Paper-1-20-2009. 121KB Jun 04 2011 12:03:09 AM

ACTA UNIVERSITATIS APULENSIS

No 20/2009

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 Σnp (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.
2000 Mathematics Subject Classification: 30C45.
1. Introduction
Let Σp denote the class of functions of the form:
f (z) = z −p +

∞
X

ak z k

(p ∈ N = {1, 2, ...}) ,

(1.1)

k=0

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) and g(z) ∈ Σp given
by
∞
X
g(z) = z −p +
bk z k (p ∈ N ),
(1.2)
k=0

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

(f ∗ g)(z) = z −p +

∞
X

ak bk z k = (g ∗ f )(z).

(1.3)

k=0

For complex parameters α1 , α2 , ..., αq and β1 , β2 , ..., βs (βj ∈
/ Z0− = {0, −1,
−2, ...}; j = 1, 2, ..., s), we now define the generalized hypergeometric function
q Fs (α1 , α2 , ..., αq ; β1 , β2 , ..., βs ; z) by (see, for example, [10] and [11])
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M.K. Aouf, A.Shamandy, A. O. Mostafa, S.M. Madian - Properties of some ...

q Fs (α1 , α2 , ..., αq ; β1 , β2 , ..., βs ; z) =

∞
X
k=0

(α1 )k ...(αq )k k
z
(β1 )k ...(βs )k (1)k

(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

Γ(θ + v)
1
(v = 0; θ ∈ C ∗ = C\{0}),
(θ)v =

=
θ(θ + 1)....(θ + v − 1)
(v ∈ N ; θ ∈ C).
Γ(θ)
Corresponding to the function hp (α1 , α2 , ..., αq ; β1 , β2 , ..., βs ; z), defined by
hp (α1 , α2 , ..., αq ; β1 , β2 , ..., βs ; z) = z −p q Fs (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

Γk ak z k ,

(1.7)

k=0

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))′ = α1 Hp,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, α1 = ν + p (ν > −p; p ∈
N ) and α2 = β1 = p.
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M.K. Aouf, A.Shamandy, A. O. Mostafa, S.M. Madian - Properties of some ...

n (f ∗ g)(z) : Σ −→
For functions f , g ∈ Σp , we define the linear operator Dλ,p
p
Σp (λ ≥ 0; p ∈ N ; n ∈ N0 ) by
0
Dλ,p
(f ∗ g)(z) = (f ∗ g)(z),

(1.10)

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

=z

−p

+

∞
X

[1 + λ(k + p)]ak bk z k (λ ≥ 0; p ∈ N ),

(1.11)

k=0

2
(f ∗ g)(z) = Dλ,p (Dλ,p (f ∗ g))(z),
Dλ,p

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

−p

+

∞
X

[1 + λ(k + p)]2 ak bk z k (λ ≥ 0; p ∈ N )

(1.12)

k=0

and ( in general )
n−1
n
Dλ,p
(f ∗ g)(z) = Dλ,p (Dλ,p
(f ∗ g)(z))

= z −p +

∞
X

[1 + λ(k + p)]n ak bk z k (λ ≥ 0; p ∈ N ; n ∈ N0 ). (1.13)

k=0

From (1.13) it is easy to verify that:
n

z(Dλ,p
(f ∗ g)(z))′ =

1 n+1
1 n
Dλ,p (f ∗ g)(z) − (p + )Dλ,p
(f ∗ g)(z) (λ > 0).
λ
λ

(1.14)

In this paper, we introduce the class Σnp (f, g; λ, β) of the functions f , g ∈
Σp , which satisfy the condition:

Re

(

n+1

Dλ,p
(f ∗ g)(z)

)
1
n (f ∗ g)(z) − (p + λ ) < −β
λDλ,p

(z ∈ U ∗ ; λ > 0; 0 ≤ β < p; p ∈ N ; n ∈ N0 ).
(1.15)

We note that:




1
in (1.15), the class
(i) If λ = 1 and the coefficients bk = 1 or g(z) = p
z (1 − z)
n
Σp (f, g; λ, β) reduces to the class Bn (β) studied by Aouf and Hossen [3];

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M.K. Aouf, A.Shamandy, A. O. Mostafa, S.M. Madian - Properties of some ...
1
, (ν > −p; p ∈ N ) in (1.15), the class
− z)ν+p
Σnp (f, g; λ, β) reduces to the class Mν+p−1 (β), studied by Aouf [1];


1
(iii) If bk = 1 or g(z) = p
in (1.15), the class Σnp (f, g; λ, β) reduces to
z (1 − z)
the class Kpn (λ, β), where Kpn (λ, β) is defined by

(ii) If n = 0, λ = 1 and g(z) =

Re

(

n+1
Dλ,p
f (z)

z p (1

)
1
n f (z) − (p + λ ) < −β
λDλ,p

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

where
n
Dλ,p
f (z) = z −p +

∞
X

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

(1.17)

k=0

(a)k
(c 6= 0, −1, ...) in (1.15), the class
(c)k
Σnp (f, g; λ, β) reduces to Σnp (a, c; λ, β), where Σnp (a, c; λ, β) is defined by

(iv) If n = 0, λ = 1 and the coefficients bk =

Re




a ℓp (a + 1, c)f (z)
− (a + p) < −β
ℓp (a, c)f (z)

(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 Σnp,q,s (α1 , β1 ; λ, β), where
Σnp,q,s (α1 , β1 ; λ, β) is defined by


α1 Hp,q,s (α1 + 1)f (z)
− (α1 + p) < −β
Re
Hp,q,s (α1 )f (z)
(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 Σnp (f, g; λ, β)
We begin by recalling the following result (Jack’s Lemma), which we shall apply
in proving our first inclusion theorems (Theorem 1 and Theorem 2 below).

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M.K. Aouf, A.Shamandy, A. O. Mostafa, S.M. Madian - Properties of some ...

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
′
z0 w (z0 ) = ξw(z0 ) ,
(2.1)
where ξ is a real number and ξ ≥ 1.
Theorem 1. For λ > 0, 0 ≤ β < p, p ∈ N and n ∈ N0 ,
(f, g; λ, β) ⊂ Σnp (f, g; λ, β) .
Σn+1
p

(2.2)

Proof. Let f (z) ∈ Σn+1
(f, g; λ, β). Then
p
( n+2
)
Dλ,p (f ∗ g)(z)
1
Re
− (p + ) < −β, |z| < 1.
n+1
λ
λDλ,p
(f ∗ g)(z)
We have to show that (2.3) implies the inequality
( n+1
)
Dλ,p (f ∗ g)(z)
1
Re
n (f ∗ g)(z) − (p + λ ) < −β.
λDλ,p

(2.3)

(2.4)

Define a regular function w(z) in U by
n+1
(f ∗ g)(z)
Dλ,p
n (f ∗ g)(z)
λDλ,p

− (p +

1
[p + (2β − p)w(z)]
)=−
.
λ
1 + w(z)

(2.5)

Clearly w(0) = 0. Equation (2.5) may be written as
n+1
Dλ,p
(f ∗ g)(z)
n (f
Dλ,p

∗ 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
n+2
Dλ,p
(f ∗g)(z)
n+1
λDλ,p
(f ∗g)(z)

− (p + λ1 ) + 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 for z ∈ U . Otherwise there exists a point z0 ∈ U such that
′
max |w(z)| = |w(z0 )| = 1. Applying Jack’s Lemma, we have z0 w (z0 ) = ξw(z0 )(ξ ≥
|z|≤|z0 |

1). Writing w(z0 ) = eiθ (0 ≤ θ ≤ 2π) and putting z = z0 in (2.7), we get

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M.K. Aouf, A.Shamandy, A. O. Mostafa, S.M. Madian - Properties of some ...

n+2
Dλ,p
(f ∗g)(z0 )
n+1
λDλ,p
(f ∗g)(z0 )

− (p + λ1 ) + 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

 n+2

Dλ,p (f ∗g)(z0 )


1


)
+
β
−
(p
+
 λDn+1 (f ∗g)(z0 )

λ
λ,p

p−β





≥





1
> 0,
2[1 + λ (p − β)]

(2.9)

which obviously contradicts our hypothesis that f (z) ∈ Σn+1
(f, g; λ, β). Thus we
p
must have |w(z)| < 1 (z ∈ U ), and so from (2.5), we conclude that f (z) ∈
Σnp (f, g; λ, β), which evidently completes the proof of Theorem 1.
Theorem 2. Let f, g ∈ Σp satisfy the condition

Re

(

n+1
Dλ,p
(f ∗ g)(z)

)
1
(p − β)
− (p + ) < −β +
(z ∈ U )
n
λDλ,p (f ∗ g)(z)
λ
2(p − β + c)

(2.10)

for λ > 0, 0 ≤ β < p, p ∈ N, n ∈ N0 and c > 0. Then
Fc,p (f ∗ g)(z) =

c
z c+p

Zz

tc+p−1 (f ∗ g)(t)dt

(2.11)

0

Σnp (f, g; λ, β).

belongs to
Proof. From the definition of Fc,p (f ∗ g)(z), we have
n
n
n
z(Dλ,p
Fc,p (f ∗ g)(z))′ = cDλ,p
(f ∗ g)(z) − (c + p)Dλ,p
Fc,p (f ∗ g)(z)

(2.12)

and also
1 n
1 n+1
Dλ,p Fc,p (f ∗ g)(z) − (p + )Dλ,p
Fc,p (f ∗ g)(z) (λ > 0).
λ
λ
(2.13)
Using (2.12) and (2.13), the condition (2.10) may be written as
n
z(Dλ,p
Fc,p (f ∗ g)(z))′ =

Re






n+2
Dλ,p
Fc,p (f ∗g)(z)
n+1
λDλ,p
Fc,p (f ∗g)(z)



 1 + (λc − 1)

+ (c −

1
λ)

n F
Dλ,p
c,p (f ∗g)(z)
n+1
Dλ,p Fc,p (f ∗g)(z)




p−β
1 
.
− (p + ) < −β +
λ 
2(p − β + c)


12

(2.14)

M.K. Aouf, A.Shamandy, A. O. Mostafa, S.M. Madian - Properties of some ...

We have to prove that (2.14) implies the inequality
( n+1
)
Dλ,p Fc,p (f ∗ g)(z)
1
Re
n F (f ∗ g)(z) − (p + λ ) < −β.
λDλ,p
c,p

(2.15)

Define a regular function w(z) in U by
n+1
Dλ,p
Fc,p (f ∗ g)(z)
n F (f
λDλ,p
c,p

∗ g)(z)

− (p +

1
[p + (2β − p)w(z)]
)=−
.
λ
1 + w(z)

(2.16)

Clearly w(0) = 0. The equation (2.16) may be written as
n+1
Dλ,p
Fc,p (f ∗ g)(z)
n F (f
Dλ,p
c,p

∗ 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
n+2
Dλ,p
Fc,p (f ∗ g)(z)
n+1
λDλ,p
Fc,p (f

∗ g)(z)

−

n+1
Dλ,p
Fc,p (f ∗ g)(z)
n F (f
λDλ,p
c,p

∗ g)(z)

=

2λ(p−β)zw′ (z)
(1+w(z))[1+(1+2λ(p−β))w(z)] .

(2.18)

The above equation may be written as
n+2
Dλ,p
Fc,p (f ∗g)(z)
n+1
λDλ,p
Fc,p (f ∗g)(z)

+ (c − λ1 )

Dn Fc,p (f ∗g)(z)

λ,p
1 + (λc − 1) Dn+1
F
λ,p

+

"

− (p +

c,p (f ∗g)(z)

n+1
Dλ,p
Fc,p (f ∗ g)(z)
1
1
)=
− (p + )
n
λ
λDλ,p Fc,p (f ∗ g)(z)
λ

#

β)zw′ (z)





1
2λ(p −



,
n F
D
(f
∗g)(z)
c,p
(1 + w(z))[1 + (1 + 2λ(p − β))w(z)]
λ,p
1 + (λc − 1) Dn+1
F (f ∗g)(z)
λ,p

c,p

which, by using (2.16) and (2.17), reduces to
n+2
Dλ,p
Fc,p (f ∗g)(z)
n+1
λDλ,p
Fc,p (f ∗g)(z)

+ (c − λ1 )

Dn Fc,p (f ∗g)(z)

λ,p
1 + (λc − 1) Dn+1
F
λ,p

+

c,p (f ∗g)(z)



1 − w(z)
1
− (p + ) = − β + (p − β)
λ
1 + w(z)

2λ(p − β)zw′ (z)
.
(1 + w(z)){c + [c + 2(p − β)]w(z)}
13

(2.19)

M.K. Aouf, A.Shamandy, A. O. Mostafa, S.M. Madian - Properties of some ...

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 = 1 and n = β = 0, we note that Theorem 2 extends a
results of Goel and Sohi [ 6, Corollary 1 ].
Theorem 3. (f ∗ g)(z) ∈ Σnp (f, g; λ, β) if and only if
F (f ∗ g)(z) =

1
z 1+p

Zz

tp (f ∗ g)(t)dt ∈ Σn+1
(f, g; λ, β).
p

(2.20)

0

Proof.From the definition of F (f ∗ g)(z) we have
n
n
n
Dλ,p
(zF ′ (f ∗ g)(z)) + (1 + p)Dλ,p
F (f ∗ g)(z) = Dλ,p
(f ∗ g)(z),

that is,
n
n
n
F (f ∗ g)(z) = Dλ,p
(f ∗ g)(z).
z(Dλ,p
F (f ∗ g)(z))′ + (1 + p)Dλ,p

(2.21)

n (f ∗g)(z) = D n+1 F (f ∗g)(z). Hence
By using the identity (1.14), (2.21) reduces to Dλ,p
λ,p
n+1
n+2
Dλ,p
(f ∗ g)(z) = Dλ,p
F (f ∗ g)(z), therefore,
n+1
Dλ,p
(f ∗ g)(z)
n (f ∗ g)(z)
Dλ,p

=

n+2
Dλ,p
F (f ∗ g)(z)
n+1
Dλ,p
F (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 associated 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.
[4] M.K. Aouf and H.M. Srivastava, A new criterion for meromorphically p-valent
convex functions of order alpha, Math. Sci. Res. Hot-Line, 1 (1997), no. 8, 7-12.
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M.K. Aouf, A.Shamandy, A. O. Mostafa, S.M. Madian - Properties of some ...

[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 functions associated with the generalized hypergeometric function, Math. Comput. Modelling , 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 Series, 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
Mansoura University
Mansoura 35516, Egypt.
email: adelaeg254@yahoo.com
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M.K. Aouf, A.Shamandy, A. O. Mostafa, S.M. Madian - Properties of some ...

S. M. Madian
Department of Mathematics
Faculty of Science
Mansoura University
Mansoura 35516, Egypt.
email: samar math@yahoo.com

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