HADAMARD PRODUCT OF CERTAIN
MEROMORPHIC p−VALENT STARLIKE AND
p−VALENT CONVEX FUNCTIONS
Hadamard Product
M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
M. K. AOUF
Department of Mathematics,
Faculty of Science,
Mansoura University,
Mansoura 35516, Egypt.
EMail: mkaouf127@yahoo.com
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Received:
12 April, 2008
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Accepted:
30 April, 2009
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Communicated by:
G. Kohr
2000 AMS Sub. Class.:
30C45.
Key words:
Meromorphic, p-valent, Hadamard product.
Abstract:
In this paper, we establish some results concerning the Hadamard product of certain meromorphic p-valent starlike and meromorphic p-valent convex functions
analogous to those obtained by Vinod Kumar (J. Math. Anal. Appl. 113(1986),
230-234) and M. L. Mogra (Tamkang J. Math. 25(1994), no. 2, 157-162).
Acknowledgements:
The author is thankful to the referee for his comments and suggestions.
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Contents
1
Introduction
3
2
The Main Theorems
8
Hadamard Product
M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
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1.
Introduction
Throughout this paper, let p ∈ N = {1, 2, . . . } and let the functions of the form:
p
ϕ(z) = cp z −
Ψ(z) = dp z p −
∞
X
n=1
∞
X
cp+n z p+n
(cp > 0; cp+n ≥ 0),
dp+n z p+n
(dp > 0; dp+n ≥ 0)
n=1
Hadamard Product
M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
be regular and p−valent in the unit disc U = {z : |z| < 1}. Also, let
∞
(1.1)
f (z) =
fi (z) =
ap−1 X
+
ap+n−1 z p+n−1
zp
n=1
∞
X
ap−1,i
+
ap+n−1,i z p+n−1
zp
n=1
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(ap−1 > 0; ap+n−1 ≥ 0),
(ap−1,i > 0; ap+n−1,i ≥ 0),
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∞
bp−1 X
g(z) = p +
bp+n−1 z p+n−1
z
n=1
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(bp−1 > 0; bp+n−1 ≥ 0)
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and
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∞
bp−1,j X
gj (z) =
+
bp+n−1,j z p+n−1
zp
n=1
(bp−1,j > 0; bp+n−1,j ≥ 0).
be regular and p−valent in the punctured disc D = {z : 0 < |z| < 1}.
Let S0∗ (p, α, β) denote the class of functions ϕ(z) which satisfy the condition
zϕ0 (z)
−
p
ϕ(z)
<β
0
zϕ (z)
ϕ(z) + p − 2α for some α, β (0 ≤ α < p, 0 < β ≤ 1, p ∈ N) and for all z ∈ U ; and let C0 (p, α, β)
0
be the class of functions ϕ(z) for which zf p(z) ∈ S0∗ (p, α, β). It is well known that
the functions in S0∗ (p, α, β) and C0∗ (p, α, β) are, respectively, p−valent starlike and
p−valent convex functions of order α and type β with negative coefficients in U (see
Aouf [1]).
Denote by ΣS0∗ (p, α, β), the class of functions f (z) which satisfy the condition
zf 0 (z)
+
p
f
(z)
0
<β
(1.2)
zf (z)
f (z) + 2α − p for some α, β (0 ≤ α < p, 0 < β ≤ 1, p ∈ N) and for all z ∈ D, and ΣC0∗ (p, α, β)
0
be the class of functions f (z) for which −zfp (z) ∈ ΣS0∗ (p, α, β). The functions
in ΣS0∗ (p, α, β) and ΣC0∗ (p, α, β) are, respectively, called meromorphic p−valent
starlike and meromorphic p−valent convex functions of order α and type β with
positive coefficients in D. The class ΣS0∗ (p, α, β) with ap−1 = 1 has been studied by
Aouf [2] and Mogra [9].
Using similar arguments as given in ([2] and [9]), we can prove the following
result for functions in ΣS0∗ (p, α, β).
A function f (z) ∈ ΣS0∗ (p, α, β) if and only if
(1.3)
∞
X
n=1
{[(n + 2p − 1) + β(n + 2α − 1)] an+p−1 } ≤ 2β(p − α)ap−1 .
Hadamard Product
M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
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The result is sharp for the function f (z) given by
f (z) =
1
2β(p − α)ap−1
+
z p+n−1
z p (n + 2p − 1) + β(n + 2α − 1)
(p, n ∈ N).
Proof Outline. Let f (z) ∈ ΣS0∗ (p, α, β) be given by (1.1). Then, from (1.2) and
(1.1), we have
P∞
2p+n−1
(n
+
2p
−
1)a
z
p+n−1
n=1
(1.4)
2(p − α)ap−1 − P∞ (n + 2α − 1)ap+n−1 z 2p+n−1 < β (z ∈ U ).
n=1
vol. 10, iss. 2, art. 43, 2009
Since |Re(z)| ≤ |z| (z ∈ C), choosing z to be real and letting z → 1− through real
values, (1.4) yields
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∞
X
(n + 2p − 1)ap+n−1 ≤ 2β(p − α)ap−1 −
n=1
∞
X
β(n + 2α − 1)ap+n−1 ,
n=1
which leads us to (1.3).
In order to prove the converse, we assume that the inequality (1.3) holds true.
Then, if we let z ∈ ∂U , we find from (1.1) and (1.3) that
zf 0 (z)
P∞
+
p
n=1 (n + 2p − 1)ap+n−1
0 f (z)
≤
zf (z)
(p − α)ap−1 − P∞ (n + 2α − 1)ap+n−1
n=1
+ 2α − p f (z)
<β
(z ∈ ∂U = {z : z ∈ C and |z| = 1}) .
Hence, by the maximum modulus theorem, we have f (z) ∈ ΣS0∗ (p, α, β). This
completes the proof of (1.3).
Hadamard Product
M. K. Aouf
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Also we can prove that f (z) ∈ ΣC0∗ (p, α, β) if and only if
(1.5)
∞ X
p+n−1
n=1
p
[(n + 2p − 1) + β(n + 2α − 1)] an+p−1
≤ 2β(p − α)ap−1 .
The result is sharp for the function f (z) given by
Hadamard Product
2β(p − α)ap−1
1
z p+n−1
f (z) = p + n+2p−1
z
[(n + 2p − 1) + β(n + 2α − 1)]
M. K. Aouf
(p, n ∈ N) .
vol. 10, iss. 2, art. 43, 2009
p
The quasi-Hadamard product of two or more functions has recently been defined and
used by Owa ([11], [12] and [13]), Kumar ([6], [7] and [8]), Aouf et al. [3], Hossen
[5], Darwish [4] and Sekine [14]. Accordingly, the quasi-Hadamard product of two
functions ϕ(z) and Ψ(z) is defined by
p
ϕ ∗ Ψ(z) = cp dp z −
∞
X
cp+n dp+n z
p+n
.
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n=1
Let us define the Hadamard product of two functions f (z) and g(z) by
∞
ap−1 bp−1 X
f ∗ g(z) =
+
ap+n−1 bp+n−1 z p+n−1 .
zp
n=1
Similarly, we can define the Hadamard product of more than two meromorphic
p−valent functions.
We now introduce the following class of meromorphic p−valent functions in D.
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A function f (z) ∈ Σ∗k (p, α, β) if and only if
(
)
k
∞
X
p+n−1
(1.6)
[(n + 2p − 1) + β(n + 2α − 1)] an+p−1
p
n=1
≤ 2β(p − α)ap−1 .
where 0 ≤ α < p, 0 < β ≤ 1, p ∈ N, and k is any fixed nonnegative real number.
Evidently, Σ∗0 (p, α, β) ≡ ΣS0∗ (p, α, β) and Σ∗1 (p, α, β) ≡ ΣC0∗ (p, α, β). Further,
∗
Σk (p, α, β) ⊂ Σ∗h (p, α, β) if k > h ≥ 0, the containment being proper. Moreover,
for any positive integer k, we have the following inclusion relation
Hadamard Product
M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
Σ∗k (p, α, β) ⊂ Σ∗k−1 (p, α, β) ⊂ · · · ⊂ Σ∗2 (p, α, β) ⊂ ΣC0∗ (p, α, β) ⊂ ΣS0∗ (p, α, β) .
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We also note that for every nonnegative real number k, the class Σ∗k (p, α, β) is
nonempty as the functions
Contents
∞
ap−1 X
f (z) = p +
z
n=1
−k
p+n−1
p
2β(p − α)
×
ap−1 λp+n−1 z p+n−1 ,
(n + 2p − 1) + β(n + 2α − 1)
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where
P∞ ap−1 > 0, 0 ≤ α < p, 0 < β ≤ 1, p ∈ N, ap−1 > 0, λp+n−1 ≥ 0 and
n=1 λp+n−1 ≤ 1, satisfy the inequality (1.1).
In this paper we establish certain results concerning the Hadamard product of
meromorphic p−valent starlike and meromorphic p−valent convex functions of order α and type β analogous to Kumar [7] and Mogra [10].
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2.
The Main Theorems
Theorem 2.1. Let the functions fi (z) belong to the class ΣC0∗ (p, α, β) for every
i = 1, 2, . . . , m. Then the Hadamard product f1 ∗ f2 ∗ · · · ∗ fm (z) belongs to the
class Σ∗2m−1 (p, α, β).
Proof. It is sufficient to show that
)
(
2m−1
m
∞
X
Y
p+n−1
[(n + 2p − 1) + β(n + 2α − 1)]
ap+n−1,i
p
n=1
i=1
"m
#
Y
≤ 2β(p − α)
ap−1,i .
Hadamard Product
M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
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i=1
Contents
Since fi (z) ∈ ΣC0∗ (p, α, β), we have
(2.1)
∞ X
n=1
p+n−1
p
[(n + 2p − 1) + β(n + 2α − 1)] ap+n−1,i
for i = 1, 2, . . . , m. Therefore,
p+n−1
[(n + 2p − 1) + β(n + 2α − 1)] ap+n−1,i ≤ 2β(p − α)ap−1,i
p

ap+n−1,i ≤  2β(p − α)
p+n−1
p
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≤ 2β(p − α)ap−1,i ,
or
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[(n + 2p − 1) + β(n + 2α − 1)]

 ap−1,i ,
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for every i = 1, 2, . . . , m. The right-hand expression of the last inequality is not
−2
greater than p+n−1
ap−1,i . Hence
p
ap+n−1,i ≤
(2.2)
p+n−1
p
−2
ap−1,i ,
for every i = 1, 2, . . . , m.
Using (2.2) for i = 1, 2, . . . , m − 1, and (2.1) for i = m, we obtain
(
)
2m−1
∞
m
X
Y
p+n−1
[(n + 2p − 1) + β(n + 2α − 1)]
ap+n−1,i
p
n=1
i=1
(
2m−1
∞
X
p+n−1
≤
[(n + 2p − 1) + β(n + 2α − 1)]
p
n=1
!
)
−2(m−1) m−1
Y
p+n−1
·
ap−1,i ap+n−1,m
×
p
i=1
"m−1
# ∞ X
Y
p+n−1
=
ap−1,i
[(n + 2p − 1) + β(n + 2α − 1)] an+p−1,m
p
n=1
i=1
"m
#
Y
≤ 2β(p − α)
ap−1,i .
i=1
Hence f1 ∗ f2 ∗ · · · ∗ fm (z) ∈ Σ∗2m−1 (p, α, β).
Theorem 2.2. Let the functions fi (z) belong to the class ΣS0∗ (p, α, β) for every
i = 1, 2, . . . , m. Then the Hadamard product f1 ∗ f2 ∗ · · · ∗ fm (z) belongs to the
class Σ∗m−1 (p, α, β).
Hadamard Product
M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
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Proof. Since fi (z) ∈ ΣS0∗ (p, α, β), we have
∞
X
(2.3)
{[(n + 2p − 1) + β(n + 2α − 1)] ap+n−1,i } ≤ 2β(p − α)ap−1,i ,
n=1
for i = 1, 2, . . . , m. Therefore,
2β(p − α)
ap+n−1,i ≤
ap−1,i ,
[(n + 2p − 1) + β(n + 2α − 1)]
and hence
−1
p+n−1
(2.4)
ap+n−1,i ≤
ap−1,i ,
p
for every i = 1, 2, . . . , m.
Using (2.4) for i = 1, 2, . . . , m − 1, and (2.3) for i = m, we get
(
)
m−1
m
∞
X
Y
p+n−1
[(n + 2p − 1) + β(n + 2α − 1)]
ap+n−1,i
p
n=1
i=1
(
∞
m−1
X
p+n−1
≤
[(n + 2p − 1) + β(n + 2α − 1)]
p
n=1
!
)
−(m−1) m−1
Y
p+n−1
×
·
ap−1,i ap+n−1,m
p
i=1
"m−1
# ∞
Y
X
=
ap−1,i
{[(n + 2p − 1) + β(n + 2α − 1)] an+p−1,m }
i=1
"
≤ 2β(p − α)
n=1
m
Y
#
ap−1,i .
i=1
Hadamard Product
M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
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Hence f1 ∗ f2 ∗ · · · ∗ fm (z) ∈ Σ∗m−1 (p, α, β).
Theorem 2.3. Let the functions fi (z) belong to the class ΣC0∗ (p, α, β) for every
i = 1, 2, . . . , m, and let the functions gj (z) belong to the class ΣS0∗ (p, α, β) for every
j = 1, 2, . . . , q. Then the Hadamard product f1 ∗ f2 ∗ · · · ∗ fm ∗ g1 ∗ g2 ∗ · · · ∗ gq (z)
belongs to the class Σ∗2m+q−1 (p, α, β).
Proof. It is sufficient to show that
(
2m+q−1
∞
X
p+n−1
[(n + 2p − 1) + β(n + 2α − 1)]
p
n=1
!)
q
m
Y
Y
×
ap+n−1,i ·
bp+n−1,i
i=1
Hadamard Product
M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
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i=1
≤ 2β(p − α)
m
Y
i=1
ap−1,i
q
Y
!
bp−1,i .
i=1
Since fi (z) ∈ ΣC0∗ (p, α, β), the inequalities (2.1) and (2.2) hold for every i =
1, 2, . . . , m. Further, since gj (z) ∈ ΣS0∗ (p, α, β), we have
(2.5)
∞
X
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{[(n + 2p − 1) + β(n + 2α − 1)] bp+n−1,j } ≤ 2β(p − α)bp−1,j ,
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for every j = 1, 2, . . . , q. Whence we obtain
−1
p+n−1
(2.6)
bp+n−1,j ≤
bp−1,j ,
p
for every j = 1, 2, . . . , q.
Using (2.2) for i = 1, 2, . . . , m, (2.6) for j = 1, 2, . . . , q − 1, and (2.5) for j = q,
we get
(
2m+q−1
∞
X
p+n−1
[(n + 2p − 1) + β(n + 2α − 1)]
p
n=1
!)
q
m
Y
Y
×
ap+n−1,i ·
bp+n−1,j
i=1
M. K. Aouf
(
2m+q−1
∞
X
p+n−1
≤
[(n + 2p − 1) + β(n + 2α − 1)]
p
n=1
!
)
−2m −(q−1) Y
q−1
m
Y
p+n−1
p+n−1
ap−1,i
×
bp−1,j bp+n−1,q
p
p
i=1
j=1
! ∞
q−1
m
Y
Y
X
=
ap−1,i
bp−1,j
{[(n + 2p − 1) + β(n + 2α − 1)] bp+n−1,q }
i=1
≤ 2β(p − α)
Hadamard Product
j=1
n=1
j=1
m
Y
i=1
ap−1,i
q
Y
!
bp−1,j
vol. 10, iss. 2, art. 43, 2009
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.
j=1
Hence f1 ∗ f2 ∗ · · · ∗ fm ∗ g1 ∗ g2 ∗ · · · ∗ gq (z) ∈ Σ∗2m+q−1 (p, α, β).
We note that the required estimate can also be obtained by using (2.2) for i =
1, 2, . . . , m − 1, (2.6) for j = 1, 2, . . . , q, and (2.1) for i = m.
Remark 1. Putting p = 1 in the above results, we obtain the results obtained by
Mogra [10].
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References
[1] M.K. AOUF, p−Valent regular functions with negative coefficients of order α,
Bull. Inst. Math. Acad. Sinica, 17(3) (1989), 255–267.
[2] M.K. AOUF, Certain classes of meronorphic multivalent functions with positive coefficients, Math. Comput. Modelling, 47 (2008), 328–340.
[3] M.K. AOUF, A. SHAMANDY AND M.F. YASSEN, Quasi-Hadamard product
of p−valent functions, Commun. Fac. Sci. Univ. Ank. Series A1 , 44 (1995),
35–40.
[4] H.E. DARWISH, The quasi-Hadamard product of certain starlike and convex
functions, Applied Math. Letters, 20 (2007), 692–695.
[5] H.M. HOSSEN, Quasi-Hadamard product of certain p-valent functions,
Demonstratio Math., 33(2) (2000), 277–281.
[6] V. KUMAR, Hadamard product of certain starlike functions, J. Math. Anal.
Appl., 110 (1985), 425–428.
[7] V. KUMAR, Hadamard product of certain starlike functions II, J. Math. Anal.
Appl., 113 (1986), 230–234.
[8] V. KUMAR, Quasi-Hadamard product of certain univalent functions, J. Math.
Anal. Appl., 126 (1987), 70–77.
[9] M.L. MOGRA, Meromorphic multivalent functions with positive coefficients.
I, Math. Japon. 35(1) (1990), 1–11.
[10] M.L. MOGRA, Hadamard product of certain meromorphic starlike and convex
functions, Tamkang J. Math., 25(2) (1994), 157–162.
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M. K. Aouf
vol. 10, iss. 2, art. 43, 2009
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[11] S. OWA, On the classes of univalent functions with negative coefficients, Math.
Japon., 27(4) (1982), 409–416.
[12] S. OWA, On the starlike functions of order α and type β, Math. Japon., 27(6)
(1982), 723–735.
[13] S. OWA, On the Hadamard products of univalent functions, Tamkang J. Math.,
14 (1983), 15–21.
[14] T. SEKINE, On quasi-Hadamard products of p−valent functions with negative
coefficients in: H. M. Srivastava and S. Owa (Editors), Univalent Functions,
Fractional Calculus, and Their Applications, Halsted Press (Ellis Horwood
Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane
and Toronto, 1989, 317-328.
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vol. 10, iss. 2, art. 43, 2009
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