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PROPERTIES OF SOME FAMILIES OF MEROMORPHIC P-VALENT FUNCTIONS INVOLVING CERTAIN DIFFERENTIAL

OPERATOR

M. K. Aouf, A. Shamandy, A. O. Mostafa and S. M. Madian

Abstract. Making use of a differential operator, which is defined here by means of the Hadamard product (or convolution), we introduce the class Σn

p(f, g;λ, β) of

meromorphically p-valent functions. The main object of this paper is to investigate various important properties and characteristics for this class. Also a property preserving integrals is considered.

2000Mathematics Subject Classification: 30C45. 1. Introduction

Let Σp denote the class of functions of the form:

f(z) =z−p+ ∞

X

k=0

akzk (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}=U\{0}. For functions f(z)∈Σp given by (1.1) andg(z) ∈Σp given

by

g(z) =z−p+

X

k=0

bkzk (p∈N), (1.2)

we define the Hadamard product (or convolution) of f(z) and g(z) by

(f∗g)(z) =z−p+ ∞

X

k=0

akbkzk = (g∗f)(z). (1.3)

For complex parametersα1, α2, ..., αq and β1, β2, ..., βs (βj ∈/ Z0−={0,−1,

−2, ...}; j = 1,2, ..., s), we now define the generalized hypergeometric function

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qFs(α1, α2, ..., αq; β1, β2, ..., βs; z) = ∞

X

k=0

(α1)k...(αq)k

(β1)k...(βs)k(1)k zk

(q ≤s+ 1; s, q∈N0=N∪ {0}; z∈U), (1.4) where (θ)k , is the Pochhammer symbol defined in terms of the Gamma function

Γ,by (θ)v =

Γ(θ+v) Γ(θ) =

1 (v= 0; θ∈C∗=C\{0}),

θ(θ+ 1)....(θ+v−1) (v∈N; θ∈C).

Corresponding to the functionhp(α1, α2, ..., αq; β1, β2, ..., βs; z),defined by hp(α1, α2, ..., αq; β1, β2, ..., βs; z) =z−p qFs (α1, α2, ..., αq; β1, β2, ..., βs; z), (1.5) we consider a linear operator

Hp(α1, α2, ..., αq; β1, β2, ..., βs) : Σp −→Σp,

which is defined by the following Hadamard product:

Hp(α1, α2, ..., αq; β1, β2, ..., βs)f(z) =hp(α1, α2, ..., αq; β1, β2, ..., βs; z)∗f(z) (q≤s+ 1; s, q ∈N0;z∈U). (1.6) We observe that, for a function f(z) of the form (1.1), we have

Hp(α1, α2, ..., αq; β1, β2, ..., βs)f(z) =Hp,q,s(α1) =z−p+ ∞

X

k=0

Γkakzk, (1.7)

where

Γk =

(α1)k+p...(αq)k+p

(β1)k+p...(βs)k+p(1)k+p

. (1.8)

Then one can easily verify from (1.7) that

z(Hp,q,s(α1)f(z))′ =α1Hp,q,s(α1+ 1)f(z)−(α1+p)Hp,q,s(α1)f(z). (1.9) The linear operator Hp,q,s(α1) was investigated recently by Liu and Srivastava [9] and Aouf [2]. The operator Hp,q,s(α1) contains the operator ℓp(a, c) ( see [8] ) for

q = 2, s = 1, α1 = a > 0, β1 = c (c 6= 0,−1, ...) and α2 = 1 and also contains the operator Dν+p−1 ( see [1] and [4]) for q = 2, s = 1, α

1 = ν+p (ν > −p; p ∈

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For functions f, g ∈ Σp,we define the linear operator Dλ,pn (f ∗g)(z) : Σp −→

Σp (λ≥0; p∈N; n∈N0) by

Dλ,p0 (f∗g)(z) = (f∗g)(z), (1.10)

Dλ,p1 (f∗g)(z) =Dλ,p(f∗g)(z) = (1−λ)(f ∗g)(z) +λz−p (zp+1(f ∗g)(z))′,

=z−p+ ∞

X

k=0

[1 +λ(k+p)]akbkzk (λ≥0; p∈N), (1.11)

D2λ,p(f∗g)(z) =Dλ,p(Dλ,p(f∗g))(z),

= (1−λ)Dλ,p(f∗g)(z) +λz−p (zp+1Dλ,p(f ∗g)(z))′ =z−p+

X

k=0

[1 +λ(k+p)]2akbkzk (λ≥0; p∈N) (1.12)

and ( in general )

Dnλ,p(f ∗g)(z) =Dλ,p(Dnλ,p−1(f ∗g)(z))

=z−p+ ∞

X

k=0

[1 +λ(k+p)]nakbkzk (λ≥0; p∈N; n∈N0). (1.13) From (1.13) it is easy to verify that:

z(Dλ,pn (f ∗g)(z))′ = 1

λD n+1

λ,p (f∗g)(z)−(p+

1

λ)D n

λ,p(f ∗g)(z) (λ >0). (1.14)

In this paper, we introduce the class Σn

p(f, g;λ, β) of the functions f, g ∈

Σp,which satisfy the condition:

Re

(

Dλ,pn+1(f∗g)(z)

λDn

λ,p(f∗g)(z)

−(p+ 1

λ)

)

<−β (z∈U∗; λ >0; 0≤β < p; p∈N; n∈N0). (1.15) We note that:

(i) If λ= 1 and the coefficients bk = 1

org(z) = 1

zp(1z)

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(ii) If n= 0, λ= 1 and g(z) = 1

zp(1z)ν+p, (ν >−p; p∈N) in (1.15), the class

Σn

p(f, g;λ, β) reduces to the classMν+p−1(β),studied by Aouf [1]; (iii) If bk = 1

or g(z) = 1

zp(1z)

in (1.15), the class Σn

p(f, g;λ, β) reduces to

the class Kn

p(λ, β),whereKpn(λ, β) is defined by

Re

(

Dλ,pn+1f(z)

λDn λ,pf(z)

−(p+ 1

λ)

)

<−β (z∈U∗; λ >0; 0≤β < p; p∈N; n∈N0), (1.16) where

Dnλ,pf(z) =z−p+ ∞

X

k=0

[1 +λ(k+p)]nakzk (λ >0; p∈N; n∈N0); (1.17)

(iv) If n= 0, λ= 1 and the coefficientsbk = (a)k (c)k

(c6= 0,−1, ...) in (1.15), the class Σn

p(f, g;λ, β) reduces to Σnp(a, c;λ, β),where Σnp(a, c;λ, β) is defined by

Re

a ℓp(a+ 1, c)f(z)

ℓp(a, c)f(z) −(a+p)

<−β (z∈U∗; 0≤β < p; p∈N); (1.18) (v) If n = 0, λ = 1 and the coefficients bk in (1.15) is replaced by Γk, where Γk is given by (1.8), the class Σnp(f, g;λ, β) reduces to Σp,q,sn (α1, β1;λ, β), where Σn

p,q,s(α1, β1;λ, β) is defined by Re

α1Hp,q,s(α1+ 1)f(z)

Hp,q,s(α1)f(z)

−(α1+p)

<−β

(z∈U∗; α1∈C∗; 0≤β < p; p∈N). (1.19) In this paper known results of Bajpai [5], Goel and Sohi [6], Uralegaddi and Somanatha [12] and Aouf and Hossen [3] are extended.

2.Basic properties of the class Σn

p(f, g;λ, β)

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Lemma 1 [7]. Let the (nonconstant) function w(z) be analytic in U,with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r < 1 at a point z0 ∈ U, then

z0w ′

(z0) =ξw(z0), (2.1)

where ξ is a real number andξ ≥1.

Theorem 1. For λ >0, 0β < p, pN and nN0, Σn+1

p (f, g;λ, β)⊂Σnp(f, g;λ, β). (2.2)

Proof. Let f(z)∈Σn+1

p (f, g;λ, β).Then

Re

(

Dnλ,p+2(f ∗g)(z)

λDnλ,p+1(f ∗g)(z) −(p+ 1

λ)

)

<−β, |z|<1. (2.3) We have to show that (2.3) implies the inequality

Re

(

Dnλ,p+1(f ∗g)(z)

λDn

λ,p(f∗g)(z)

−(p+ 1

λ)

)

<−β. (2.4)

Define a regular functionw(z) in U by

Dλ,pn+1(f∗g)(z)

λDn

λ,p(f∗g)(z)

−(p+ 1

λ) =−

[p+ (2β−p)w(z)]

1 +w(z) . (2.5) Clearly w(0) = 0. Equation (2.5) may be written as

Dnλ,p+1(f ∗g)(z)

Dn

λ,p(f∗g)(z)

= 1 + [1 + 2λ(p−β)]w(z)

1 +w(z) . (2.6)

Differentiating (2.6) logarithmically with respect to z and using (1.14), we obtain

Dn+2λ,p (f∗g)(z)

λDn+1λ,p(f∗g)(z)−(p+ 1

λ) +β

(p−β) =

2λzw′(z)

(1+w(z)){1+[1+2λ(p−β)]w(z)} − 1−w(z)

1+w(z). (2.7) We claim that |w(z)|<1 forz∈U. Otherwise there exists a pointz0∈U such that

max |z|≤|z0|

|w(z)|=|w(z0)|= 1. Applying Jack’s Lemma, we havez0w ′

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Dλ,pn+2(f∗g)(z0)

λDn+1λ,p (f∗g)(z0)−(p+

1

λ) +β

p−β =

2λξw(z0)

(1+w(z0)){1+[1+2λ(p−β)]w(z)} −

1−w(z0)

1+w(z0). (2.8)

Thus Re       

Dn+2λ,p (f∗g)(z0)

λDn+1λ,p(f∗g)(z0)−(p+

1

λ) +β p−β

       ≥ 1

2[1 +λ(p−β)] >0, (2.9)

which obviously contradicts our hypothesis that f(z) ∈ Σn+1

p (f, g;λ, β). Thus we

must have |w(z)| < 1 (z ∈ U), and so from (2.5), we conclude that f(z) ∈

Σn

p(f, g;λ, β), which evidently completes the proof of Theorem 1.

Theorem 2. Let f, gΣp satisfy the condition

Re

(

Dnλ,p+1(f ∗g)(z)

λDn

λ,p(f∗g)(z)

−(p+ 1

λ)

)

<−β+ (p−β)

2(p−β+c) (z∈U) (2.10) for λ >0, 0≤β < p, p∈N, n∈N0 and c >0.Then

Fc,p(f∗g)(z) = c

zc+p z

Z

0

tc+p−1(f ∗g)(t)dt (2.11)

belongs to Σn

p(f, g;λ, β).

Proof. From the definition of Fc,p(f∗g)(z), we have

z(Dλ,pn Fc,p(f ∗g)(z))′=cDλ,pn (f∗g)(z)−(c+p)Dλ,pn Fc,p(f ∗g)(z) (2.12)

and also

z(Dnλ,pFc,p(f∗g)(z))′ = 1

λD n+1

λ,p Fc,p(f ∗g)(z)−(p+

1

λ)D n

λ,pFc,p(f ∗g)(z) (λ >0).

(2.13) Using (2.12) and (2.13), the condition (2.10) may be written as

Re     

Dn+2λ,p Fc,p(f∗g)(z)

λDλ,pn+1Fc,p(f∗g)(z)+ (c−

1

λ)

1 + (λc−1)D

n

λ,pFc,p(f∗g)(z)

−(p+ 1

λ)     

<−β+ p−β

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We have to prove that (2.14) implies the inequality

Re

(

Dnλ,p+1Fc,p(f∗g)(z)

λDn

λ,pFc,p(f∗g)(z)

−(p+ 1

λ)

)

<−β. (2.15)

Define a regular functionw(z) in U by

Dnλ,p+1Fc,p(f ∗g)(z)

λDn

λ,pFc,p(f ∗g)(z)

−(p+ 1

λ) =−

[p+ (2β−p)w(z)]

1 +w(z) . (2.16) Clearly w(0) = 0. The equation (2.16) may be written as

Dnλ,p+1Fc,p(f∗g)(z)

Dn

λ,pFc,p(f ∗g)(z)

= 1 + [1 + 2λ(p−β)]w(z)

1 +w(z) . (2.17)

Differentiating (2.17) logarithmically with respect to z and using (2.13), we obtain

Dnλ,p+2Fc,p(f∗g)(z)

λDλ,pn+1Fc,p(f∗g)(z) − D

n+1

λ,p Fc,p(f ∗g)(z) λDn

λ,pFc,p(f ∗g)(z)

= (1+w(z))[1+(1+22λ(p−β)zwλ(p(z)β))w(z)]. (2.18)

The above equation may be written as

Dn+2λ,p Fc,p(f∗g)(z)

λDn+1λ,pFc,p(f∗g)(z)+ (c−

1

λ)

1 + (λc−1)D

n

λ,pFc,p(f∗g)(z)

Dn+1λ,p Fc,p(f∗g)(z)

−(p+ 1

λ) =

Dλ,pn+1Fc,p(f∗g)(z) λDn

λ,pFc,p(f∗g)(z)

−(p+ 1

λ)

+

"

2λ(p−β)zw′(z)

(1 +w(z))[1 + (1 + 2λ(p−β))w(z)]

# 

 

1 1 + (λc−1)D

n

λ,pFc,p(f∗g)(z)

Dλ,pn+1Fc,p(f∗g)(z) 

 ,

which, by using (2.16) and (2.17), reduces to

Dn+2λ,p Fc,p(f∗g)(z)

λDn+1λ,pFc,p(f∗g)(z)+ (c−

1

λ)

1 + (λc−1)D

n

λ,pFc,p(f∗g)(z)

Dn+1λ,p Fc,p(f∗g)(z)

−(p+ 1

λ) =−

β+ (p−β)1−w(z) 1 +w(z)

+ 2λ(p−β)zw ′(z)

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The remaining part of the proof is similar to that Theorem 1, so we omit it. Remark 1. (i) For λ=p=c=ak =bk= 1 and n=β= 0,we note that Theorem 2 extends a results of Bajpai [ 5, Theorem 1 ];

(ii) For λ=p=ak=bk = 1and n=β = 0,we note that Theorem 2 extends a

results of Goel and Sohi [ 6, Corollary 1 ].

Theorem 3.(fg)(z)Σnp(f, g;λ, β) if and only if

F(f ∗g)(z) = 1

z1+p z

Z

0

tp(f∗g)(t)dt∈Σnp+1(f, g;λ, β). (2.20)

Proof.From the definition ofF(f∗g)(z) we have

Dn

λ,p(zF′(f∗g)(z)) + (1 +p)Dλ,pn F(f ∗g)(z) =Dλ,pn (f ∗g)(z),

that is,

z(Dnλ,pF(f∗g)(z))′+ (1 +p)Dλ,pn F(f ∗g)(z) =Dλ,pn (f ∗g)(z). (2.21) By using the identity (1.14), (2.21) reduces toDλ,pn (f∗g)(z) =Dλ,pn+1F(f∗g)(z).Hence

Dλ,pn+1(f ∗g)(z) =Dλ,pn+2F(f ∗g)(z),therefore,

Dλ,pn+1(f∗g)(z)

Dn

λ,p(f ∗g)(z)

= D

n+2

λ,p F(f ∗g)(z) Dnλ,p+1F(f ∗g)(z) and the result follows.

References

[1] M.K. Aouf, New certeria for multivalent meromorphic starlike functions of order alpha, Proc. Japan. Acad., 69 (1993), 66-70.

[2] M.K. Aouf, Certain subclasses of meromorphically multivalent functions as-sociated with generalized hypergeometric function, Comput. Math. Appl., 55 (2008), 494-509.

[3] M.K. Aouf and H.M. Hossen,New certeria for meromorphic p-valent starlike functions, Tsukuba J. Math., 17 (1993), no. 2, 481-486.

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[5] S.K. Bajpai, A note on a class of meromorphic univalent functions, Rev. Roum. Math. Pures Appl., 22 (1977), 295-297.

[6] R.M. Goel and N.S. Sohi,On a class of meromorphic functions, Glas. Mat., 17 (1981), 19-28.

[7] I.S. Jack, Functions starlike and convex of order α, J. London Math. Soc., (2) 3 (1971), 469-474.

[8] J.-L. Liu and H.M. Srivastava, A linear operator and associated with the generalized hypergeometric function, J. Math. Anal. Appl., 259 (2000), 566-581.

[9] J.-L. Liu and H.M. Srivastava,Classes of meromorphically multivalent func-tions associated with the generalized hypergeometric function, Math. Comput. Mod-elling , 39 (2004), 21-34.

[10] R.K. Raina and H.M. Srivastava,A new class of meromorphiclly multivalent functions with applications to generalized hypergeometric functions, Math. Comput. Modelling, 43 (2006), 350-356.

[11] H.M. Srivastava and P.W. Karlsson,Multiple Gaussian Hypergeometric Se-ries, Halsted Press, Ellis Horwood Limited, Chichester, John Wiley and Sons, New York, Chichester, Brisbane, Toronto, 1985.

[12] B.A. Uralegaddi and C. Somanatha,New certeria for meromorphic starlike univalent functions, Bull. Austral. Math. Soc., 43 (1991), 137-140.

M. K. Aouf

Department of Mathematics Faculty of Science

Mansoura University Mansoura 35516, Egypt. email: mkaouf127@yahoo.com A. Shamandy

Department of Mathematics Faculty of Science

Mansoura University Mansoura 35516, Egypt.

email: shamandy16@hotmail.com A. O. Mostafa

Department of Mathematics Faculty of Science

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Department of Mathematics Faculty of Science

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