AN APPLICATION OF HÖLDER’S INEQUALITY
FOR CONVOLUTIONS
JUNICHI NISHIWAKI AND SHIGEYOSHI OWA
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577 - 8502
Japan
EMail: jerjun2002@yahoo.co.jp
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
Title Page
owa@math.kindai.ac.jp
Received:
30 March, 2009
Accepted:
16 July, 2009
Communicated by:
N.E. Cho
2000 AMS Sub. Class.:
30C45.
Key words:
Analytic function, Multivalent starlike, Multivalent convex.
Abstract:
Let Ap (n) be the class of analytic and multivalent functions f (z) in the open
unit disk U. Furthermore, let Sp (n, α) and Tp (n, α) be the subclasses of Ap (n)
consisting of multivalent starlike functions f (z) of order α and multivalent convex functions f (z) of order α, respectively. Using the coefficient inequalities
for f (z) to be in Sp (n, α) and Tp (n, α), new subclasses Sp∗ (n, α) and Tp∗ (n, α)
are introduced. Applying the Hölder inequality, some interesting properties of
generalizations of convolutions (or Hadamard products) for functions f (z) in the
classes Sp∗ (n, α) and Tp∗ (n, α) are considered.
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1
Introduction
3
2
Convolution Properties for the Classes Sp∗ (n, α) and Tp∗ (n, α)
5
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
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1.
Introduction
Let Ap (n) be the class of functions f (z) of the form
f (z) = z p +
∞
X
ak z k
k=p+n
which are analytic in the open unit disk U = {z ∈ C||z| < 1} for some natural
numbers p and n. Let Sp (n, α) be the subclass of Ap (n) consisting of functions f (z)
which satisfy
0 zf (z)
Re
>α
(z ∈ U)
f (z)
for some α (0 5 α < p). Also let Tp (n, α) be the subclass of Ap (n) consisting of
functions f (z) satisfying zf 0 (z)/p ∈ Sp (n, α), that is,
zf 00 (z)
Re 1 + 0
>α
(z ∈ U)
f (z)
for some α (0 5 α < p). These classes, Ap (n), Sp (n, α) and Tp (n, α), were studied
by Owa [3]. It is easy to derive the following lemmas, which provide the sufficient
conditions for functions f (z) ∈ Ap (n) to be in the classes Sp (n, α) and Tp (n, α),
respectively.
Lemma 1.1. If f (z) ∈ Ap (n) satisfies
(1.1)
∞
X
(k − α)|ak | 5 p − α
k=p+n
for some α (0 5 α < p), then f (z) ∈ Sp (n, α).
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
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Lemma 1.2. If f (z) ∈ Ap (n) satisfies
(1.2)
∞
X
k(k − α)|ak | 5 p(p − α)
k=p+n
for some α (0 5 α < p), then f (z) ∈ Tp (n, α).
Remark 1. We note that Silverman [4] has given Lemma 1.1 and Lemma 1.2 in the
case of p = 1 and n = 1. Also, Srivastava, Owa and Chatterjea [5] have given the
coefficient inequalities in the case of p = 1.
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
Sp∗ (n, α)
In view of Lemma 1.1 and Lemma 1.2, we introduce the subclass
consisting of functions f (z) which satisfy the coefficient inequality (1.1), and the subclass Tp∗ (n, α) consisting of functions f (z) which satisfy the coefficient inequality
(1.2).
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2.
Convolution Properties for the Classes Sp∗ (n, α) and Tp∗ (n, α)
For functions fj (z) ∈ Ap (n) given by
∞
X
fj (z) = z p +
ak,j z k
(j = 1, 2, . . . , m),
k=p+n
we define
Gm (z) = z p +
and
Hm (z) = z p +
∞
X
m
Y
k=p+n
j=1
∞
X
m
Y
k=p+n
j=1
!
ak,j
zk
vol. 10, iss. 4, art. 98, 2009
!
pj
ak,j
zk
(pj > 0).
k=p+n
j=1
j=1
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Then Gm (z) denotes the convolution of fj (z) (j = 1, 2, . . . , m). Therefore, Hm (z)
is the generalization of the convolutions. In the case of pj = 1, we have Gm (z) =
Hm (z). The generalization of the convolution was considered by Choi, Kim and
Owa [1].
In the present paper, we discuss an application of the Hölder inequality for Hm (z)
to be in the classes Sp∗ (n, α) and Tp∗ (n, α).
For fj (z) ∈ Ap (n), the Hölder inequality is given by
!
! p1
∞
m
m
∞
j
X
Y
Y
X
|ak,j | 5
|ak,j |pj
,
Pm
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
k=p+n
where pj > 1 and j=1 p1j = 1.
Recently, Nishiwaki, Owa and Srivastava [2] have given some results of Höldertype inequalities for a subclass of uniformly starlike functions.
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Theorem 2.1. If fj (z) ∈ Sp∗ (n, αj ) for each j = 1, 2, . . . , m, then Hm (z) ∈ Sp∗ (n, β)
with
)
(
Q
pj
(k − p) m
j=1 (p − αj )
Qm
,
β = inf
p − Qm
pj −
pj
(k
−
α
)
k=p+n
j
j=1
j=1 (p − αj )
P
1
where pj = q1j , qj > 1 and m
j=1 qj = 1.
Proof. For fj (z) ∈ Sp∗ (n, αj ), Lemma 1.1 gives us that
∞ X
k − αj
|ak,j | 5 1
(j = 1, 2, . . . , m),
p − αj
k=p+n
which implies
) q1
∞ j
X
k − αj
51
|ak,j |
p
−
α
j
k=p+n
(
with qj > 1 and
Pm
1
j=1 qj
= 1. Applying the Hölder inequality, we have:
(m )
1
∞
X
Y k − αj qj
1
|ak,j | qj 5 1.
p
−
α
j
k=p+n j=1
Note that we have to find the largest β such that
!
Y
m
∞ X
k−β
|ak,j |pj 5 1,
p
−
β
j=1
k=p+n
that is,
!
(m )
1
Y
∞ m
∞
X
X
Y k − αj qj
1
k−β
|ak,j | qj .
|ak,j |pj 5
p
−
β
p
−
α
j
j=1
k=p+n
k=p+n j=1
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
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Therefore, we need to find the largest β such that
!
Y
1
m
m Y
1
k − αj qj
k−β
pj
|ak,j |
5
|ak,j | qj ,
p−β
p − αj
j=1
j=1
which is equivalent to
!
Y
q1
m
m Y
j
k
−
α
k−β
pj − q1
j
j
|ak,j |
5
p−β
p − αj
j=1
j=1
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
for all k = p + n. Since
p − 1
m Y
k − αj j qj
pj − 1
|ak,j | qj 5 1
p − αj
j=1
we see that
m
Y
j=1
pj − q1
j
|ak,j |
5
1
pj −
=0 ,
qj
1
.
Qm k−αj pj − q1j
j=1
m
k − αj
p − αj
This completes the proof of the theorem.
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pj
for all k = p + n. Therefore, β should be
Q
pj
(k − p) m
j=1 (p − αj )
Q
Q
β 5p− m
m
pj
pj
j=1 (k − αj ) −
j=1 (p − αj )
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p−αj
This implies that
k−β Y
5
p−β
j=1
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(k = p + n).
Taking pj = 1 in Theorem 2.1, we obtain
Corollary 2.2. If fj (z) ∈ Sp∗ (n, αj ) for each j = 1, 2, . . . , m, then Gm (z) ∈
Sp∗ (n, β) with
Q
n m
j=1 (p − αj )
Qm
.
β = p − Qm
j=1 (p − αj )
j=1 (p + n − αj ) −
Proof. In view of Theorem 2.1, we have
(
)
Q
(k − p) m
(p
−
α
)
j
j=1
Qm
.
β 5 inf
p − Qm
k=p+n
j=1 (k − αj ) −
j=1 (p − αj )
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
Let F (k; m) be the right hand side of the above inequality. Further, let us define
G(k; m) by the numerator of F 0 (k; m). When m = 2,
G(k; 2) = −(p − α1 )(p − α2 ){(k − α1 )(k − α2 ) − (p − α1 )(p − α2 )}
+ (k − p)(p − α1 )(p − α2 ){(k − α1 ) + (k − α2 )}
= (p − α1 )(p − α2 )(k − p)2 > 0.
n(p − α1 )(p − α2 )
.
(p + n − α1 )(p + n − α2 ) − (p − α1 )(p − α2 )
Therefore, the corollary is true for m = 2. Let us suppose that Gm−1 (z) ∈
and fm (z) ∈ Sp∗ (n, αm ), where
Qm−1
n
j=1 (p − αj )
β ∗ = p − Qm−1
.
Qm−1
j=1 (p + n − αj ) −
j=1 (p − αj )
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F (k; 2) = F (p + n; 2)
=p−
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Since F (k; 2) is an increasing function of k, we see that
(2.1)
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Sp∗ (n, β ∗ )
Then replacing α1 by β ∗ and α2 by αm from (2.1), we see that
n(p − β ∗ )(p − αm )
(p + n − β ∗ )(p + n − αm ) − (p − β ∗ )(p − αm )
Q
n m
j=1 (p − αj )
Qm
= p − Qm
.
j=1 (p + n − αj ) −
j=1 (p − αj )
β =p−
Therefore, the corollary is true for the integer m. Using mathematical induction, we
complete the proof of the corollary.
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
Taking αj = α in Theorem 2.1, we have:
Corollary 2.3. If fj (z) ∈ Sp∗ (n, α) for all j = 1, 2, . . . , m, then Hm (z) ∈ Sp∗ (n, β)
with
n(p − α)s
β =p−
,
(p + n − α)s − (p − α)s
where
s=
m
X
pj = 1 +
j=1
p−α
,
n
pj =
1
,
qj
qj > 1 and
m
X
1
= 1.
q
j=1 j
Proof. By means of Theorem 2.1, we obtain that
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s
β 5p−
(k − p)(p − α)
(k − α)s − (p − α)s
(k = p + n).
Let us define F (k) by
F (k) = p −
(k − p)(p − α)s
(k − α)s − (p − α)s
(k = p + n).
Close
Then the numerator of F 0 (k) can be written as
(p − α)s (k − α)s {s(k − p) − (k − α)} + (p − α)s .
Since s = 1+ p−α
, we easily see that the numerator of F 0 (k) is positive for k = p+n.
n
Therefore, F (k) is increasing for k = p + n. This gives the value of β in the
corollary.
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
We consider the example for Corollary 2.3.
Example 2.1. Let us define fj (z) by
fj (z) = z p +
vol. 10, iss. 4, art. 98, 2009
p−α
p−α
z p+n +
j z p+n+j
p+n−α
p+n+j−α
(|| + |j | 5 1)
Sp∗ (n, α).
for each j = 1, 2, . . . , m, which is equivalent to fj (z) ∈
Sp∗ (n, β) with
n(p − α)s
β =p−
.
(p + n − α)s − (p − α)s
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Then Hm (z) ∈
Because, for functions
(2.2)
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fj (z) = z p +
p−α
p−α
z p+n +
j z p+n+j
p+n−α
p+n+j−α
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for each j = 1, 2, . . . , m, we have
∞
X
k−α
p+n−α
p+n+j−α
|ak | =
|||ap+n | +
|j ||ap+n+j |
p−α
p−α
p−α
k=p+n
= || + |j | 5 1
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from Lemma 1.1 which implies fj (z) ∈ Sp∗ (n, αj ). From (2.2), we see that
s
p−α
p
Hm (z) = z +
z p+n .
p+n−α
Therefore Hm (z) ∈ Sp∗ (n, β).
We also derive other results about Sp∗ and Tp∗ .
Sp∗ (n, αj ) for each j
Theorem 2.4. If fj (z) ∈
with
(
= 1, 2, . . . , m, then Hm (z) ∈
)
Q
pj
k(k − p) m
j=1 (p − αj )
Q
,
β = inf
p − Qm
pj
p j=1 (k − αj )pj − k m
k=p+n
j=1 (p − αj )
P
1
where pj = q1j , qj > 1 and m
j=1 qj = 1.
Tp∗ (n, β)
Proof. Using the same method as the proof in Theorem 2.1, we have to find the
largest β such that
!
Y
1
m
m Y
1
k(k − β)
k − αj qj
pj
|ak,j |
5
|ak,j | qj ,
p(p − β)
p − αj
j=1
j=1
which implies that
vol. 10, iss. 4, art. 98, 2009
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Qm
pj
k(k − p) j=1 (p − αj )
Q
β 5 p − Qm
pj
p j=1 (k − αj )pj − k m
j=1 (p − αj )
for all k = p + n.
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
Close
Corollary 2.5. If fj (z) ∈ Sp∗ (n, αj ) for each j = 1, 2, . . . , m, then Gm (z) ∈
Tp∗ (n, β) with
Q
n(p + n) m
j=1 (p − αj )
Q
β = p − Qm
.
p j=1 (p + n − αj ) − (p + n) m
j=1 (p − αj )
Theorem 2.6. If fj (z) ∈ Tp∗ (n, αj ) for each j = 1, 2, . . . , m, then Hm (z) ∈
Tp∗ (n, β) with
(
)
Q
pj
pj
k(k − p) m
p
(p
−
α
)
j
j=1
Qm p
,
β = inf
p − Qm pj
p
j
p j=1 k (k − αj ) − k j=1 p j (p − αj )pj
k=p+n
where pj =
1
,
qj
qj > 1 and
Pm
1
j=1 qj
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
Title Page
= 1.
Proof. To prove the theorem, we have to find the largest β such that
!
1
m
m Y
1
k(k − β) Y
k(k − αj ) qj
pj
|ak,j | qj
|ak,j |
5
p(p − β) j=1
p(p − αj )
j=1
for all k = p + n.
Corollary 2.7. If fj (z) ∈ Tp∗ (n, αj ) for each j = 1, 2, . . . , m, then Gm (z) ∈
Tp∗ (n, β) with
Q
npm−1 m
j=1 (p − αj )
Qm
Qm
.
β =p−
m−1
m−1
(p + n)
j=1 (p + n − αj ) − p
j=1 (p − αj )
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Theorem 2.8. If fj (z) ∈ Tp∗ (n, αj ) for each j = 1, 2, . . . , m, then Hm (z) ∈
Sp∗ (n, β) with
)
(
Q
pj
pj
p
(p
−
α
)
(k − p) m
j
j=1
Qm p
,
β = inf
p − Qm pj
pj −
pj
j
k
(k
−
α
)
k=p+n
j
j=1 p (p − αj )
j=1
where pj =
1
,
qj
qj > 1 and
Pm
1
j=1 qj
= 1.
Proof. We note that, we need to find the largest β such that
!
q1
m
m Y
1
j
k(k
−
α
)
k−β Y
j
|ak,j |pj 5
|ak,j | qj
p − β j=1
p(p − αj )
j=1
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
Title Page
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for all k = p + n.
Corollary 2.9. If fj (z) ∈ Tp∗ (n, αj ) for each j = 1, 2, . . . , m, then Gm (z) ∈
Sp∗ (n, β) with
Q
npm m
j=1 (p − αj )
Q
Q
β =p−
.
m
(p + n)m j=1 (p + n − αj ) − pm m
j=1 (p − αj )
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References
[1] J.H. CHOI, Y.C. KIM AND S. OWA, Generalizations of Hadamard products of
functions with negative coefficients, J. Math. Anal. Appl., 199 (1996), 495–501.
[2] J. NISHIWAKI, S. OWA AND H.M. SRIVASTAVA, Convolution and Höldertype inequalities for a certain class of analytic functions, Math. Inequal. Appl.,
11 (2008), 717–727.
[3] S. OWA, On certain classes of p−valent functions with negative coefficients,
Simon. Stevin, 59 (1985), 385–402.
[4] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc., 51 (1975), 109–116.
[5] H.M. SRIVASTAVA, S. OWA AND S.K. CHATTERJEA, A note on certain
classes of starlike functions, Rend. Sem. Mat. Univ. Padova, 77 (1987), 115–
124.
Hölder’s Inequality for Convolutions
Junichi Nishiwaki and
Shigeyoshi Owa
vol. 10, iss. 4, art. 98, 2009
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