Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 2(2012), Pages 120-128.
APPLICATION OF HÖLDER’S INEQUALITY AND
CONVOLUTIONS
(COMMUNICATED BY R.K. RAINA)
S.P. GOYAL, ONKAR SINGH
Abstract. In this paper we introduce a new subclass Mp (n, α, c) of analytic
and multivalent functions in the unit disk which includes the class Sp (n, α) of
multivalent starlike functions of order α and the class Tp (n, α) of multivalent
convex functions of order α . Using generalized Bernardi Libera integral operator and Hölder’s inequality, some interesting properties of convolution for
the class Mp (n, α, c) are considered
1. Introduction
Let Ap (n) be class of functions f (z) of the form
f (z) = z p +
∞
X
ak z k ,
(p, n ∈ N )
k=p+n
which are analytic in the open unit disk U = {z : z ∈ C; |z| < 1}. Let Sp (n, α) be
the subclass of Ap (n) consisting of functions f (z) which satisfy
′ zf (z)
Re
> α (z ∈ U)
f (z)
for some α(0 ≤ α < p). A function f (z) ∈ Sp (n, α) is known as starlike of order
α in U .
Further, let Tp (n, α) be the subclass of Ap (n) consisting of functions f (z) satis′
fying zf p(z) ∈ Sp (n, α), that is,
zf ′′ (z)
Re 1 + ′
> α (z ∈ U)
f (z)
2000 Mathematics Subject Classification. 30C45.
Key words and phrases. Multivalent starlike function, Multivalent convex function, Convolutions, Hölder’s inequality, Generalized Libera integral operator.
c
2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted April 1, 2012. Published May 22, 2012.
The author (S.P.G) is thankful to CSIR, New Delhi, India for awarding Emeritus Scientistship,
under scheme number 21(084)/10/EMR-II. The second author (O.S.)is also thankful to CSIR for
JRF under the same scheme.
120
APPLICATION OF HÖLDER’S INEQUALITY AND CONVOLUTIONS
121
for some α (o ≤ α < p). A function f (z) in Tp (n, α) is known as convex of order
α in U .
These classes Ap (n), Sp (n, α) and Tp (n, α) were studied earlier by Owa [9] respectively. Also Nishiwaki and Owa [6] have given 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
(k − α) |ak | ≤ p − α
(1.1)
k=p+n
for some α (o ≤ α < p), then f (z) ∈ Sp (n, α)
Lemma 1.2. If f (z) ∈ Ap (n) satisfies
∞
X
k(k − α) |ak | ≤ p(p − α)
(1.2)
k=p+n
for some α (o ≤ α < p), then f (z) ∈ Tp (n, α).
Remark 1. We note that Silverman [10] has given Lemma (1.1) and Lemma (1.2)
in the case of p = 1 and n = 1. Also Srivastava, Owa and Chatterjea [11] have
given the coefficient inequalities in the case of p = 1.
In view of lemmas (1.1) and (1.2) Nishiwaki and Owa [6] introuced the subclasses
Sp∗ (n, α), Tp∗ (n, α) consisting of functions f (z) which satisfy the coefficient inequalities (1.1) and (1.2) respectively.
Now, we introduce subclass Mp∗ (n, α, c) consisting of functions f (z) ∈ Ap (n)
which satisfy the following coefficient inequality.
c
∞
X
k
(k − α) |ak | ≤ p − α
p
(1.3)
k=p+n
for some c ≥ 0 and α (o ≤ α < p).
Obviously Mp∗ (n, α, 0) ≡ Sp∗ (n, α) and Mp∗ (n, α, 1) ≡ Tp∗ (n, α).
For functions fj (z) ∈ Ap (n) given by
fj (z) = z p +
∞
X
ak,j z k (j = 1, 2, ..., m),
(1.4)
k=p+n
we define
Gm (z) = z p +
∞
X
k=p+n
and
Hm (z) = z p +
∞
X
k=p+n


m
Y
j=1


m
Y
j=1

ak,j z k ,

(ak,j )pj )z k (pj > 0)
(1.5)
(1.6)
where Gm (z) denotes the convolution of fj (z) (j = 1, 2, ..., m). Therefore Hm (z) is
the generalization of convolutions. In the case pj = 1, we have Gm (z) := Hm (z).
The generalization of the convolutions was considered by Choi, Kim and Owa [2],
and Nishiwaki and Owa [6].
122
S.P. GOYAL, ONKAR SINGH
Further for functions fj (z) ∈ Ap (n) given by (1.4), generalized Libera integral
operator is defined as follows :
Zz
p + cj
Bj,p (z) =
z cj
tcj −1 f (t) dt (cj > −p)
(1.7)
0
(For cj = 1, we obtain multivalent Libera operator. For p = 1, we get generalized
Bernardi-Libera- Livingston integral operator defined recently by Gordji et al. [4]
and for p = 1, cj = 1, Libera [5] studied the above operator. The operator (1.7)
also includes the Alexander operator [1] for p = 1 and cj = 0)
Using the operator (1.7), we find that the convolution integral of B1,p and
B2,p as
p + c1
(B1,p ∗ B2,p )(z) =
z c1
Zz
t
c1 −1
p + c2
f (t)dt∗
z c2
0
Zz
tc2 −1 f (t)dt
0
∞
X
(p + c1 )(p + c2 )
ak,1 ak,2 z k
= zp +
(k + c1 )(k + c2 )
k=p+n
The convolution integral was studied by Duren [3].
Hence the convolution integral of B1,p , B2,p , ..., Bm,p is given by
p + c1
(B1,p ∗ B2,p ∗ ... ∗ Bm,p )(z) =
z c1
Zz
t
c1 −1
p + cm
f (t)dt∗ ... ∗
z cm
0
= zp +
∞
X
k=p+n
Zz
tcm −1 f (t)dt
0


m
Y
p
+
c
j

ak,j  z k
k
+
c
j
j=1
(1.8)
For functions fj (z) ∈ Ap (n) (j = 1, 2, ..., m) given by (1.4), the familiar Hölder’s
inequality assumes the form
∞
X
k=p+n
where pj > 1 and
m
P
j=1
1
pj


m
Y
j=1

|ak,j | ≤
m
Y
j=1


≥ 1 (j = 1, 2, ..., m)
∞
X
k=p+n
 p1
j
|ak,j |
pj 
(1.9)
Nishiwaki, Owa and Srivastava [8] have given some results of Hölder’s-type inequalities for subclass of uniformly starlike functions. Also applying these inequalities, Nishiwaki and Owa [6] have obtained some interesting properties of generalizations of convolutions for functions f (z) in the classes Sp∗ (n, α) and Tp∗ (n, α)
. Again Nishiwaki and Owa [7] have given application of convolution integral for
certain subclasses by using Hölder’s inequalities . Motivated essentially by these
papers, we discuss some applications of Hölder’s inequalities for Hm (z) defined by
(1.6) and convolution integral defined by (1.8) for the subclass Mp∗ (n, α, c).
APPLICATION OF HÖLDER’S INEQUALITY AND CONVOLUTIONS
123
2. Main Results
Theorem 1. If fj (z) ∈ Mp∗ (n, αj , c) for each j = 1, 2, ....., m, then Hm (z) ∈
with


m
Q
p


k c (k − p)
[pc (p − αj )] j




j=1
β = inf
p−
(2.1)
m
m
Q
Q
p
p 
k≥p+n 


pc
[k c (k − αj )] j − k c
[pc (p − αj )] j 

Mp∗ (n, β, c)
j=1
where pj ≥
1
qj ,
qj > 1 and
j=1
m
P
1
qj ≥ 1.
j=1
Mp∗ (n, αj , c),
Proof . Since fj (z) ∈
therefore by equation (1.3), we get
c ∞
X
k
k − αj
|ak,j | ≤ 1 (j = 1, 2, ......., m)
p
p − αj
k=p+n
which implies

 q1
c ∞
 X
 j
k
k − αj
|ak,j |
≤ 1)


p
p − αj
(2.2)
k=p+n
with qj > 1 and
m
P
1
qj
j=1
≥ 1.
From (2.2), we have

 q1
c m  X
∞
 j
Y
k
k − αj
≤1
|ak,j |


p
p − αj
j=1
k=p+n
Applying Hölder’s inequality (1.9), we find that

1
∞
m qc Y
X
k j k − αj qj
k=p+n

j=1
p
p − αj
|ak,j |
1
qj


≤1
(2.3)

Note that we have to find the largest β such that


m
c ∞
X
k − β Y
k
pj 
|ak,j |
≤1
p
p−β
j=1
k=p+n
that is,




c Y
q1
∞
m
∞ Y
m qc 
X
X
1
j
j
k
k−β 
k
k − αj
p
|ak,j | j  ≤
|ak,j | qj


p
p−β
p
p − αj
j=1
j=1
k=p+n
k=p+n
Therefore we need to find largest β such that


m
1
c m qc Y
1
k
k − β Y
k j k − αj qj
pj 
|ak,j |
≤
|ak,j | qj
p
p−β
p
p − αj
j=1
j=1
124
S.P. GOYAL, ONKAR SINGH
which is equivalent to


c m
1
m qc Y
k
k − β Y
k j k − αj qj
pj − q1
j

|ak,j |
≤
p
p−β
p
p − αj
j=1
j=1
for all k ≥ p + n. Since
pj − q1
m c Y
j
k
k − αj
1
|ak,j |
≥0
≤1
pj −
p
p − αj
qj
j=1
,
therefore
m
Y
1
pj − q1
j ≤
|ak,j |
1
n
opj − qj
m
c
j=1
Q
k−αj
k
p
j=1
p−αj
This implies that
c pj
m c Y
k
k−β
k
k − αj
≤
p
p−β
p
p − αj
j=1
for all k ≥ p + n. Therefore β should be
k c (k − p)
m
Q
pj
(pc (p − αj ))
j=1
β ≤p−
pc
m
Q
(k c (k − αj ))
pj
− kc
j=1
m
Q
(pc (p − αj ))
pj
j=1
for all k ≥ p + n This completes the proof of the theorem.
Taking pj = 1 in Theorem 1, we obtain
Corollary 2.1. If fj (z) ∈ Mp∗ (n, αj , c) for each j = 1, 2, ....., m, then Gm (z) ∈
Mp∗ (n, β, c) with
np(m−1)c
m
Q
(p − αj )
j=1
β =p−
(p + n)(m−1)c
m
Q
(p + n − αj ) − p(m−1)c
j=1
β ≤ inf
k≥p+n 


k c (k − p)
m
Q
pc (p − αj )
j=1
p−
pc
m
Q
(2.4)
(p − αj )
j=1
Proof . In view of Theorem 1, we obtain




m
Q
k c (k
− αj ) − k c
j=1
m
Q
j=1





pc (p − αj ) 

Let F (c, k, m) be the right-hand side of the above inequality. Further let us define
G(c, k, m) be the numerator of F ′ (c, k, m). Then


m
m
m


Y
Y
Y
G(c, k, m) = −p(m−1)c
(p − αj ) k (m−1)c
(k − αj ) − p(m−1)c
(p − αj )


j=1
j=1
j=1
APPLICATION OF HÖLDER’S INEQUALITY AND CONVOLUTIONS
+ (k − p)(pk)(m−1)c
m
Q
j=1
(p − αj )[ (m−1)c
k
m
Q
125
(k − αj )
j=1
+ {(k − α2 )(k − α3 ) · · · (k − αm ) + · · · + ( k − α1 )(k − α2 ) · · · (k − αm−1 )}]
(
)
m
m
m
Q
Q
Q
= −p(m−1)c
(p − αj ) k (m−1)c
(k − αj ) − p(m−1)c
(p − αj )
j=1
j=1
j=1



m
Q
P
(k−αj ) 
m
m
m
Q
Q
j=1
.
(k − αj ) +
+(k−p)(pk)(m−1)c
(p − αj )  (m−1)c
k
j=1 (k−αj ) 
j=1
j=1
= (pk)(m−1)c
m
Y
{(p − αj )(k − αj )}
j=1
+ p2(m−1)c
m
Q
(p − αj )2 ≥ 0




(k − p) 
m
X
j=1



1
(m − 1)c 
+
−1

k − αj
k
j=1
Thus F (c, k, m) is increasing function for all k ≥ p + n.
This means that
m
Q
np(m−1)c
(p − αj )
j=1
β = F (p + n) = p −
(p + n)(m−1)c
m
Q
(p + n − αj ) − p(m−1)c
j=1
m
Q
(p − αj )
j=1
On taking c = 0 and c = 1 in Theorem 1, we get the Theorem 2.1, 2.6 and 2.8
obtained recently by Nishiwaki and Owa [6]. Corollaries 2.2, 2.3, 2.5, 2.7 and 2.9
due to them are also special cases of our Theorem 1 and corollary 2.1
Theorem 2. If fj (z) ∈ Mp∗ (n, αj , c) for each j = 1, 2, ....., m then (B1 ∗ B2 ..... ∗
Bm )(z) ∈ Mp∗ (n, β, c) with
np(m−1)c
m
Q
(p + cj )(p − αj )
j=1
β =p−
(p + n)(m−1)c
m
Q
j=1
(p + n + cj )(p + n − αj ) − p(m−1)c
m
Q
(p + cj )(p − αj )
j=1
(2.5)
Proof . Since fj (z) ∈ Mp∗ (n, αj , c), then from (2.3) , we get


1
∞
m qc Y

X
1
k j k − αj qj
|ak,j | qj ≤ 1


p
p − αj
j=1
k=p+n
Note that we have to find the largest β such that
m ∞ c X
k
k − β Y p + cj
|ak,j | ≤ 1
p
p − β j=1 k + cj
k=p+n
that is


c m 1
∞
m Y
m qc 
X
X
1
k
k − β Y p + cj
k j k − αj qj
|ak,j | ≤
|ak,j | qj


p
p − β j=1 k + cj
p
p − αj
j=1
j=1
k=p+n
126
S.P. GOYAL, ONKAR SINGH
Using the same procedure of Theorem 1, we can easily prove that for all k ≥ p + n,
β should be
k c (k − p)
m
Q
(p + cj )pc (p − αj )
j=1
β ≤p−
pc
m
Q
(k + cj )k c (k − αj ) − k c
j=1
(2.6)
m
Q
(p + cj )pc (p − αj )
j=1
Let F (c, k, m) be the right hand side of the above inequality. Further let us
define G(c, k, m) be the numerator of F ′ (c, k, m). Then
G(c, k, m) = −p(m−1)c
m
Y
(p − αj )(p + cj )

j=1
+(k − p)(pk)(m−1)c
m
Y


k (m−1)c
(p − αj )(p − cj )[
j=1
m
Y
(k − αj )(k + cj ) − p(m−1)c
j=1
m
Y
j=1
m
(m − 1) c Y
(k − αj )(k + cj )
k
j=1


(p − αj )(p + cj )

+ {(k − α2 )(k − α3 ) · · · (k − αm )(k + c1 ) · · · (k + cm ) + · · · + (k − α1 )
(k−α2 ) · · · (k−αm−1 )(k+c1 ) · · · (k+cm )+(k−α1 ) · · · (k−αm )(k+c2 ) · · · (k+cm )+
· · · +(k − α1 ) · · · (k − αm )(k + c1 ) · · · (k + cm−1 )}]
= −p(m−1)c
(
m
Q
m
Q
)
m
Q
(p − αj )(p + cj ) k (m−1)c
(k − αj )(k + cj ) − p(m−1)c
(p − αj )(p + cj )
j=1
j=1
"
m
m
Q
Q
+(k − p)(pk)(m−1)c
(k−αj )(k + cj )
(p − αj )(p + cj ) (m−1)c
k
j=1
j=1


m
m
Q
Q
P
(k−αj )(k+cj )
(k−αj )(k+cj ) 
m
m
P
j=1
j=1

+
+
k−αj
k+cj
j=1

j=1
j=1
= (pk)(m−1)c
m
Y
(p − αj )(p + cj )(k − αj )(k + cj )
j=1




m
m
m


X
X
Y
1
1
(m − 1)c 
× (k − p) 
+
− 1 +p2(m−1)c
(p − αj )2 (p + cj )2 ≥ 0.


k
−
α
k
+
c
k
j
j
j=1
j=1
j=1
Thus F (c, k, m) is increasing function for all k ≥ p + n.
This means that
np(m−1)c
m
Q
(p + cj )(p − αj )
j=1
β = F (p+n) = p−
(p + n)(m−1)c
m
Q
j=1
(p + n + cj )(p + n − αj ) − p(m−1)c
m
Q
j=1
(p + cj )(p − αj )
APPLICATION OF HÖLDER’S INEQUALITY AND CONVOLUTIONS
127
If we take cj = 1 in Theorem 2, we get
Corollary 2.2. If fj (z) ∈ Mp∗ (n, αj , c) for each j = 1, 2, ....., m then (B1 ∗ B2 ..... ∗
Bm )(z) ∈ Mp∗ (n, β, c) with
np(m−1)c (p + 1)m
m
Q
(p − αj )
j=1
β = p−
(p + n)(m−1)c (p + n + 1)m
m
Q
(p + n − αj ) − p(m−1)c (p + 1)m
j=1
m
Q
(p − αj )
j=1
If we take p = 1 in Theorem 2 and Corollary 2.2, we deduce that
Corollary 2.3. If fj (z) ∈ M ∗ (n, αj , c) for each j = 1, 2, ....., m then (B1 ∗ B2 ..... ∗
Bm )(z) ∈ M ∗ (n, β, c) with
m
Q
n
(1 + cj )(1 − αj )
j=1
β =1−
(n +
1)(m−1)c
m
Q
(n + 1 + cj )(n + 1 − αj ) −
j=1
m
Q
(1 + cj )(1 − αj )
j=1
Corollary 2.4. If fj (z) ∈ M ∗ (n, αj , c) for each j = 1, 2, ....., m then (B1 ∗ B2 ..... ∗
Bm )(z) ∈ M ∗ (n, β, c) with
m
Q
n2m
(1 − αj )
j=1
β =1−
(n + 1)(m−1)c (n + 2)m
m
Q
(n + 1 − αj ) − 2m
j=1
m
Q
(1 − αj )
j=1
Further taking cj = 0 in Corollary 2.3, we get
Corollary 2.5. If fj (z) ∈ Mp∗ (n, αj , c) for each j = 1, 2, ....., m then (B1 ∗ B2 ..... ∗
Bm )(z) ∈ Mp∗ (n, β, c) with
n
β =1−
(n + 1)(m−1)c+m
m
Q
(1 − αj )
j=1
m
Q
(n + 1 − αj ) −
j=1
m
Q
(1 − αj )
j=1
Remark. All results due to Nishiwaki and Owa [7] can be deduced as special cases
if we put c = 0 and c = 1 in Theorem 2 and Corollaries 2.2-2.5.
Acknowledgments. The authors would like to thank the anonymous referee for
his/her comments that helped us improve this article.
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128
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Department of Mathematics, University of Rajasthan, Jaipur-302055, India.
E-mail address:
somprg@gmail.com
E-mail address: onkarbhati@gmail.com