Volume 10 (2009), Issue 2, Article 43, 7 pp.
HADAMARD PRODUCT OF CERTAIN MEROMORPHIC p−VALENT STARLIKE
AND p−VALENT CONVEX FUNCTIONS
M. K. AOUF
D EPARTMENT OF M ATHEMATICS ,
FACULTY OF S CIENCE , M ANSOURA U NIVERSITY,
M ANSOURA 35516, E GYPT.
mkaouf127@yahoo.com
Received 12 April, 2008; accepted 30 April, 2009
Communicated by G. Kohr
A BSTRACT. 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).
Key words and phrases: Meromorphic, p-valent, Hadamard product.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION
Throughout this paper, let p ∈ N = {1, 2, . . . } and let the functions of the form:
∞
X
p
ϕ(z) = cp z −
cp+n z p+n
(cp > 0; cp+n ≥ 0),
Ψ(z) = dp z p −
n=1
∞
X
dp+n z p+n
(dp > 0; dp+n ≥ 0)
n=1
be regular and p−valent in the unit disc U = {z : |z| < 1}. Also, let
∞
ap−1 X
ap+n−1 z p+n−1
(ap−1 > 0; ap+n−1 ≥ 0),
(1.1)
f (z) = p +
z
n=1
∞
fi (z) =
ap−1,i X
+
ap+n−1,i z p+n−1
zp
n=1
(ap−1,i > 0; ap+n−1,i ≥ 0),
∞
bp−1 X
g(z) = p +
bp+n−1 z p+n−1
z
n=1
The author is thankful to the referee for his comments and suggestions.
117-08
(bp−1 > 0; bp+n−1 ≥ 0)
2
M. K. AOUF
and
∞
bp−1,j X
gj (z) =
+
bp+n−1,j z p+n−1
p
z
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
0 ϕ(z)
<β
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) be the
0
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) be
0
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 + 2p − 1) + β(n + 2α − 1)] an+p−1 } ≤ 2β(p − α)ap−1 .
n=1
The result is sharp for the function f (z) given by
f (z) =
1
2β(p − α)ap−1
+
z p+n−1
p
z
(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
< β (z ∈ U ).
P
(1.4)
∞
2(p − α)ap−1 −
2p+n−1 n=1 (n + 2α − 1)ap+n−1 z
Since |Re(z)| ≤ |z| (z ∈ C), choosing z to be real and letting z → 1− through real values,
(1.4) yields
∞
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).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 43, 7 pp.
http://jipam.vu.edu.au/
H ADAMARD P RODUCT
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)
(p − α)ap−1 − P∞ (n + 2α − 1)ap+n−1
zf (z)
f (z) + 2α − p n=1
<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).
Also we can prove that f (z) ∈ ΣC0∗ (p, α, β) if and only if
∞ X
p+n−1
(1.5)
[(n + 2p − 1) + β(n + 2α − 1)] an+p−1 ≤ 2β(p − α)ap−1 .
p
n=1
The result is sharp for the function f (z) given by
f (z) =
2β(p − α)ap−1
1
+
z p+n−1
p
n+2p−1
z
[(n + 2p − 1) + β(n + 2α − 1)]
(p, n ∈ N) .
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
∞
X
p
ϕ ∗ Ψ(z) = cp dp z −
cp+n dp+n z p+n .
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.
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 ≤ 2β(p − α)ap−1 .
p
n=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, Σ∗k (p, α, β) ⊂
∗
Σh (p, α, β) if k > h ≥ 0, the containment being proper. Moreover, for any positive integer k,
we have the following inclusion relation
Σ∗k (p, α, β) ⊂ Σ∗k−1 (p, α, β) ⊂ · · · ⊂ Σ∗2 (p, α, β) ⊂ ΣC0∗ (p, α, β) ⊂ ΣS0∗ (p, α, β) .
We also note that for every nonnegative real number k, the class Σ∗k (p, α, β) is nonempty as the
functions
−k ∞ ap−1 X p + n − 1
2β(p − α)
f (z) = p +
ap−1 λp+n−1 z p+n−1 ,
z
p
(n + 2p − 1) + β(n + 2α − 1)
n=1
P
where 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).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 43, 7 pp.
http://jipam.vu.edu.au/
4
M. K. AOUF
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].
2. T HE M AIN T HEOREMS
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
"
≤ 2β(p − α)
m
Y
#
ap−1,i .
i=1
Since fi (z) ∈ ΣC0∗ (p, α, β), we have
∞ X
p+n−1
(2.1)
[(n + 2p − 1) + β(n + 2α − 1)] ap+n−1,i ≤ 2β(p − α)ap−1,i ,
p
n=1
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
or

ap+n−1,i ≤  
2β(p − α)
p+n−1
p
[(n + 2p − 1) + β(n + 2α − 1)]
 ap−1,i ,
for every i = 1, 2, . . . , m. The right-hand expression of the last inequality is not greater than
−2
p+n−1
ap−1,i . Hence
p
(2.2)
ap+n−1,i ≤
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
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 43, 7 pp.
http://jipam.vu.edu.au/
H ADAMARD P RODUCT
=
"m−1
Y
#
ap−1,i
i=1
≤ 2β(p − α)
∞ X
p+n−1
p
n=1
"m
Y
5
[(n + 2p − 1) + β(n + 2α − 1)] an+p−1,m
#
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).
Proof. Since fi (z) ∈ ΣS0∗ (p, α, β), we have
(2.3)
∞
X
{[(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,
ap+n−1,i ≤
2β(p − α)
[(n + 2p − 1) + β(n + 2α − 1)]
ap−1,i ,
and hence
ap+n−1,i ≤
(2.4)
p+n−1
p
−1
ap−1,i ,
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
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).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 43, 7 pp.
http://jipam.vu.edu.au/
6
M. K. AOUF
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
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
∞
X
(2.5)
{[(n + 2p − 1) + β(n + 2α − 1)] bp+n−1,j } ≤ 2β(p − α)bp−1,j ,
n=1
for every j = 1, 2, . . . , q. Whence we obtain
−1
p+n−1
bp−1,j ,
(2.6)
bp+n−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
∞
X
j=1
(
2m+q−1
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)
n=1
j=1
m
Y
i=1
ap−1,i
q
Y
!
bp−1,j
.
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].
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 43, 7 pp.
http://jipam.vu.edu.au/
H ADAMARD P RODUCT
7
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J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 43, 7 pp.
http://jipam.vu.edu.au/