AN APPLICATION OF HÖLDER’S INEQUALITY FOR CONVOLUTIONS

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Hölder’s Inequality for Convolutions Junichi Nishiwaki and

Shigeyoshi Owa vol. 10, iss. 4, art. 98, 2009

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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 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: LetAp(n)be the class of analytic and multivalent functionsf(z)in the open unit diskU. Furthermore, letSp(n, α)andTp(n, α)be the subclasses ofAp(n) consisting of multivalent starlike functionsf(z)of orderαand multivalent convex functionsf(z)of order α, respectively. Using the coefficient inequalities forf(z)to be inSp(n, α)andTp(n, α), new subclassesSp^∗(n, α)andTp^∗(n, α) are introduced. Applying the Hölder inequality, some interesting properties of generalizations of convolutions (or Hadamard products) for functionsf(z)in the classesSp^∗(n, α)andTp^∗(n, α)are considered.

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1. Introduction

LetA_p(n)be the class of functionsf(z)of the form

f(z) =z^p+

∞

X

k=p+n

akz^k

which are analytic in the open unit disk U = {z ∈ C||z| < 1} for some natural numberspandn. LetS_p(n, α)be the subclass ofA_p(n)consisting of functionsf(z) which satisfy

Re

zf⁰(z) f(z)

> α (z ∈U)

for someα (0 5 α < p). Also letT_p(n, α)be the subclass of A_p(n)consisting of functionsf(z)satisfyingzf⁰(z)/p∈ S_p(n, α), that is,

Re

1 + zf⁰⁰(z) f⁰(z)

> α (z ∈U)

for someα(05α < p). These classes,A_p(n),S_p(n, α)andT_p(n, α), were studied by Owa [3]. It is easy to derive the following lemmas, which provide the sufficient conditions for functions f(z) ∈ A_p(n)to be in the classes S_p(n, α)and T_p(n, α), respectively.

Lemma 1.1. Iff(z)∈ A_p(n)satisfies

(1.1)

∞

X

k=p+n

(k−α)|ak|5p−α

for someα(05α < p), thenf(z)∈ Sp(n, α).

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Lemma 1.2. Iff(z)∈ A_p(n)satisfies

(1.2)

∞

X

k=p+n

k(k−α)|a_k|5p(p−α)

for someα(05α < p), thenf(z)∈ T_p(n, α).

Remark 1. We note that Silverman [4] has given Lemma1.1and Lemma1.2in the case ofp = 1andn = 1. Also, Srivastava, Owa and Chatterjea [5] have given the coefficient inequalities in the case ofp= 1.

In view of Lemma1.1 and Lemma1.2, we introduce the subclassS_p^∗(n, α)consisting of functionsf(z)which satisfy the coefficient inequality (1.1), and the sub- classT_p^∗(n, α)consisting of functions f(z)which satisfy the coefficient inequality (1.2).

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2. Convolution Properties for the Classes S

_p^∗

(n, α) and T

_p^∗

(n, α)

For functionsf_j(z)∈ A_p(n)given by f_j(z) =z^p+

∞

X

k=p+n

a_k,jz^k (j = 1,2, . . . , m),

we define

G_m(z) =z^p +

∞

X

k=p+n m

Y

j=1

a_k,j

! z^k

and

H_m(z) =z^p+

∞

X

k=p+n m

Y

j=1

a^p_k,j^j

!

z^k (p_j >0).

ThenG_m(z)denotes the convolution off_j(z) (j = 1,2, . . . , m). Therefore, H_m(z) is the generalization of the convolutions. In the case of p_j = 1, we have G_m(z) = H_m(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 forH_m(z) to be in the classesS_p^∗(n, α)andT_p^∗(n, α).

Forf_j(z)∈ A_p(n), the Hölder inequality is given by

∞

X

k=p+n m

Y

j=1

|a_k,j|

! 5

m

Y

j=1

∞

X

k=p+n

|a_k,j|^p^j

!_pj¹ , wherep_j >1andPm

j=1 1 pj =1.

Recently, Nishiwaki, Owa and Srivastava [2] have given some results of Hölder- type inequalities for a subclass of uniformly starlike functions.

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Theorem 2.1. Iff_j(z)∈ S_p^∗(n, α_j)for eachj = 1,2, . . . , m, thenH_m(z)∈ S_p^∗(n, β) with

β = inf

k=p+n

(

p− (k−p)Qm

j=1(p−α_j)^p^j Qm

j=1(k−α_j)^p^j−Qm

j=1(p−α_j)^p^j )

, wherep_j = _q¹_j,q_j >1andPm

j=1 1 qj =1.

Proof. Forf_j(z)∈ S_p^∗(n, α_j), Lemma1.1gives us that

∞

X

k=p+n

k−α_j p−αj

|a_k,j|51 (j = 1,2, . . . , m),

which implies

( _∞ X

k=p+n

k−α_j p−α_j

|ak,j| )_qj¹

51

withq_j >1andPm j=1

1

qj =1. Applying the Hölder inequality, we have:

∞

X

k=p+n

( _m Y

j=1

k−α_j p−α_j

_qj¹

|a_k,j|

1 qj

) 51.

Note that we have to find the largestβsuch that

∞

X

k=p+n

k−β p−β

^m Y

j=1

|ak,j|^p^j

! 51,

that is,

∞

X

k=p+n

k−β p−β

m

Y

j=1

|a_k,j|^p^j

! 5

∞

X

k=p+n

( _m Y

j=1

k−α_j p−α_j

_qj¹

|a_k,j|

1 qj

) .

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Therefore, we need to find the largestβ such that k−β

p−β ^m

Y

j=1

|a_k,j|^p^j

! 5

m

Y

j=1

k−α_j p−αj

_qj¹

|a_k,j|

1 qj,

which is equivalent to k−β

p−β

^m Y

j=1

|a_k,j|^p^j⁻

1 qj

! 5

m

Y

j=1

k−α_j p−α_j

_qj¹

for allk=p+n. Since

m

Y

j=1

k−α_j p−α_j

pj−¹

qj |a_k,j|^p^j⁻

1 qj 51

p_j− 1 q_j =0

, we see that

m

Y

j=1

|a_k,j|^p^j⁻

1

qj 5 1

Qm j=1

_k−α

j

p−α_j

pj−¹

qj

.

This implies that

k−β p−β 5

m

Y

j=1

k−α_j p−αj

pj

for allk=p+n. Therefore,βshould be β5p− (k−p)Qm

j=1(p−α_j)^p^j Qm

j=1(k−α_j)^p^j −Qm

j=1(p−α_j)^p^j (k=p+n).

This completes the proof of the theorem.

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Takingp_j = 1in Theorem2.1, we obtain

Corollary 2.2. If f_j(z) ∈ S_p^∗(n, α_j) for each j = 1,2, . . . , m, then G_m(z) ∈ S_p^∗(n, β)with

β =p− nQm

j=1(p−α_j) Qm

j=1(p+n−αj)−Qm

j=1(p−αj). Proof. In view of Theorem2.1, we have

β 5 inf

k=p+n

(

p− (k−p)Qm

j=1(p−α_j) Qm

j=1(k−α_j)−Qm

j=1(p−α_j) )

.

Let F(k;m) be the right hand side of the above inequality. Further, let us define G(k;m)by the numerator ofF⁰(k;m). Whenm= 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.

SinceF(k; 2)is an increasing function ofk, we see that F(k; 2)=F(p+n; 2)

(2.1)

=p− n(p−α₁)(p−α₂)

(p+n−α₁)(p+n−α₂)−(p−α₁)(p−α₂).

Therefore, the corollary is true form= 2. Let us suppose thatGm−1(z)∈ S_p^∗(n, β^∗) andf_m(z)∈ S_p^∗(n, α_m), where

β^∗ =p− nQm−1

j=1 (p−αj) Qm−1

j=1 (p+n−αj)−Qm−1

j=1 (p−αj).

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Then replacingα₁byβ^∗ andα₂byα_mfrom (2.1), we see that β =p− n(p−β^∗)(p−α_m)

(p+n−β^∗)(p+n−αm)−(p−β^∗)(p−αm)

=p− nQm

j=1(p−α_j) Qm

j=1(p+n−α_j)−Qm

j=1(p−α_j).

Therefore, the corollary is true for the integerm. Using mathematical induction, we complete the proof of the corollary.

Takingα_j =αin Theorem2.1, we have:

Corollary 2.3. Iffj(z) ∈ S_p^∗(n, α)for allj = 1,2, . . . , m, thenHm(z) ∈ S_p^∗(n, β) with

β =p− n(p−α)^s

(p+n−α)^s−(p−α)^s, where

s =

m

X

j=1

p_j =1 + p−α

n , p_j = 1 qj

, q_j >1 and

m

X

j=1

1 qj =1.

Proof. By means of Theorem2.1, we obtain that

β 5p− (k−p)(p−α)^s

(k−α)^s−(p−α)^s (k =p+n).

Let us defineF(k)by

F(k) =p− (k−p)(p−α)^s

(k−α)^s−(p−α)^s (k =p+n).

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Then the numerator ofF⁰(k)can be written as

(p−α)^s(k−α)^s{s(k−p)−(k−α)}+ (p−α)^s.

Sinces=1+^p−α_n , we easily see that the numerator ofF⁰(k)is positive fork =p+n.

Therefore, F(k) is increasing for k = p +n. This gives the value of β in the corollary.

We consider the example for Corollary2.3.

Example 2.1. Let us definef_j(z)by f_j(z) =z^p + p−α

p+n−αz^p+n+ p−α

p+n+j−α_jz^p+n+j (||+|_j|51) for eachj = 1,2, . . . , m, which is equivalent tof_j(z) ∈ S_p^∗(n, α). ThenH_m(z) ∈ S_p^∗(n, β)with

β =p− n(p−α)^s

(p+n−α)^s−(p−α)^s. Because, for functions

(2.2) f_j(z) = z^p+ p−α

p+n−αz^p+n+ p−α

p+n+j−α_jz^p+n+j for eachj = 1,2, . . . , m, we have

∞

X

k=p+n

k−α

p−α|a_k|= p+n−α

p−α |||a_p+n|+p+n+j−α

p−α |_j||a_p+n+j|

=||+|_j|51

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from Lemma1.1which impliesf_j(z)∈ S_p^∗(n, α_j). From (2.2), we see that H_m(z) =z^p+

p−α p+n−α

s

z^p+n.

ThereforeH_m(z)∈ S_p^∗(n, β).

We also derive other results aboutS_p^∗ andT_p^∗.

Theorem 2.4. Iff_j(z)∈ S_p^∗(n, α_j)for eachj = 1,2, . . . , m, thenH_m(z)∈ T_p^∗(n, β) with

β = inf

k=p+n

(

p− k(k−p)Qm

j=1(p−α_j)^p^j pQm

j=1(k−α_j)^p^j −kQm

j=1(p−α_j)^p^j )

, wherep_j = _q¹_j,q_j >1andPm

j=1 1 qj =1.

Proof. Using the same method as the proof in Theorem 2.1, we have to find the largestβsuch that

k(k−β) p(p−β)

m

Y

j=1

|a_k,j|^p^j

! 5

m

Y

j=1

k−α_j p−α_j

¹

qj |a_k,j|

1 qj,

which implies that

β 5p− k(k−p)Qm

j=1(p−αj)^p^j pQm

j=1(k−α_j)^p^j −kQm

j=1(p−α_j)^p^j for allk=p+n.

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Corollary 2.5. If f_j(z) ∈ S_p^∗(n, α_j) for each j = 1,2, . . . , m, then G_m(z) ∈ T_p^∗(n, β)with

β =p− n(p+n)Qm

j=1(p−α_j) pQm

j=1(p+n−α_j)−(p+n)Qm

j=1(p−α_j).

Theorem 2.6. If f_j(z) ∈ T_p^∗(n, α_j) for each j = 1,2, . . . , m, then H_m(z) ∈ T_p^∗(n, β)with

β = inf

k=p+n

(

p− k(k−p)Qm

j=1p^p^j(p−αj)^p^j pQm

j=1k^p^j(k−α_j)^p^j−kQm

j=1p^p^j(p−α_j)^p^j )

,

wherep_j = _q¹_j,q_j >1andPm j=1

1 qj =1.

Proof. To prove the theorem, we have to find the largestβsuch that k(k−β)

p(p−β)

m

Y

j=1

|a_k,j|^p^j

! 5

m

Y

j=1

k(k−αj) p(p−α_j)

_qj¹

|a_k,j|

1 qj

for allk=p+n.

Corollary 2.7. If f_j(z) ∈ T_p^∗(n, α_j) for each j = 1,2, . . . , m, then G_m(z) ∈ T_p^∗(n, β)with

β=p− np^m−1Qm

j=1(p−α_j) (p+n)^m−1Qm

j=1(p+n−αj)−p^m−1Qm

j=1(p−αj).

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Theorem 2.8. If f_j(z) ∈ T_p^∗(n, α_j) for each j = 1,2, . . . , m, then H_m(z) ∈ S_p^∗(n, β)with

β = inf

k=p+n

(

p− (k−p)Qm

j=1p^p^j(p−α_j)^p^j Qm

j=1k^p^j(k−α_j)^p^j −Qm

j=1p^p^j(p−α_j)^p^j )

,

wherepj = _q¹_j,qj >1andPm j=1

1 qj =1.

Proof. We note that, we need to find the largestβsuch that k−β

p−β

m

Y

j=1

|a_k,j|^p^j

! 5

m

Y

j=1

k(k−α_j) p(p−α_j)

_qj¹

|a_k,j|^qj¹

for allk=p+n.

Corollary 2.9. If f_j(z) ∈ T_p^∗(n, α_j) for each j = 1,2, . . . , m, then G_m(z) ∈ S_p^∗(n, β)with

β =p− np^mQm

j=1(p−α_j) (p+n)^mQm

j=1(p+n−α_j)−p^mQm

j=1(p−α_j).

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References

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[2] J. NISHIWAKI, S. OWA AND H.M. SRIVASTAVA, Convolution and Hölder- type 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.