Volume 10 (2009), Issue 4, Article 98, 7 pp.
AN APPLICATION OF HÖLDER’S INEQUALITY FOR CONVOLUTIONS
JUNICHI NISHIWAKI AND SHIGEYOSHI OWA
D EPARTMENT OF M ATHEMATICS
K INKI U NIVERSITY
H IGASHI -O SAKA , O SAKA 577 - 8502
JAPAN
jerjun2002@yahoo.co.jp
owa@math.kindai.ac.jp
Received 30 March, 2009; accepted 16 July, 2009
Communicated by N.E. Cho
A BSTRACT. 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.
Key words and phrases: Analytic function, Multivalent starlike, Multivalent convex.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION
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)
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J UNICHI N ISHIWAKI AND S HIGEYOSHI OWA
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
∞
X
(1.1)
(k − α)|ak | 5 p − α
k=p+n
for some α (0 5 α < p), then f (z) ∈ Sp (n, α).
Lemma 1.2. If f (z) ∈ Ap (n) satisfies
∞
X
(1.2)
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.
In view of Lemma 1.1 and Lemma 1.2, we introduce the subclass Sp∗ (n, α) 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).
2. C ONVOLUTION P ROPERTIES FOR THE C LASSES Sp∗ (n, α) AND Tp∗ (n, α)
For functions fj (z) ∈ Ap (n) given by
∞
X
p
fj (z) = z +
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
!
p
j
ak,j
zk
(pj > 0).
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
,
k=p+n
j=1
j=1
k=p+n
Pm
1
j=1 pj
where pj > 1 and
= 1.
Recently, Nishiwaki, Owa and Srivastava [2] have given some results of Hölder-type inequalities for a subclass of uniformly starlike functions.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 98, 7 pp.
http://jipam.vu.edu.au/
H ÖLDER ’ S I NEQUALITY FOR C ONVOLUTIONS
3
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
(p
−
α
)
j
j=1
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
|ak,j |
51
p − αj
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 |pj 5
|ak,j | qj .
p
−
β
p
−
α
j
j=1
k=p+n
k=p+n j=1
Therefore, we need to find the largest β such that
!
Y
1
m
m Y
1
k−β
k − αj qj
pj
|ak,j |
5
|ak,j | qj ,
p−β
p − αj
j=1
j=1
which is equivalent to
k−β
p−β
Y
m
pj − q1
|ak,j |
1
!
5
j
j=1
m Y
k − αj qj
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
pj − q1
|ak,j |
j=1
p − αj
j=1
j
5
1
pj −
=0 ,
qj
1
.
Qm k−αj pj − q1j
j=1
p−αj
This implies that
m
k−β Y
5
p−β
j=1
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 98, 7 pp.
k − αj
p − αj
pj
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J UNICHI N ISHIWAKI AND S HIGEYOSHI OWA
for all k = p + n. Therefore, β should be
Q
pj
(k − p) m
j=1 (p − αj )
Qm
β 5 p − Qm
pj
pj
j=1 (k − αj ) −
j=1 (p − αj )
(k = p + n).
This completes the proof of the theorem.
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 + n − αj ) −
j=1 (p − αj )
Proof. In view of Theorem 2.1, we have
(
β 5 inf
k=p+n
)
Q
(k − p) m
j=1 (p − αj )
Qm
.
p − Qm
j=1 (k − αj ) −
j=1 (p − αj )
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.
Since F (k; 2) is an increasing function of k, we see that
(2.1)
F (k; 2) = F (p + n; 2)
=p−
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) ∈ Sp∗ (n, β ∗ ) and
fm (z) ∈ Sp∗ (n, αm ), where
Q
n m−1
j=1 (p − αj )
∗
β = p − Qm−1
.
Qm−1
j=1 (p + n − αj ) −
j=1 (p − αj )
Then replacing α1 by β ∗ and α2 by αm from (2.1), we see that
n(p − β ∗ )(p − αm )
β =p−
(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 )
Therefore, the corollary is true for the integer m. Using mathematical induction, we complete
the proof of the corollary.
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
β =p−
n(p − α)s
,
(p + n − α)s − (p − α)s
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 98, 7 pp.
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H ÖLDER ’ S I NEQUALITY FOR C ONVOLUTIONS
where
m
X
p−α
s=
pj = 1 +
,
n
j=1
1
pj = ,
qj
qj > 1 and
5
m
X
1
= 1.
q
j=1 j
Proof. By means of Theorem 2.1, we obtain that
(k − p)(p − α)s
β 5p−
(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).
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.
We consider the example for Corollary 2.3.
Example 2.1. Let us define fj (z) by
p−α
p−α
z p+n +
j z p+n+j
fj (z) = z p +
p+n−α
p+n+j−α
(|| + |j | 5 1)
for each j = 1, 2, . . . , m, which is equivalent to fj (z) ∈ Sp∗ (n, α). Then Hm (z) ∈ Sp∗ (n, β)
with
n(p − α)s
β =p−
.
(p + n − α)s − (p − α)s
Because, for functions
p−α
p−α
z p+n +
j z p+n+j
(2.2)
fj (z) = z p +
p+n−α
p+n+j−α
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
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∗ .
Theorem 2.4. If fj (z) ∈ Sp∗ (n, αj ) for each j = 1, 2, . . . , m, then Hm (z) ∈ Tp∗ (n, β) with
(
)
Q
pj
k(k − p) m
(p
−
α
)
j
j=1
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.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 98, 7 pp.
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6
J UNICHI N ISHIWAKI AND S HIGEYOSHI OWA
Proof. Using the same method as the proof in Theorem 2.1, we have to find the largest β such
that
!
1
Y
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
Q
pj
k(k − p) m
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.
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
Q
β = inf
p − Qm pj
,
pj
pj
p j=1 k (k − αj )pj − k m
k=p+n
j=1 p (p − αj )
P
1
where pj = q1j , qj > 1 and m
j=1 qj = 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 |
5
|ak,j | qj
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 )
Q
Q
β =p−
.
m
(p + n)m−1 j=1 (p + n − αj ) − pm−1 m
j=1 (p − αj )
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
(k − p) m
j=1 p (p − αj )
Qm p
β = inf
p − Qm pj
,
pj
pj
j
k=p+n
j=1 k (k − αj ) −
j=1 p (p − αj )
P
1
where pj = q1j , qj > 1 and m
j=1 qj = 1.
Proof. We note that, we need to find the largest β such that
!
q1
m
m Y
1
j
k−β Y
k(k
−
α
)
j
|ak,j | qj
|ak,j |pj 5
p − β j=1
p(p − αj )
j=1
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 )
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 98, 7 pp.
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H ÖLDER ’ S I NEQUALITY FOR C ONVOLUTIONS
7
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(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.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 98, 7 pp.
http://jipam.vu.edu.au/