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

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.

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:zC 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 fg or f(z)g(z) (zU), 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)) (zU). 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)andf(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 (pN),
(1.2)

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

(fg)(z)= z p + k = 1 a k p b k p z k p =(gf)(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 (pN)
(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;pN).
(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 (zU).

For 0η, 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) (0η, γ<p and ϕ,ψM) 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 (α0, μ>0, β>1; pN) and for ϕ,ψM, 0η, 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<A1)

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<A1).

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 leth(z)be convex (univalent) in U withh(0)=1andRe{ςh(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)(zU),

implies

q(z)h(z)(zU).

Lemma 2[7]

Leth(z)be convex (univalent) in U andψ(z)be analytic in U withRe{ψ(z)}0. If q is analytic in U andq(0)=h(0), then

q(z)+ψ(z)z q (z)h(z)(zU)

implies

q(z)h(z)(zU).

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 α1, β>1, μ>0, 0γ, η<p and pN.

Theorem 1 Forf(z) Σ p , ϕMwith

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 ) =(pη)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 ) (zU).
(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(zU).

Applying Lemma 1 to (2.5), it follows that qϕ, 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 Forf(z) Σ p , ϕMwith

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<A1;zU)

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<A1).

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)(zU).
(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 hϕ in U, and ϕM. 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 ) =(pγ)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),zU.
(2.12)

Since hϕ in U, and

max z U Re { h ( z ) } < p + μ η p η ,

then

Re { p + μ η ( p η ) h ( z ) } >0(zU).
(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 qψ, 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)dt= 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 ϕM with

max z U ( Re { ϕ ( z ) } ) < δ + p η p η (δ>p).

Iff 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 ) =(pη)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 ) (zU).

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

Theorem 5 Let ϕM with

max z U ( Re { ϕ ( z ) } ) < δ + p η p η (δ>p).

Iff 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<A1).

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,ψM with

max z U Re { ϕ ( z ) } < δ + p η p η (δ>p).

Iff 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)(zU).
(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 qϕ 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 hψ, 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.

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Mostafa, A. Inclusion results for certain subclasses of p-valent meromorphic functions associated with a new operator. J Inequal Appl 2012, 169 (2012). https://doi.org/10.1186/1029-242X-2012-169

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Keywords

  • p-valent meromorphic functions
  • Hadamard product
  • inclusion properties