Open Access

Inclusion results for certain subclasses of p-valent meromorphic functions associated with a new operator

Journal of Inequalities and Applications20122012:169

https://doi.org/10.1186/1029-242X-2012-169

Received: 21 May 2012

Accepted: 18 July 2012

Published: 31 July 2012

Abstract

In this paper, we introduce new subclasses of p-valent starlike, p-valent convex, p-valent close-to-convex, and p-valent quasi-convex meromorphic functions and investigate some inclusion properties of these subclasses and investigate various inclusion properties and integral-preserving properties for the p-valent meromorphic function classes.

MSC:30C45.

Keywords

p-valent meromorphic functions Hadamard product inclusion properties

1 Introduction

Let Σ p denote the class of functions of the form
f ( z ) = z p + k = 1 a k p z k p ( p N = { 1 , 2 , } ) ,
(1.1)
which are analytic and p-valent in the punctured unit disc U = { z : z C  and  0 < | z | < 1 } . If f ( z ) and g ( z ) are analytic in U = U { 0 } , we say that f ( z ) is subordinate to g ( z ) , written f g or f ( z ) g ( z ) ( z U ), if there exists a Schwarz function w ( z ) in U with w ( 0 ) = 0 and | w ( z ) | < 1 , such that f ( z ) = g ( w ( z ) ) ( z U ). Furthermore, if g ( z ) is univalent in U, then the following equivalence relationship holds true (see [3] and [8]):
f ( z ) g ( z ) f ( 0 ) = g ( 0 ) and f ( U ) g ( U ) .
For functions f ( z ) Σ p , given by (1.1) and g ( z ) Σ p defined by
g ( z ) = z p + k = 1 b k p z k p ( p N ) ,
(1.2)
the Hadamard product (or convolution) of f ( z ) and g ( z ) is given by
( f g ) ( z ) = z p + k = 1 a k p b k p z k p = ( g f ) ( z ) .
(1.3)
Aqlan et al.[1] defined the operator Q β , p α : Σ p Σ p by:
Q β , p α f ( z ) = { z p + Γ ( α + β ) Γ ( β ) k = 1 Γ ( k + β ) Γ ( k + β + α ) a k p z k p ( α > 0 ; β > 1 ; p N ; f Σ p ) , f ( z ) ( α = 0 ; β > 1 ; p N ; f Σ p ) .
(1.4)

Now, we define the operator H p , β , μ α : Σ p Σ p as follows:

First, put
G β , p α ( z ) = z p + Γ ( α + β ) Γ ( β ) k = 1 Γ ( k + β ) Γ ( k + β + α ) z k p ( p N )
(1.5)
and let G β , p , μ α be defined by
G β , p α ( z ) G β , p , μ α ( z ) = 1 z p ( 1 z ) μ ( μ > 0 ; p N ) .
(1.6)
Then
H p , β , μ α f ( z ) = G β , p α ( z ) f ( z ) ( f Σ p ) .
(1.7)
Using (1.5)-(1.7), we have
H p , β , μ α f ( z ) = z p + Γ ( β ) Γ ( α + β ) k = 1 Γ ( k + β + α ) ( μ ) k Γ ( k + β ) ( 1 ) k a k p z k p ,
(1.8)
where f Σ p is in the form (1.1) and ( ν ) n denotes the Pochhammer symbol given by
( ν ) n = Γ ( ν + n ) Γ ( ν ) = { 1 ( n = 0 ) , ν ( ν + 1 ) ( ν + n 1 ) ( n N ) .
It is readily verified from (1.8) that
z ( H p , β , μ α f ( z ) ) = ( α + β ) H p , β , μ α + 1 f ( z ) ( α + β + p ) H p , β , μ α f ( z )
(1.9)
and
z ( H p , β , μ α f ( z ) ) = μ H p , β , μ + 1 α f ( z ) ( μ + p ) H p , β , μ α f ( z ) .
(1.10)
It is noticed that, putting μ = 1 in (1.8), we obtain the operator
H p , β , 1 α f ( z ) = z p + Γ ( β ) Γ ( α + β ) k = 1 Γ ( k + α + β ) Γ ( k + β ) a k p z k p .
(1.11)

Let M be the class of analytic functions h ( z ) with h ( 0 ) = 1 , which are convex and univalent in U and satisfy Re { h ( z ) } > 0 ( z U ).

For 0 η , γ < p , we denote by Σ p S ( η ) , Σ p K ( η ) , Σ p C ( η , γ ) , and Σ p C ( η , γ ) , the subclasses of Σ p consisting of all p-valent meromorphic functions which are, respectively, starlike of order η, convex of order η, close-to-convex functions of order γ and type η, and quasi-convex functions of order γ and type η in U.

Making use of the principle of subordination between analytic functions, we introduce the subclasses Σ p S ( η ; ϕ ) , Σ p C ( η ; ϕ ) , Σ p K ( η , γ ; ϕ , ψ ) , and Σ p K ( η , γ ; ϕ , ψ ) ( 0 η , γ < p and ϕ , ψ M ) of the class Σ p which are defined by:
and

From these definitions, we can obtain some well-known subclasses of Σ p by special choices of the functions ϕ and ψ as well as special choices of η, γ, and p (see [2, 5], and [10]).

Now, by using the linear operator H p , β , μ α ( α 0 , μ > 0 , β > 1 ; p N ) and for ϕ , ψ M , 0 η , γ < p , we define new subclasses of meromorphic functions of Σ p by:
and
Σ p C β , μ α ( η , γ ; ϕ , ψ ) = { f Σ p : H p , β , μ α f Σ p C ( η , γ ; ϕ , ψ ) } .
We also note that
f ( z ) Σ p K β , μ α ( η ; ϕ ) z f ( z ) p Σ p S β , μ α ( η ; ϕ ) ,
(1.12)
and
f ( z ) Σ p C β , μ α ( η , γ ; ϕ , ψ ) z f ( z ) p Σ p C β , μ α ( η , γ ; ϕ , ψ ) .
(1.13)
In particular, we set
Σ p S β , μ α ( η ; 1 + A z 1 + B z ) = Σ p S β , μ α ( η ; A , B ) ( 1 < B < A 1 )
and
Σ p K β , μ α ( η ; 1 + A z 1 + B z ) = Σ p K β , μ α ( η ; A , B ) ( 1 < B < A 1 ) .

In this paper, we investigate several inclusion properties of the classes Σ p S β , μ α ( η ; ϕ ) , Σ p K β , μ α ( η ; ϕ ) , Σ p C β , μ α ( η , γ ; ϕ , ψ ) , and Σ p C β , μ α ( η , γ ; ϕ , ψ ) associated with the operator H p , β , μ α . Some applications involving integral operators are also considered.

In order to establish our main results, we need the following lemmas.

Lemma 1[4]

Let ς and υ be complex constants and let h ( z ) be convex (univalent) in U with h ( 0 ) = 1 and Re { ς h ( z ) + υ } > 0 . If
q ( z ) = 1 + q 1 z +
(1.14)
is analytic in U, then
q ( z ) + z q ( z ) ς q ( z ) + υ h ( z ) ( z U ) ,
implies
q ( z ) h ( z ) ( z U ) .

Lemma 2[7]

Let h ( z ) be convex (univalent) in U and ψ ( z ) be analytic in U with Re { ψ ( z ) } 0 . If q is analytic in U and q ( 0 ) = h ( 0 ) , then
q ( z ) + ψ ( z ) z q ( z ) h ( z ) ( z U )
implies
q ( z ) h ( z ) ( z U ) .

2 Some inclusion results

In this section, we give some inclusion properties for meromorphic function classes, which are associated with the operator H p , β , μ α . Unless otherwise mentioned, we assume that α 1 , β > 1 , μ > 0 , 0 γ , η < p and p N .

Theorem 1 For f ( z ) Σ p , ϕ M with
max z U ( Re { ϕ ( z ) } ) < min z U [ p + μ η p η , α + β + p η p η ] ,
then we have
Σ p S β , μ + 1 α ( η , ϕ ) Σ p S β , μ α ( η , ϕ ) Σ p S β , μ α 1 ( η , ϕ ) .
(2.1)
Proof We will first show that
Σ p S β , μ + 1 α ( η , ϕ ) Σ p S β , μ α ( η , ϕ ) .
(2.2)
Let f Σ p S β , μ + 1 α ( η ; ϕ ) and put
q ( z ) = 1 p η ( z ( H p , β , μ α f ( z ) ) H p , β , μ α f ( z ) η ) ,
(2.3)
where q ( z ) is analytic in U with q ( 0 ) = 1 . Applying (1.10) in (2.3), we have
μ H p , β , μ + 1 α f ( z ) H p , β , μ α f ( z ) = ( p η ) q ( z ) + η ( p + μ ) .
(2.4)
Differentiating (2.4) logarithmically with respect to z, we have
1 p η ( z ( H p , β , μ + 1 α f ( z ) ) H p , β , μ + 1 α f ( z ) η ) = q ( z ) + z q ( z ) ( p + μ ) η ( p η ) q ( z ) ( z U ) .
(2.5)
Since
max z U ( Re { ϕ ( z ) } ) < min z U p + μ η p η ,
we see that
Re { ( p + μ ) η ( p η ) ϕ ( z ) } > 0 ( z U ) .

Applying Lemma 1 to (2.5), it follows that q ϕ , that is, that f Σ p S β , μ α ( η ; ϕ ) . The proof of the second part will follow by using arguments similar to those used in the first part with f Σ p S β , μ α ( η ; ϕ ) and using the identity (1.9) instead of (1.10). This completes the proof of Theorem 1. □

Theorem 2 For f ( z ) Σ p , ϕ M with
Proof Applying (1.10) and using Theorem 1, we have
f ( z ) Σ p K β , μ + 1 α ( η ; ϕ ) H p , β , μ + 1 α f ( z ) Σ p K ( η ; ϕ ) z ( H p , β , μ + 1 α f ( z ) ) p Σ p S ( η ; ϕ ) H p , β , μ + 1 α ( z f ( z ) p ) Σ p S ( η ; ϕ ) z f ( z ) p Σ p S β , μ + 1 α ( η ; ϕ ) z f ( z ) p Σ p S β , μ α ( η ; ϕ ) H p , β , μ α ( z f ( z ) p ) Σ p S ( η ; ϕ ) z ( H p , β , μ α f ( z ) ) p Σ p S ( η ; ϕ ) H p , β , μ α f ( z ) Σ p K ( η ; ϕ ) f ( z ) Σ p K β , μ α ( η ; ϕ ) .
Also,
f ( z ) Σ p K β , μ α ( η ; ϕ ) z f ( z ) p Σ p S β , μ α ( η ; ϕ ) z f ( z ) p Σ p S β , μ α 1 ( η ; ϕ ) z ( H p , β , μ α 1 f ( z ) ) p Σ p S ( η ; ϕ ) H p , β , μ α 1 f ( z ) Σ p K ( η ; ϕ ) f ( z ) Σ p K β , μ α 1 ( η ; ϕ ) .

This completes the proof of Theorem 2. □

Taking
ϕ ( z ) = 1 + A z 1 + B z ( 1 < B < A 1 ; z U )

in Theorem 1 and Theorem 2, we have the following corollary.

Corollary 1 Let f ( z ) Σ p and
1 + A 1 + B < min z U ( p + μ η p η , α + β + p η p η ) ( 1 < B < A 1 ) .
Then we have
Σ p S β , μ + 1 α ( η ; A , B ) Σ p S β , μ α ( η ; A , B ) Σ p S β , μ α 1 ( η ; A , B )
and
Σ p K β , μ + 1 α ( η ; A , B ) Σ p K β , μ α ( η ; A , B ) Σ p K β , μ α 1 ( η ; A , B ) .

Now, using Lemma 2, we obtain similar inclusion relations for the subclass Σ p C β , μ α ( η ; γ ; ϕ , ψ ) .

Theorem 3 Let f ( z ) Σ p and
max z U Re { ϕ ( z ) } < min z U ( p + μ η p η , α + β + p η p η ) .
Then we have
Proof First, we will prove that
Σ p C β , μ + 1 α ( η ; γ ; ϕ , ψ ) Σ p C β , μ α ( η ; γ ; ϕ , ψ ) .
Let f Σ p C β , μ + 1 α ( η ; γ ; ϕ , ψ ) . Then, from the definition of the class Σ p C β , μ α ( η ; γ ; ϕ , ψ ) , there exists a function g Σ p S β , μ α ( η ; ϕ ) such that
1 p γ ( z ( H p , β , μ + 1 α f ( z ) ) H p , β , μ + 1 α g ( z ) γ ) ψ ( z ) ( z U ) .
(2.6)
Now, let
q ( z ) = 1 p γ ( z ( H p , β , μ α f ( z ) ) H p , β , μ α g ( z ) γ ) ,
(2.7)
where q ( z ) is analytic in U with q ( 0 ) = 1 . Applying (1.10) in (2.6), we have
(2.8)
Since, by Theorem 1,
g ( z ) Σ p S β , μ + 1 α ( η ; ϕ ) Σ p S β , μ α ( η ; ϕ ) ,
set
h ( z ) = 1 p η ( z ( H p , β , μ α g ( z ) ) H p , β , μ α g ( z ) η ) ,
where h ϕ in U, and ϕ M . Then, using (2.7) and (2.8), we have
H p , β , μ α ( z f ( z ) p ) = [ ( p γ ) q ( z ) + γ ] H p , β , μ α g ( z )
(2.9)
and
(2.10)
Differentiating both sides of (2.9) with respect to z and multiplying by z, we have
z ( H p , β , μ α ( z f ( z ) p ) ) H p , β , μ α g ( z ) = ( p γ ) z q ( z ) [ ( p γ ) q ( z ) + γ ] [ ( p η ) h ( z ) + η ] .
(2.11)
Making use of (2.6), (2.10), and (2.11), we have
1 p γ ( z ( H p , β , μ + 1 α f ( z ) ) H p , β , μ + 1 α g ( z ) γ ) = q ( z ) + z q ( z ) p + μ η ( p η ) h ( z ) ψ ( z ) , z U .
(2.12)
Since h ϕ in U, and
max z U Re { h ( z ) } < p + μ η p η ,
then
Re { p + μ η ( p η ) h ( z ) } > 0 ( z U ) .
(2.13)
Hence, putting
χ ( z ) = 1 { p + μ η ( p η ) h ( z ) } ,

in Eq. (2.12) and applying Lemma 2, we can show that q ψ , that is, that f Σ p C β , μ α ( η , γ ; ϕ , ψ ) . The second part can be proved by using similar arguments and using (1.9). This completes the proof of Theorem 3. □

3 Inclusion properties involving the integral operator F p , δ

Now, we consider the generalized Libera integral operator F p , δ ( f ) (see [6] and [9]), defined by
F p , δ ( f ) ( z ) = δ z δ + p 0 z t δ + p 1 f ( t ) d t = z p + k = 1 δ δ + k a k p z k p ( δ > p ) .
(3.1)
From (3.1), we have
z ( H p , β , μ σ F p , δ ( f ) ( z ) ) = δ H p , β , μ σ f ( z ) ( δ + p ) H p , β , μ σ F p , δ ( f ) ( z ) .
(3.2)
Theorem 4 Let ϕ M with
max z U ( Re { ϕ ( z ) } ) < δ + p η p η ( δ > p ) .

If f Σ p S β , μ α ( η ; ϕ ) , then F p , δ ( f ) Σ p S β , μ α ( η ; ϕ ) .

Proof Let f Σ p S β , μ α ( η ; ϕ ) and put
h ( z ) = 1 p η ( z ( H p , β , μ σ F p , δ ( f ) ( z ) ) H p , β , μ σ F p , δ ( f ) ( z ) η ) ,
(3.3)
where h is analytic in U with h ( 0 ) = 1 . Then, by using (3.2) and (3.3), we have
δ H p , β , μ σ f ( z ) H p , β , μ σ F p , δ ( f ) ( z ) = ( p η ) h ( z ) + η ( p + δ ) .
(3.4)
Differentiating (3.4) logarithmically with respect to z, we have
1 p η ( z ( H p , β , μ σ f ( z ) ) H p , β , μ σ f ( z ) η ) = h ( z ) + z h ( z ) p + δ η ( p η ) h ( z ) ( z U ) .

Applying Lemma 1, we conclude that h ϕ ( z U ) , which implies that F p , δ ( f ) Σ p S β , μ α ( η ; ϕ ) . □

Theorem 5 Let ϕ M with
max z U ( Re { ϕ ( z ) } ) < δ + p η p η ( δ > p ) .

If f Σ p K β , μ α ( η ; ϕ ) , then F p , δ ( f ) Σ p K β , μ α ( η ; ϕ ) .

Proof Applying Theorem 4 and (1.12), we have
f ( z ) Σ p K β , μ α ( η ; ϕ ) z f ( z ) p Σ p S β , μ α ( η ; ϕ ) F p , δ ( z f ( z ) p ) ( z ) Σ p S β , μ α ( η ; ϕ ) z p F p , δ ( f ) ( z ) Σ p S β , μ α ( η ; ϕ ) F p , δ ( f ) ( z ) K p , λ σ ( η ; ϕ ) .

This completes the proof of Theorem 5. □

From Theorem 4 and Theorem 5, we have the following corollary.

Corollary 2 Suppose that
1 + A 1 + B < δ + p η p η ( δ > p ; 1 < B < A 1 ) .
Then, for the classes Σ p S β , μ α ( η ; ϕ ) and Σ p K β , μ α ( η ; ϕ ) , the following inclusion relations hold true:
f Σ p S β , μ α ( A , B ) F p , δ ( f ) Σ p S β , μ α ( A , B )
and
f Σ p K β , μ α ( A , B ) F p , δ ( f ) Σ p K β , μ α ( A , B ) .
Theorem 6 Let ϕ , ψ M with
max z U Re { ϕ ( z ) } < δ + p η p η ( δ > p ) .

If f Σ p C β , μ α ( η , γ ; ϕ , ψ ) , then F p , δ ( f ) Σ p C β , μ α ( η , γ ; ϕ , ψ ) .

Proof Let f Σ p C β , μ α ( η , γ ; ϕ , ψ ) . Then, from the definition of the class Σ p C β , μ α ( η , γ ; ϕ , ψ ) , there exists a function g Σ p S β , μ α ( η ; ϕ ) such that
1 p γ ( z ( H p , β , μ σ f ( z ) ) H p , β , μ σ g ( z ) γ ) ψ ( z ) ( z U ) .
(3.5)
Now, let
h ( z ) = 1 p γ ( z ( H p , β , μ σ F p , δ ( f ) ( z ) ) H p , β , μ σ F p , δ ( g ) ( z ) γ ) ,
(3.6)
where h ( z ) is analytic in U with h ( 0 ) = 1 . Applying (3.2) in (3.6), we have
(3.7)
Since g Σ p S β , μ α ( η ; ϕ ) , then by Theorem 4, we have F p , δ ( g ) ( z ) Σ p S β , μ α ( η ; ϕ ) . Let
q ( z ) = 1 p η ( z ( H p , β , μ σ F p , δ ( g ) ( z ) ) H p , β , μ σ F p , δ g ( z ) η ) ,
(3.8)
where q ϕ in U. Then, using the same techniques as in the proof of Theorem 3 and using (3.5) and (3.7), we have
1 p γ ( z ( H p , β , μ σ f ( z ) ) H p , β , μ σ g ( z ) γ ) = h ( z ) + z h ( z ) δ + p η ( p η ) q ( z ) ψ ( z ) .
(3.9)

Since Re { 1 δ + p η ( p η ) q ( z ) } > 0 , then applying Lemma 2, we find that h ψ , which yields F p , δ ( f ) ( z ) Σ p C β , μ α ( η , γ ; ϕ , ψ ) . This completes the proof of Theorem 6. □

Remark Putting μ = 1 in the above results, we obtain the results corresponding to the operator H p , β , 1 α defined by (1.11).

Author’s contributions

The author read and approved the final manuscript.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Mansoura University

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© Mostafa; licensee Springer 2012

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