LetAp(n)be the class of analytic and multivalent functionsf(z)in the open unit diskU

AN APPLICATION OF HÖLDER’S INEQUALITY FOR CONVOLUTIONS

JUNICHI NISHIWAKI AND SHIGEYOSHI OWA DEPARTMENT OFMATHEMATICS

KINKIUNIVERSITY

HIGASHI-OSAKA, OSAKA577 - 8502 JAPAN

jerjun2002@yahoo.co.jp owa@math.kindai.ac.jp

Received 30 March, 2009; accepted 16 July, 2009 Communicated by N.E. Cho

ABSTRACT. LetAp(n)be the class of analytic and multivalent functionsf(z)in the open unit diskU. Furthermore, letS_p(n, α)andT_p(n, α)be the subclasses ofA_p(n)consisting of mul- tivalent starlike functionsf(z)of order αand multivalent convex functionsf(z)of order α, respectively. Using the coefficient inequalities forf(z)to be inS_p(n, α)andT_p(n, α), new sub- classesS_p^∗(n, α)andT_p^∗(n, α)are introduced. Applying the Hölder inequality, some interesting properties of generalizations of convolutions (or Hadamard products) for functionsf(z)in the classesS_p^∗(n, α)andT_p^∗(n, α)are considered.

Key words and phrases: Analytic function, Multivalent starlike, Multivalent convex.

2000 Mathematics Subject Classification. 30C45.

1. INTRODUCTION

LetAp(n)be the class of functionsf(z)of the form f(z) =z^p+

∞

X

k=p+n

a_kz^k

which are analytic in the open unit diskU ={z ∈ C||z| <1}for some natural numberspand n. LetSp(n, α)be the subclass ofAp(n)consisting of functionsf(z)which satisfy

Re

zf⁰(z) f(z)

> α (z ∈U)

for some α (0 5 α < p). Also let T_p(n, α)be the subclass of A_p(n) consisting of functions f(z)satisfyingzf⁰(z)/p∈ S_p(n, α), that is,

Re

1 + zf⁰⁰(z) f⁰(z)

> α (z ∈U)

088-09

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 classesS_p(n, α)andT_p(n, α), respectively.

Lemma 1.1. Iff(z)∈ A_p(n)satisfies (1.1)

∞

X

k=p+n

(k−α)|a_k|5p−α for someα(05α < p), thenf(z)∈ S_p(n, α).

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 Lemma 1.1 and Lemma 1.2 in the case ofp= 1andn = 1. Also, Srivastava, Owa and Chatterjea [5] have given the coefficient inequalities in the case ofp= 1.

In view of Lemma 1.1 and Lemma 1.2, we introduce the subclassS_p^∗(n, α)consisting of func- tionsf(z)which satisfy the coefficient inequality (1.1), and the subclassT_p^∗(n, α)consisting of functionsf(z)which satisfy the coefficient inequality (1.2).

2. CONVOLUTION PROPERTIES FOR THECLASSESS_p^∗(n, α)ANDT_p^∗(n, α) For functionsfj(z)∈ Ap(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).

Then G_m(z) denotes the convolution of f_j(z) (j = 1,2, . . . , m). Therefore, H_m(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 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|^pj

!_pj¹ , wherep_j >1andPm

j=1 1 pj =1.

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

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−αj)^pj Qm

j=1(k−α_j)^pj−Qm

j=1(p−α_j)^pj )

, wherep_j = _q¹_j,q_j >1andPm

j=1 1 qj =1.

Proof. Forf_j(z)∈ S_p^∗(n, α_j), Lemma 1.1 gives us that

∞

X

k=p+n

k−αj

p−α_j

|ak,j|51 (j = 1,2, . . . , m), which implies

( _∞ X

k=p+n

k−α_j p−α_j

|a_k,j| )_qj¹

51 withqj >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|^pj

! 51, that is,

∞

X

k=p+n

k−β p−β

^m Y

j=1

|a_k,j|^pj

! 5

∞

X

k=p+n

( _m Y

j=1

k−αj

p−α_{j qj}¹

|a_k,j|

1 qj

) .

Therefore, we need to find the largestβsuch that k−β

p−β ^m

Y

j=1

|a_k,j|^pj

! 5

m

Y

j=1

k−α_j p−α_j

_qj¹

|a_k,j|

1 qj,

which is equivalent to

k−β p−β

^m Y

j=1

|a_k,j|^pj−qj1

! 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|^pj−

1 qj 51

p_j− 1 q_j =0

, we see that

m

Y

j=1

|a_k,j|^pj−

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−α_j

pj

for allk =p+n. Therefore,βshould be β 5p− (k−p)Qm

j=1(p−α_j)^pj Qm

j=1(k−α_j)^pj −Qm

j=1(p−α_j)^pj (k=p+n).

This completes the proof of the theorem.

Takingp_j = 1in Theorem 2.1, we obtain

Corollary 2.2. Iff_j(z)∈ S_p^∗(n, α_j)for eachj = 1,2, . . . , m, thenG_m(z)∈ S_p^∗(n, β)with

β =p− nQm

j=1(p−α_j) Qm

j=1(p+n−α_j)−Qm

j=1(p−α_j). Proof. In view of Theorem 2.1, 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) )

.

LetF(k;m)be the right hand side of the above inequality. Further, let us defineG(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 for m = 2. Let us suppose that Gm−1(z) ∈ S_p^∗(n, β^∗) and fm(z)∈ S_p^∗(n, αm), where

β^∗ =p− nQm−1

j=1 (p−α_j) Qm−1

j=1 (p+n−α_j)−Qm−1

j=1 (p−α_j). Then replacingα₁byβ^∗andα₂byα_mfrom (2.1), we see that

β =p− n(p−β^∗)(p−α_m)

(p+n−β^∗)(p+n−α_m)−(p−β^∗)(p−α_m)

=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 integer m. Using mathematical induction, we complete

the proof of the corollary.

Takingαj =αin Theorem 2.1, we have:

Corollary 2.3. Iff_j(z)∈ S_p^∗(n, α)for allj = 1,2, . . . , m, thenH_m(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

q_j, q_j >1 and

m

X

j=1

1 q_j =1.

Proof. By means of Theorem 2.1, 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).

Then the numerator ofF⁰(k)can be written as

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

Since s = 1 + ^p−α_n , we easily see that the numerator of F⁰(k) is positive for k = p +n.

Therefore,F(k)is increasing fork =p+n. This gives the value ofβin the corollary.

We consider the example for Corollary 2.3.

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 to f_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|ak|= p+n−α

p−α |||ap+n|+p+n+j−α

p−α |j||ap+n+j|

=||+|_j|51

from Lemma 1.1 which 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. Iffj(z)∈ S_p^∗(n, αj)for eachj = 1,2, . . . , m, thenHm(z)∈ T_p^∗(n, β)with β = inf

k=p+n

(

p− k(k−p)Qm

j=1(p−α_j)^pj pQm

j=1(k−αj)^pj −kQm

j=1(p−αj)^pj )

, 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|^pj

! 5

m

Y

j=1

k−α_j p−α_j

_qj¹

|a_k,j|^qj1,

which implies that

β 5p− k(k−p)Qm

j=1(p−αj)^pj pQm

j=1(k−α_j)^pj −kQm

j=1(p−α_j)^pj

for allk =p+n.

Corollary 2.5. Iff_j(z)∈ S_p^∗(n, α_j)for eachj = 1,2, . . . , m, thenG_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. Iff_j(z)∈ T_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=1p^pj(p−α_j)^pj pQm

j=1k^pj(k−α_j)^pj−kQm

j=1p^pj(p−α_j)^pj )

, wherepj = _q¹_j,qj >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

|ak,j|^pj

! 5

m

Y

j=1

k(k−α_j) p(p−α_j)

_qj¹

|ak,j|

1 qj

for allk =p+n.

Corollary 2.7. Iff_j(z)∈ T_p^∗(n, α_j)for eachj = 1,2, . . . , m, thenG_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−α_j)−p^m−1Qm

j=1(p−α_j).

Theorem 2.8. Iff_j(z)∈ T_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=1p^pj(p−α_j)^pj Qm

j=1k^pj(k−α_j)^pj −Qm

j=1p^pj(p−α_j)^pj )

, wherep_j = _q¹_j,q_j >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|^pj

! 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. Iff_j(z)∈ T_p^∗(n, α_j)for eachj = 1,2, . . . , m, thenG_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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[3] S. OWA, On certain classes of p−valent functions with negative coefficients, Simon. Stevin, 59 (1985), 385–402.

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