CERTAIN SUBCLASSES OF p-VALENT
MEROMORPHIC FUNCTIONS INVOLVING
CERTAIN OPERATOR
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
vol. 9, iss. 2, art. 45, 2008
M.K. AOUF AND A.O. MOSTAFA
Department of Mathematics
Faculty of Science
Mansoura University
Mansoura 35516, Egypt.
EMail: mkaouf127@yahoo.com
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adelaeg254@yahoo.com
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Received:
25 March, 2008
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Accepted:
20 May, 2008
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Communicated by:
N.K. Govil
2000 AMS Sub. Class.:
30C45.
Key words:
Analytic, p−valent, Meromorphic, Integral operator.
Abstract:
P
In this paper, a new subclass α
p,β (η, δ, µ, λ) of p−valent meromorphic functions defined by certain integral operator is introduced. Some interesting properties of this class are obtained.
Acknowledgements:
The authors would like to thank the referees of the paper for their helpful suggestions.
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Contents
1
Introduction
3
2
Main Results
6
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
vol. 9, iss. 2, art. 45, 2008
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1.
Introduction
Let
P
p
be the class of functions f of the form:
f (z) = z
(1.1)
−p
+
∞
X
ak−p z k−p
(p ∈ N = {1, 2, ...}),
k=1
which are analytic and p−valent in the punctured unit disc U ∗ = {z : z ∈ C and
0 < |z| < 1} = U \{0}.
P
Similar to [1], we define the following family of integral operators Qαβ,p : p →
P
p (α ≥ 0, β > −1; p ∈ N) as follows:
(1.2)
Qαβ,p f (z)
(1.3)
vol. 9, iss. 2, art. 45, 2008
Z z
α+β−1
t
−(p+β)
=
αz
(1 − )α−1 tβ+p−1 f (t)dt
β−1
z
0
(α > 0; β > −1; p ∈ N; f ∈ Σp );
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and
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(ii)
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(1.4)
Q0β,p f (z) = f (z) (for α = 0; β > −1; p ∈ N; f ∈ Σp ).
∞
Qαβ,p f (z)
=z
−p
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From (1.2) and (1.4), we have
(1.6)
M.K. Aouf and A.O. Mostafa
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(i)
(1.5)
p-Valent Meromorphic Functions
Γ(α + β) X Γ(k + β)
+
ak−p z k−p
Γ(β) k=1 Γ(k + β + α)
(α ≥ 0; β > −1; p ∈ N; f ∈ Σp ).
Using the relation (1.5), it is easy to show that
α
z(Qαβ,p f (z))0 = (α + β − 1)Qα−1
β,p f (z) − (α + β + p − 1)Qβ,p f (z).
P
Pα
Definition 1.1. Let
p,β (η, δ, µ, λ) be the class of functions f ∈
p which satisfy:
(1.7)
(1.8)


Re (1 − λ)

Qαβ,p f (z)
Qαβ,p g(z)
!µ
+λ
Qα−1
β,p f (z)
Qα−1
β,p g(z)
Qαβ,p f (z)
Qαβ,p g(z)
!µ−1 

> η,

p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
vol. 9, iss. 2, art. 45, 2008
where g ∈
(1.9)
P
p satisfies the following condition:
(
)
Qαβ,p g(z)
Re
> δ (0 ≤ δ < 1; z ∈ U ),
Qα−1
β,p g(z)
and η and µ are real numbers such that 0 ≤ η < 1, µ > 0 and λ ∈ C with
Re{λ} > 0.
To establish our main results we need the following lemmas.
Lemma 1.2 ([2]). Let Ω be a set in the complex plane C and let the function ψ :
1+r 2
C2 → C satisfy the condition ψ(ir2 , s1 ) ∈
/ Ω for all real r2 , s1 ≤ − 2 2 . If q is
analytic in U with q(0) = 1 and ψ(q(z), zq 0 (z)) ∈ Ω, z ∈ U, then Re{q(z)} >
0 (z ∈ U ).
Lemma 1.3 ([3]). If q is analytic in U with q(0) = 1, and if λ ∈ C\{0} with
Re{λ} ≥ 0, then Re{q(z) + λzq 0 (z)} > α (0 ≤ α < 1) implies Re{q(z)} >
α + (1 − α)(2γ − 1), where γ is given by
Z 1
γ = γ(Re λ) =
(1 + tRe{λ} )−1 dt
0
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which is increasing function of Re{λ} and
sense that the bound cannot be improved.
1
2
≤ γ < 1. The estimate is sharp in the
For real or complex numbers a, b and c (c 6= 0, −1, −2, . . . ), the Gauss hypergeometric function is defined by
(1.10)
2 F1 (a, b; c; z) = 1 +
a·b z
a(a + 1) · b(b + 1) z 2
+
+ ··· .
c 1!
c(c + 1)
2!
We note that the series (1.10) converges absolutely for z ∈ U and hence represents
an analytic function in U (see, for details, [4, Ch. 14]). Each of the identities (asserted by Lemma 1.4 below) is fairly well known (cf., e.g., [4, Ch. 14]).
Lemma 1.4 ([4]). For real or complex numbers a, b and c (c 6= 0, −1, −2, . . . ),
Z 1
Γ(b)Γ(c − b)
(1.11)
tb−1 (1 − t)c−b−1 (1 − tz)−a dt =
2 F1 (a, b; c; z)
Γ(c)
0
(Re(c) > Re(b) > 0);
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
vol. 9, iss. 2, art. 45, 2008
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(1.12)
2 F1 (a, b; c; z)
= (1 − z)−a 2 F1
z
a, c − b; c;
z−1
;
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(1.13)
2 F1 (a, b; c; z)
=2 F1 (b, a; c; z);
and
(1.14)
2 F1
1
1, 1; 2;
2
= 2 ln 2.
Close
2.
Main Results
Unless otherwise mentioned, we assume throughout this paper that
α ≥ 0; β > −1; α + β 6= 1; µ > 0; 0 ≤ η < 1; p ∈ N and λ ≥ 0.
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Theorem 2.1. Let f ∈ αp,β (η, δ, µ, λ). Then
!µ
Qαβ,p f (z)
2µη(α + β − 1) + λδ
(2.1)
Re
>
,
(z ∈ U ),
α
Qβ,p g(z)
2µ(α + β − 1) + λδ
P
where the function g ∈ p satisfies the condition (1.9).
Proof. Let γ =
(2.2)
2µη(α+β−1)+λδ
,
2µ(α+β−1)+λδ
and we define the function q by
"
!µ
#
Qαβ,p f (z)
1
q(z) =
−γ .
1−γ
Qαβ,p g(z)
Then q is analytic in U and q(0) = 1. If we set
(2.3)
h(z) =
Qαβ,p g(z)
M.K. Aouf and A.O. Mostafa
vol. 9, iss. 2, art. 45, 2008
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,
Qα−1
β,p g(z)
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then by the hypothesis (1.9), Re{h(z)} > δ. Differentiating (2.2) with respect to z
and using the identity (1.7), we have
!µ
!µ−1
α
Qα−1
f
(z)
Qαβ,p f (z)
Q
f
(z)
β,p
β,p
+ λ α−1
(2.4) (1 − λ)
α
Qβ,p g(z)
Qβ,p g(z) Qαβ,p g(z)
= [(1 − γ)q(z) + γ] +
p-Valent Meromorphic Functions
λ(1 − γ)
zq 0 (z)h(z).
µ(α + β − 1)
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Let us define the function ψ(r, s) by
(2.5)
ψ(r, s) = [(1 − γ)r + γ] +
Using (2.5) and the fact that f ∈
Pα
λ(1 − γ)
sh(z).
µ(α + β − 1)
p,β (η, δ, µ, λ),
we obtain
{ψ(q(z), zq 0 (z)); z ∈ U )} ⊂ Ω = {w ∈ C : Re(w) > η}.
Now for all real r2 , s1 ≤ −
1+r22
,
2
p-Valent Meromorphic Functions
we have
M.K. Aouf and A.O. Mostafa
λ(1 − γ)s1
Re h(z)
µ(α + β − 1)
λ(1 − γ)δ(1 + r22 )
≤γ−
2µ(α + β − 1)
λ(1 − γ)δ
≤γ−
= η.
2µ(α + β − 1)
vol. 9, iss. 2, art. 45, 2008
Re{ψ(ir2 , s1 )} = γ +
Hence for each z ∈ U, ψ(ir2 , s1 ) ∈
/ Ω. Thus by Lemma 1.2, we have Re{q(z)} >
0 (z ∈ U ) and hence
!µ
Qαβ,p f (z)
> γ (z ∈ U ).
Re
Qαβ,p g(z)
This proves Theorem 2.1.
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P
Corollary 2.2. Let the functions f and g be in p and let g satisfy the condition
(1.9). If λ ≥ 1 and
(
)
Qα−1
Qαβ,p f (z)
β,p f (z)
(2.6)
Re (1 − λ) α
+ λ α−1
> η (0 ≤ η < 1; z ∈ U ),
Qβ,p g(z)
Qβ,p g(z)
then
(
(2.7)
Re
Qα−1
β,p f (z)
)
Qα−1
β,p g(z)
>γ=
η[2(α + β − 1) + δ] + δ(λ − 1)
2(α + β − 1) + δλ
(z ∈ U ).
Proof. We have
(
)
Qα−1
Qα−1
Qαβ,p f (z)
Qαβ,p f (z)
β,p f (z)
β,p f (z)
= (1 − λ) α
+ λ α−1
+ (λ − 1) α
λ α−1
Qβ,p g(z)
Qβ,p g(z)
Qβ,p g(z)
Qβ,p g(z)
p-Valent Meromorphic Functions
(z ∈ U ).
Since λ ≥ 1, making use of (2.6) and (2.1) (for µ = 1), we deduce that
( α−1
)
Qβ,p f (z)
η[2(α + β − 1) + δ] + δ(λ − 1)
Re
>γ=
(z ∈ U ).
α−1
2(α + β − 1) + δλ
Qβ,p g(z)
Corollary 2.3. Let λ ∈ C\{0} with Re{λ} ≥ 0. If f ∈
condition:
P
p
satisfies the following
(z ∈ U ),
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then
Re{(z p Qαβ,p f (z))µ } >
Further, if λ ≥ 1 and f ∈
(2.9)
vol. 9, iss. 2, art. 45, 2008
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p α
µ−1
Re{(1 − λ)(z p Qαβ,p f (z))µ + λz p Qα−1
}>η
β,p f (z)(z Qβ,p f (z))
(2.8)
M.K. Aouf and A.O. Mostafa
P
p
2µη(α + β − 1) + Re{λ}
2µ(α + β − 1) + Re{λ}
(z ∈ U ).
satisfies
Re{(1 − λ)z p Qαβ,p f (z) + λz p Qα−1
β,p f (z)} > η
(z ∈ U ),
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then
(2.10)
Re{z p Qα−1
β,p f (z)} >
2η(α + β − 1) + λ − 1
2(α + β − 1) + λ
(z ∈ U ).
Proof. The results (2.8) and (2.10) follow by putting g(z) = z −p in Theorem 2.1
and Corollary 2.2, respectively.
Remark 1. Choosing α, λ and µ appropriately in Corollary 2.3, we have,
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
(i) For α = 0, β 6= 1 and λ = 1 in Corollary 2.3, we have:
1
zf 0 (z)
µ
p
Re
β+p−1+
(z f (z)) > η
β−1
f (z)
vol. 9, iss. 2, art. 45, 2008
(z ∈ U ),
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which implies that
2µη(β − 1) + 1
Re {z f (z)} >
2µ(β − 1) + 1
p
µ
(z ∈ U ).
(ii) For α = 0, β 6= 1, µ = 1 and λ ∈ C\{0} with Re{λ} ≥ 0 in Corollary 2.3, we
have
λp
λ p+1 0
p
Re (1 +
)z f (z) +
z f (z) > η,
β−1
β−1
which implies that
Re {z p f (z)} >
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2η(β − 1) + Re{λ}
2(β − 1) + Re{λ}
(z ∈ U ).
0
(iii) Replacing f by − zfp in the result (ii), we have:
h
i z p+1 f 0 (z)
p+1 00
λ
λ
− Re 1 + β−1 (p + 1)
+ p(β−1) z f (z) > η
p
(0 ≤ η < 1; z ∈ U ),
which implies that
p+1 0 z f (z)
2η(β − 1) + Re{λ}
>
(z ∈ U ).
− Re
p
2(β − 1) + Re{λ}
P
Theorem 2.4. Let λ ∈ C with Re{λ} > 0. If f ∈
p satisfies the following
condition:
p α
µ−1
(2.11) Re{(1−λ)(z p Qαβ,p f (z))µ +λz p Qα−1
}>η
β,p f (z)(z Qβ,p f (z))
(z ∈ U ),
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
vol. 9, iss. 2, art. 45, 2008
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then
(2.12)
Contents
Re{(z
p
Qαβ,p f (z))µ }
> η + (1 − η)(2ρ − 1),
where
(2.13)
1
µ(α + β − 1)
1
ρ = 2 F1 1, 1;
+ 1;
.
2
Re{λ}
2
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Proof. Let
q(z) = (z p Qαβ,p f (z))µ .
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Then q is analytic with q(0) = 1. Differentiating (2.14) with respect to z and using
the identity (1.7), we have
Close
(2.14)
p α
µ−1
(1 − λ)(z p Qαβ,p f (z))µ + λz p Qα−1
β,p f (z)(z Qβ,p f (z))
= q(z) +
λ
zq 0 (z),
µ(α + β − 1)
so that by the hypothesis (2.11), we have
λ
0
Re q(z) +
zq (z) > η
µ(α + β − 1)
(z ∈ U ).
In view of Lemma 1.3, this implies that
Re{q(z)} > η + (1 − η)(2ρ − 1),
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
where
Z
1
ρ = ρ(Re{λ}) =
1+t
Re{λ}
µ(α+β−1)
−1
vol. 9, iss. 2, art. 45, 2008
dt.
0
Putting Re{λ} = λ1 > 0, we have
Z 1
Z
−1
λ1
µ(α+β−1)
µ(α + β − 1) 1
−1
µ(α+β−1))
ρ=
1+t
dt =
(1 + u)−1 u λ1
du
λ1
0
0
Using (1.11), (1.12), (1.13) and (1.14), we obtain
1
µ(α + β − 1)
1
ρ=
+ 1;
.
2 F1 1, 1;
2
λ1
2
Re{z
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Corollary 2.5. Let λ ∈ R with λ ≥ 1. If f ∈ p satisfies
(2.15)
Re (1 − λ)z p Qαβ,p f (z) + λz p Qα−1
f
(z)
>η
β,p
Qα−1
β,p f (z)}
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p
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This completes the proof of Theorem 2.1.
then
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1
> η + (1 − η)(2ρ1 − 1) 1 −
λ
Close
(z ∈ U ),
(z ∈ U ),
where
1
ρ1 =
2
2 F1
(α + β − 1)
1
1, 1;
+ 1;
λ
2
.
Proof. The result follows by using the identity
p α
p α−1
p α
(2.16) λz p Qα−1
β,p f (z) = (1 − λ)z Qβ,p f (z) + λz Qβ,p f (z) + (λ − 1)z Qβ,p f (z).
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
Remark 2. We note that, for α = 0, β = 2 and λ = µ > 0 in Corollary 2.3, that is, if
(2.17)
Re (1 − λ)(z p f (z))λ + λ(z p+1 f (z))0 (z p f (z))λ−1 > η
(z ∈ U ),
vol. 9, iss. 2, art. 45, 2008
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then (2.8) implies that
Re{(z p f (z))λ } >
(2.18)
2η + 1
3
(z ∈ U ),
P
whereas, if f ∈ p satisfies the condition (2.17) then by using Theorem 2.4, we
have
Re{(z p f (z))λ } > 2(1 − ln 2)η + (2 ln 2 − 1) (z ∈ U ),
which is better than (2.18).
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Theorem 2.6. Suppose that the functions f and g are in p and g satisfies the
condition (1.9). If
( α−1
)
Qβ,p f (z) Qαβ,p f (z)
(1 − η)δ
>−
(z ∈ U ),
(2.19)
Re
− α
α−1
Qβ,p g(z)
2(α + β − 1)
Qβ,p g(z)
Close
for some η (0 ≤ η < 1), then
(
Re
(2.20)
Qαβ,p f (z)
Qαβ,p g(z)
)
>η
(z ∈ U )
and
(
(2.21)
Re
Qα−1
β,p f (z)
Qα−1
β,p g(z)
)
>
η[2(α + β − 1) + δ] − δ
2(α + β − 1)
(z ∈ U ).
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
vol. 9, iss. 2, art. 45, 2008
Proof. Let
(2.22)
"
#
Qαβ,p f (z)
1
q(z) =
−η .
1 − η Qαβ,p g(z)
Then q is analytic in U with q(0) = 1. Setting
(2.23)
φ(z) =
Qαβ,p g(z)
Qα−1
β,p g(z)
(z ∈ U ),
we observe that from (1.9), we have Re{φ(z)} > δ (0 ≤ δ < 1) in U . A simple
computation shows that
Qα−1
Qαβ,p f (z)
(1 − η)zq 0 (z)
β,p f (z)
φ(z) = α−1
− α
α+β−1
Qβ,p g(z)
Qβ,p g(z)
= ψ(q(z), zq 0 (z)),
where
ψ(r, s) =
(1 − η)sφ(z)
.
α+β−1
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Using the hypothesis (2.19), we obtain
{ψ(q(z), zq (z)); z ∈ U } ⊂ Ω = w ∈ C : Re w > −
0
Now, for all real r2 , s1 ≤ −
1+r22
,
2
(1 − η)δ
2(α + β − 1)
.
we have
s1 (1 − η) Re{φ(z)}
α+β−1
−(1 − η)δ(1 + r22 )
≤
2(α + β − 1)
−(1 − η)δ
≤
.
2(α + β − 1)
p-Valent Meromorphic Functions
Re {ψ(ir2 , s1 )} =
M.K. Aouf and A.O. Mostafa
vol. 9, iss. 2, art. 45, 2008
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This shows that ψ(ir2 , s1 ) ∈
/ Ω for each z ∈ U. Hence by Lemma 1.2, we have
Re{q(z)} > 0 (z ∈ U ). This proves (2.20). The proof of (2.21) follows by using
(2.20) and (2.21) in the identity:
( α−1
)
( α−1
)
(
)
Qβ,p f (z)
Qβ,p f (z) Qαβ,p f (z)
Qαβ,p f (z)
Re
= Re
− α
− Re
.
Qβ,p g(z)
Qαβ,p g(z)
Qα−1
Qα−1
β,p g(z)
β,p g(z)
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This completes the proof of Theorem 2.6.
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Remark 3.
(i) For α = 0 and g(z) = z −p in Theorem 2.6, we have
−(1 − η)δ
Re z p+1 f 0 (z) + pz p f (z) >
2
(z ∈ U ),
which implies that
Re {z p f (z)} > η
(z ∈ U )
and
η[2(β − 1) + δ] − δ
Re z p+1 f 0 (z) + (p + β − 1)z p f (z) >
2
(ii) Putting α = 0, β = 2 in Theorem 2.6, we get that, if
0
zf (z) + (p + 1)f (z) f (z)
−(1 − η)δ
Re
−
>
0
zg (z) + (p + 1)g(z)
g(z)
2
(z ∈ U ).
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
(z ∈ U ),
vol. 9, iss. 2, art. 45, 2008
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then
Re
and
Re
f (z)
g(z)
zf 0 (z) + (p + 1)f (z)
zg 0 (z) + (p + 1)g(z)
>η
(z ∈ U )
η(2 + δ) − δ
>
2
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(z ∈ U ).
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References
[1] E. AQLAN, J.M. JAHANGIRI AND S.R. KULKARNI, Certain integral operators applied to meromorphic p-valent functions, J. of Nat. Geom., 24 (2003),
111–120.
[2] S.S. MILLER AND P.T. MOCANU, Second order differential equations in the
complex plane, J. Math. Anal. Appl., 65 (1978), 289–305.
[3] S. PONNUSAMY, Differential subordination and Bazilevic functions, Proc. Indian Acad. Sci. (Math. Sci.), 105 (1995), 169–186.
[4] E.T. WHITTAKER AND G.N. WATSON, A Course of Modern Analysis: An
Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Fourth Edition, Cambridge University Press, Cambridge, 1927.
p-Valent Meromorphic Functions
M.K. Aouf and A.O. Mostafa
vol. 9, iss. 2, art. 45, 2008
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