17)

for all |x| = 1.

Proof. From Theorem 2.4 we have that f ∈ Cλ,γ

p;q,s[a1;φ] if and only if

1zp

[Hp;q,s(a1)f (z) ∗

pzp − [p + D(p + 1)− 2] zp+1+ D(p − 1)zp+2

(1 − z)3

]= 0, z ∈ U, (2.18)

for all |x| = 1, where D is given by (2.9). Now, it can be easily shown that

zp

(1 − z)3= zp +

∞−k=p+1

(k − p + 1)(k − p + 2)2

zk, z ∈ U,

zp+1

(1 − z)3=

∞−k=p+1

(k − p)(k − p + 1)2

zk, z ∈ U,

zp+2

(1 − z)3=

∞−k=p+1

(k − p − 1)(k − p)2

zk z ∈ U.

Using these identities in (2.18), after some simple computations we get

1zp

pzp +

∞−k=p+1

k[1 + (1 − D)(k − p)]Γk−p[a1; b1]akzk

= 0, z ∈ U,

for all |x| = 1, which simplifies to (2.17). �

3. Inclusion properties

In this sectionwewill discuss some inclusion relations for the classes Sλp;q,s[a1;φ] andCλp;q,s[a1;φ]. To prove these resultswe shall require the following lemma:

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Lemma 3.1 ([8]). Let φ be convex (univalent) inU, with Re[βφ(z)+γ ] > 0 for all z ∈ U. If q is analytic inU, with q(0) = φ(0),then

q(z)+zq′(z)

βq(z)+ γ≺ φ(z) ⇒ q(z) ≺ φ(z).

Theorem 3.1. Suppose that the function φ is convex (univalent) in U and satisfies the inequality

p Reφ(z) > (p − Re a1) sec2 λ− p [Imφ(z)+ tan λ] tan λ, z ∈ U. (3.1)

If f ∈ Sλp;q,s[a1 + 1;φ] such that Hp;q,s(a1)f (z) = 0 for all z ∈ U \ {0}, then f ∈ Sλp;q,s[a1;φ].

Proof. Suppose that f ∈ Sλp;q,s[a1 + 1;φ], and let us define

q(z) =1

cos λ

eiλ

zHp;q,s(a1)f (z)

′

pHp;q,s(a1)f (z)− i sin λ

. (3.2)

Then q is analytic in U, and using the well-known relation

zHp;q,s(a1)f (z)

′= a1Hp;q,s(a1 + 1)f (z)− (a1 − p)Hp;q,s(a1)f (z),

from (3.2) we obtain

q(z) cos λ+a1 − p

peiλ + i sin λ = eiλ

a1Hp;q,s(a1 + 1)f (z)pHp;q,s(a1)f (z)

. (3.3)

Differentiating logarithmically (3.3) and then using (3.2), we deduce that

q(z)+zq′(z)

p [q(z) cos λ+ i sin λ] e−iλ + a1 − p≺ φ(z). (3.4)

Since the inequality Rep [φ(z) cos λ+ i sin λ] e−iλ

+ a1 − p> 0, z ∈ U, is equivalent to (3.1), according to Lemma 3.1

the subordination (3.4) implies q(z) ≺ φ(z), which proves that f ∈ Sλp;q,s[a1;φ]. �

From the duality formula (1.3) we have

Hp;q,s(a1)f ∈ Cλp (φ) ⇔zHp;q,s(a1)f (z)

′

p∈ Sλp (φ),

and using the fact that

Hp;q,s(a1)zf ′(z)

p

=

zHp;q,s(a1)f (z)

′

p,

in the same way we may prove the following inclusion:

Theorem 3.2. Suppose that the function φ satisfies the inequality (3.1). If f ∈ Cλp;q,s[a1 +1;φ] such that Hp;q,s(a1)

zf ′(z)p

= 0

for all z ∈ U \ {0}, then f ∈ Cλp;q,s[a1;φ].

Acknowledgments

The second author (Dr. Pranay Goswami) is grateful to Prof. S.P. Goyal, Emeritus Scientist, Department of Mathematics,University of Rajasthan, Jaipur, India, for his guidance and constant encouragement. The authors are also thankful to refereesfor their helpful comments.

References

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1–13.[4] J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec.

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