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volume 3, issue 3, article 42, 2002.

Received 5 March, 2002;

accepted 6 April, 2002.

Communicated by:G.V. Milovanovi´c

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Journal of Inequalities in Pure and Applied Mathematics

SOME GENERALIZED CONVOLUTION PROPERTIES ASSOCIATED WITH CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS

SHIGEYOSHI OWA AND H.M. SRIVASTAVA

Department of Mathematics Kinki University

Higashi-Osaka Osaka 577-8502, Japan EMail:owa@math.kindai.ac.jp

Department of Mathematics and Statistics, University of Victoria,

Victoria, British Columbia V8W 3P4, Canada.

EMail:harimsri@math.uvic.ca

URL:http://www.math.uvic.ca/faculty/harimsri/

c

2000School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

033-02

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Some Generalized Convolution Properties Associated with Certain Subclasses of Analytic

Functions

Shigeyoshi Owa,H.M. Srivastava

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J. Ineq. Pure and Appl. Math. 3(3) Art. 42, 2002

http://jipam.vu.edu.au

Abstract

For functions belonging to each of the subclassesM^∗_n(α)andN_n^∗(α)of normalized analytic functions in open unit disk U,which are introduced and investigated in this paper, the authors derive several properties involving their generalized convolution by applying certain techniques based especially upon the Cauchy-Schwarz and Hölder inequalities. A number of interesting conse- quences of these generalized convolution properties are also considered.

2000 Mathematics Subject Classification: Primary 30C45; Secondary 26D15, 30A10.

Key words: Analytic functions, Hadamard product (or convolution), Generalized con- volution, Cauchy-Schwarz inequality, Hölder inequality, Inclusion theo- rems.

The present investigation was supported, in part, by the Natural Sciences and Engi- neering Research Council of Canada under Grant OGP0007353.

1. Introduction and Definitions

LetA_ndenote the class of functionsf(z)normalized in the form:

(1.1) f(z) =z+

∞

X

k=n

a_kz^k (n∈N\ {1}; N:={1,2,3, . . .}), which are analytic in the open unit disk

U:={z :z ∈C and |z|<1}.

We denote by M_n(α) the subclass ofA_n consisting of functions f(z) which satisfy the inequality:

(1.2) R

zf⁰(z) f(z)

< α (α >1; z ∈U).

Also letN_n(α)be the subclass ofA_nconsisting of functionsf(z)which satisfy the inequality:

(1.3) R

1 + zf⁰⁰(z) f⁰(z)

< α (α >1; z ∈U).

Forn = 2and1< α < ⁴₃,the classesM₂(α)andN₂(α)were investigated earlier by Uralegaddi et al. (cf. [5]; see also [4] and [6]). In fact, following these earlier works in conjunction with those by Nishiwaki and Owa [1] (see also [3]), it is easy to derive Lemma1.1and Lemma1.2below, which provide the sufficient conditions for functions f ∈ A_nto be in the classes M_n(α)and Nn(α), respectively.

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Lemma 1.1. Iff ∈ A_ngiven by(1.1)satisfies the condition:

(1.4)

∞

X

k=n

(k−n)|a_k|5α−1

1< α < n+ 1 2

, thenf ∈ M_n(α).

Lemma 1.2. Iff ∈ A_ngiven by(1.1)satisfies the condition:

(1.5)

∞

X

k=n

k(k−α)|a_k|5α−1

1< α < n+ 1 2

, thenf ∈ N_n(α).

For examples of functions in the classes M_n(α) and N_n(α), let us first consider the functionϕ(z)defined by

(1.6) ϕ(z) := z+

∞

X

k=n

n(α−1) k(k+ 1) (k−α)

z^k, which is of the form (1.1) with

(1.7) a_k= n(α−1)

k(k+ 1) (k−α) (k =n, n+ 1, n+ 2, . . .), so that we readily have

(1.8)

∞

X

k=n

k−α α−1

|a_k|=n

∞

X

k=n

1

k − 1

k+ 1

= 1.

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Thus, by Lemma1.1,ϕ ∈ M_n(α). Furthermore, since (1.9) f(z)∈ N_n(α)⇐⇒zf⁰(z)∈ M_n(α), we observe that the functionψ(z)defined by

(1.10) ψ(z) :=z+

∞

X

k=n

n(α−1) k²(k+ 1) (k−α)

z^k belongs to the classN_n(α).

In view of Lemma1.1and Lemma1.2, we now define the subclasses M^∗_n(α)⊂ M_n(α) and N_n^∗(α)⊂ N_n(α),

which consist of functions f(z) satisfying the conditions (1.4) and (1.5), respectively.

Finally, for functionsf_j ∈A_n(j = 1, . . . , m)given by (1.11) f_j(z) = z+

∞

X

k=n

a_k,j z^k (j = 1, . . . , m), the Hadamard product (or convolution) is defined by

(1.12) (f₁∗ · · · ∗f_m) (z) :=z+

∞

X

k=n m

Y

j=1

a_k,j

! z^k.

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2. Convolution Properties of Functions in the Classes M

^∗_n

(α) and N

_n^∗

(α)

For the Hadamard product (or convolution) defined by (1.12), we first prove Theorem 2.1. Iff_j(z)∈ M^∗_n(α_j) (j = 1, . . . , m), then

(f₁ ∗ · · · ∗f_m) (z)∈ M^∗_n(β), where

(2.1) β = 1 + (n−1)Qm

j=1(α_j −1) Qm

j=1(n−α_j) +Qm

j=1(α_j −1). The result is sharp for the functionsf_j(z) (j = 1, . . . , m)given by (2.2) f_j(z) =z+

α_j −1 n−αj

zⁿ (j = 1, . . . , m).

Proof. Following the work of Owa [2], we use the principle of mathematical induction in our proof of Theorem 2.1. Let f₁(z) ∈ M^∗_n(α₁) and f₂(z) ∈ M^∗_n(α₂). Then the inequality:

∞

X

k=n

(k−α_j)|a_k,j|5α_j −1 (j = 1,2) implies that

(2.3)

∞

X

k=n

s k−α_j

α_j −1|a_k,j|51 (j = 1,2).

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Thus, by applying the Cauchy-Schwarz inequality, we have

∞

X

k=n

s

(k−α₁) (k−α₂)

(α1−1) (α2 −1)|a_k,1| |a_k,2|

2

5

∞

X

k=n

k−α₁ α₁−1

|ak,1|

! _∞ X

k=n

k−α₂ α₂−1

|ak,2|

! 51.

Therefore, if

∞

X

k=n

k−δ δ−1

|a_k,1| |a_k,2|5

∞

X

k=n

s

(k−α1) (k−α2)

(α₁−1) (α₂−1)|a_k,1| |a_k,2|, that is, if

q

|a_k,1| |a_k,2|5

δ−1 k−δ

s

(k−α₁) (k−α₂)

(α₁ −1) (α₂−1) (k=n, n+ 1, n+ 2, . . .), then(f1∗f2) (z)∈ M^∗_n(δ).

We also note that the inequality (2.3) yields q

|ak,j|5 s

αj−1

k−α_j (j = 1,2; k =n, n+ 1, n+ 2, . . .). Consequently, if

s

(α₁−1) (α₂−1)

(k−α1) (k−α2) 5 δ−1 k−δ

s

(k−α₁) (k−α₂) (α1−1) (α2−1),

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that is, if (2.4) k−δ

δ−1 5 (k−α₁) (k−α₂)

(α1−1) (α2−1) (k =n, n+ 1, n+ 2, . . .), then we have(f₁∗f₂) (z)∈ M^∗_n(δ). It follows from (2.4) that

δ =1 + (k−1) (α₁−1) (α₂−1)

(k−α₁) (k−α₂) + (α₁−1) (α₂−1) =:h(k)

(k =n, n+ 1, n+ 2, . . .). Sinceh(k)is decreasing fork=n,we have

δ=1 + (n−1) (α₁−1) (α₂−1)

(n−α₁) (n−α₂) + (α₁−1) (α₂−1), which shows that(f₁ ∗f₂) (z)∈ M^∗_n(δ), where

δ := 1 + (n−1) (α1−1) (α2−1)

(n−α₁) (n−α₂) + (α₁−1) (α₂−1). Next, we suppose that

(f₁∗ · · · ∗f_m) (z)∈ M^∗_n(γ), where

γ := 1 + (n−1)Qm

j=1(αj −1) Qm

j=1(n−α_j) +Qm

j=1(α_j −1).

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Then, by means of the above technique, we can show that (f₁∗ · · · ∗f_m+1) (z)∈ M^∗_n(β), where

(2.5) β := 1 + (n−1) (γ−1) (α_m+1−1)

(n−γ) (n−α_m+1) + (γ−1) (α_m+1−1). Since

(γ−1) (α_m+1−1) = (n−1)Qm+1

j=1 (α_j −1) Qm

j=1(n−α_j) +Qm

j=1(α_j−1) and

(n−γ) (n−α_m+1) = (n−1)Qm+1

j=1 (n−α_j) Qm

j=1(n−α_j) +Qm

j=1(α_j −1), Equation (2.5) shows that

β = 1 + (n−1)Qm+1

j=1 (α_j−1) Qm+1

j=1 (n−α_j) +Qm+1

j=1 (α_j−1).

Finally, for the functionsf_j(z) (j = 1, . . . , m)given by (2.2), we have (f₁∗ · · · ∗f_m) (z) = z+

m

Y

j=1

α_j−1 n−α_j

!

zⁿ=z+A_nzⁿ, where

A_n:=

m

Y

j=1

α_j−1 n−α_j

.

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It follows that

∞

X

k=n

k−β β−1

|A_k|= 1.

This evidently completes the proof of Theorem2.1.

By settingα_j =α(j = 1, . . . , m)in Theorem2.1, we get Corollary 2.2. Iff_j(z)∈ M^∗_n(α) (j = 1, . . . , m), then

(f₁ ∗ · · · ∗f_m) (z)∈ M^∗_n(β), where

β = 1 + (n−1) (α−1)^m (n−α)^m+ (α−1)^m.

The result is sharp for the functionsf_j(z) (j = 1, . . . , m)given by fj(z) = z+

α−1 n−α

zⁿ (j = 1, . . . , m).

Next, for the Hadamard product (or convolution) of functions in the class N_n^∗(z), we derive

Theorem 2.3. Iff_j(z)∈ N_n^∗(α_j) (j = 1, . . . , m), then (f₁∗ · · · ∗f_m) (z)∈ N_n^∗(β), where

β = 1 + (n−1)Qm

j=1(αj −1) n^m−1Qm

j=1(n−α_j) +Qm

j=1(α_j −1).

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The result is sharp for the functionsf_j(z) (j = 1, . . . , m)given by (2.6) f_j(z) = z+

α_j−1 n(n−αj)

zⁿ (j = 1, . . . , m).

Proof. As in the proof of Theorem 2.1, for f1(z) ∈ N_n^∗(α1) and f2(z) ∈ N_n^∗(α₂), the following inequality:

∞

X

k=n

k(k−δ) δ−1

|a_k,1| |a_k,2|51

implies that(f₁ ∗f₂) (z)∈ N_n^∗(δ). Also, in the same manner as in the proof of Theorem2.1, we obtain

(2.7) δ=1 + (k−1) (α₁−1) (α₂−1)

k(k−α₁) (k−α₂) + (α₁−1) (α₂−1)

(k =n, n+ 1, n+ 2, . . .). The right-hand side of (2.7) takes its maximum value fork =n,because it is a decreasing function ofk =n. This shows that(f₁∗f₂) (z)∈ N_n^∗(δ), where

δ= 1 + (n−1) (α₁ −1) (α₂−1)

n(n−α₁) (n−α₂) + (α₁−1) (α₂−1). Now, assuming that

(f₁∗ · · · ∗f_m) (z)∈ N_n^∗(γ),

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where

γ := 1 + (n−1)Qm

j=1(α_j −1) n^m−1Qm

j=1(n−α_j) +Qm

j=1(α_j −1), we have

(f₁∗ · · · ∗f_m+1) (z)∈ N_n^∗(β), where

β = 1 + (n−1) (γ−1) (α_m+1−1)

n(n−γ) (n−α_m+1) + (γ −1) (α_m+1−1)

= 1 + (n−1)Qm+1

j=1 (α_j−1) n^mQm+1

j=1 (n−α_j) +Qm+1

j=1 (α_j −1).

Moreover, by taking the functions f_j(z) given by (2.6), we can easily verify that the result of Theorem2.3is sharp.

By lettingα_j =α(j = 1, . . . , m)in Theorem2.3, we obtain Corollary 2.4. Iff_j(z)∈ N_n^∗(α) (j = 1, . . . , m), then

(f₁∗ · · · ∗f_m) (z)∈ N_n^∗(β), where

β = 1 + (n−1) (α−1)^m n^m−1(n−α)^m+ (α−1)^m.

The result is sharp for the functionsfj(z) (j = 1, . . . , m)given by fj(z) =z+

α−1 n(n−α)

zⁿ (j = 1, . . . , m).

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Now we turn to the derivation of the following lemma which will be used in our investigation.

Lemma 2.5. Iff(z)∈ M^∗_n(α)andg(z)∈ N_n^∗(β), then(f ∗g) (z)∈ M^∗_n(γ), where

γ := 1 + (n−1) (α−1) (β−1)

n(n−α) (n−β) + (α−1) (β−1). The result is sharp for the functionsf(z)andg(z)given by

f(z) =z+

α−1 n−α

zⁿ and

g(z) = z+

β−1 n(n−β)

zⁿ. Proof. Let

f(z) =z+

∞

X

k=n

a_kz^k ∈ M^∗_n(α) and

g(z) =z+

∞

X

k=n

b_kz^k ∈ N_n^∗(β). Then, by virtue of Lemma1.1, it is sufficient to show that

∞

X

k=n

k−γ γ−1

|a_k| |b_k|51

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for(f∗g) (z)∈ M^∗_n(γ). Indeed, since

∞

X

k=n

k−α α−1

|a_k|51

and ∞

X

k=n

k(k−β) β−1

|b_k|51, if we assume that

∞

X

k=n

k−γ γ−1

|a_k| |b_k|5

∞

X

k=n

s

k(k−α) (k−β)

(α−1) (β−1) |a_k| |b_k|, so that

p|a_k| |b_k|5

γ−1 k−γ

s

k(k−α) (k−β)

(α−1) (β−1) (k =n, n+ 1, n+ 2, . . .) then we prove that (f∗g) (z) ∈ M^∗_n(γ). Consequently, if γ satisfies the inequality:

γ =1 + (k−1) (α−1) (β−1)

k(k−α) (k−β) + (α−1) (β−1) (k=n, n+ 1, n+ 2, . . .), then (f ∗g) (z) ∈ M^∗_n(γ). Thus it is easy to see that (f ∗g) (z) ∈ M^∗_n(γ) withγgiven already in Lemma2.5.

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By combining Theorem2.1and Theorem2.3with Lemma2.5, we arrive at Theorem 2.6. If f_j(z) ∈ M^∗_n(α_j) (j = 1, . . . , p) and g_j(z) ∈ N_n^∗(β_j) (j = 1, . . . , q), then

(f₁∗ · · · ∗f_p∗g₁∗ · · · ∗g_q) (z)∈ M^∗_n(γ), where

γ = 1 + (n−1) (α−1) (β−1)

n(n−α) (n−β) + (α−1) (β−1),

(2.8) α= 1 + (n−1)Qp

j=1(α_j −1) Qp

j=1(n−αj) +Qp

j=1(αj−1), and

(2.9) β = 1 + (n−1)Qq

j=1(β_j−1) n^q−1Qq

j=1(n−β_j) +Qq

j=1(β_j −1).

The result is sharp for the functionsf_j(z) (j = 1, . . . , p)andg_j(z) (j = 1, . . . , q) given by

(2.10) fj(z) = z+

α_j −1 n−α_j

zⁿ (j = 1, . . . , p) and

(2.11) g_j(z) = z+

β_j−1 n(n−βj)

zⁿ (j = 1, . . . , q).

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Forα_j = α (j = 1, . . . , p)and β_j = β (j = 1, . . . , q), Theorem2.6 imme- diately yields

Corollary 2.7. If fj(z) ∈ M^∗_n(α) (j = 1, . . . , p) and gj(z) ∈ N_n^∗(β) (j = 1, . . . , q), then

(f1∗ · · · ∗fp∗g1∗ · · · ∗gq) (z)∈ M^∗_n(γ), where

γ = 1 + (n−1) (α−1)^p(β−1)^q

n^q(n−α)^p(n−β)^q+ (α−1)^p(β−1)^q.

The result is sharp for the functionsfj(z) (j = 1, . . . , p)andgj(z) (j = 1, . . . , q) given by

(2.12) f_j(z) = z+

α−1 n−α

zⁿ (j = 1, . . . , p) and

(2.13) g_j(z) = z+

β−1 n(n−β)

zⁿ (j = 1, . . . , q).

We also have the following results analogous to Theorem2.6and Corollary 2.7:

Theorem 2.8. If f_j(z) ∈ M^∗_n(α_j) (j = 1, . . . , p) and g_j(z) ∈ N_n^∗(β_j) (j = 1, . . . , q), then

(f1∗ · · · ∗fp ∗g1∗ · · · ∗gq) (z)∈ N_n^∗(γ),

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where

(2.14) γ = 1 + (n−1) (α−1) (β−1) (n−α) (n−β) + (α−1) (β−1),

α andβ are given by(2.8)and (2.9), respectively. The result is sharp for the functions f_j(z) (j = 1, . . . , p) and g_j(z) (j = 1, . . . , q) given by (2.10) and (2.11), respectively.

Corollary 2.9. If f_j(z) ∈ M^∗_n(α) (j = 1, . . . , p) and g_j(z) ∈ N_n^∗(β) (j = 1, . . . , q), then

(f₁∗ · · · ∗f_p ∗g₁∗ · · · ∗g_q) (z)∈ N_n^∗(γ), where

(2.15) γ = 1 + (n−1) (α−1)^p(β−1)^q

n^q−1(n−α)^p(n−β)^q+ (α−1)^p(β−1)^q.

The result is sharp for the functionsf_j(z) (j = 1, . . . , q)andg_j(z) (j = 1, . . . , q) given by(2.12)and(2.13), respectively.

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3. Generalizations of Convolution Properties

For functions f_j(z) (j = 1, . . . , m) given by (1.11), the generalized convolu- tion (or the generalized Hadamard product) is defined here by

(3.1) (f₁• · · · •f_m) (z) := z+

∞

X

k=n m

Y

j=1

(a_k,j)

1 pj

! z^k

m

X

j=1

1

p_j = 1; p_j >1; j = 1, . . . , m

! .

Our first result for the generalized convolution defined by (3.1) is contained in

Theorem 3.1. Iff_j(z)∈ M^∗_n(α_j) (j = 1, . . . , m), then (f1 • · · · •fm) (z)∈ M^∗_n(β), where

(3.2) β = 1 + (n−1)Qm

j=1(α_j−1)

1 pj

Qm

j=1(n−α_j)

1

pj +Qm

j=1(α_j−1)

1 pj

and

n−1 Qm

j=1(α_j −1)

m

X

j=1

α_j p_j

−1

!

=2.

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The result is sharp for the functionsf_j(z) (j = 1, . . . , m)given by (3.3) f_j(z) =z+

α_j −1 n−αj

zⁿ (j = 1, . . . , m).

Proof. We use the principle of mathematical induction once again for the proof of Theorem3.1. Since, forf₁(z)∈ M^∗_n(α₁)andf₂(z)∈ M^∗_n(α₂),

∞

X

k=n

k−α_j α_j −1

|a_k,j|51 (j = 1,2), we have

(3.4)

2

Y

j=1

∞

X

k=n

( k−α_j α_j −1

_pj¹

|ak,j|

1 pj

)pj!_pj¹ 51.

Therefore, by appealing to the Hölder inequality, we find from (3.4) that

∞

X

k=n

( ₂ Y

j=1

k−α_j αj−1

_pj¹

|a_k,j|

1 pj

) 51,

which implies that (3.5)

2

Y

j=1

|a_k,j|

1 pj 5

2

Y

j=1

α_j−1 k−α_j

¹

pj (k=n, n+ 1, n+ 2, . . .).

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Now we need to find the smallest δ 1< δ < ⁿ⁺¹₂

which satisfies the inequality:

∞

X

k=n

k−δ δ−1

² Y

j=1

|a_k,j|

1 pj

! 51.

By virtue of the inequality (3.5), this means that we find the smallest δ 1< δ < ⁿ⁺¹₂

such that

∞

X

k=n

k−δ δ−1

2

Y

j=1

|a_k,j|

1 pj

! 5

∞

X

k=n

k−δ δ−1

2

Y

j=1

α_j −1 k−α_j

¹

pj

! 51,

that is, that k−δ δ−1 5

2

Y

j=1

k−α_j α_j −1

_pj¹

(k =n, n+ 1, n+ 2, . . .), which yields

δ =1 + (k−1)Q2

j=1(α_j−1)

1 pj

Q2

j=1(k−αj)

1

pj +Q2

j=1(αj−1)

1 pj

(k =n, n+ 1, n+ 2, . . .).

Let us define

h(k) := k−1

Q2

j=1(k−α_j)

1

pj +Q2

j=1(α_j−1)

1 pj

(k =n).

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Then, for the numeratorN(k)ofh⁰(k), we have

N(k) = (α1−1)^p¹¹ (α2−1)^p¹² −(k−α1)^p¹¹⁻¹(k−α2)^p¹²⁻¹

·

k−1

p₁ (k−α₂) + k−1

p₂ (k−α₁)−(k−α₁) (k−α₂)

5(α₁−1)^p¹¹ (α₂−1)^p¹² −(k−α₁)^p¹¹⁻¹(k−α₂)^p¹²⁻¹

· 1

p₁ (k−α2) (α1−1) + 1

p₂ (k−α1) (α2−1)

.

Sincek =nand1< α_j < ⁿ⁺¹₂ ,we note thatk−α_j > α_j −1 (j = 1,2). This implies that

N(k)5−(k−α₁)^p¹¹⁻¹(k−α₂)^p¹²⁻¹

· 1

p₁ (k−α₂) (α₁−1) + 1

p₂ (k−α₁) (α₂−1)−(α₁−1) (α₂−1)

5−(k−α₁)^p¹¹⁻¹(k−α₂)^p¹²⁻¹

· 1

p₁ (n−α₂) (α₁−1) + 1

p₂ (n−α₁) (α₂−1)−(α₁−1) (α₂ −1)

=−(k−α₁)^p¹¹⁻¹(k−α₂)^p¹²⁻¹

·

(n−1) α₁

p₁ +α₂ p₂ −1

−2 (α₁−1) (α₂−1)

50,

by means of the condition of Theorem3.1. This implies thath(k)is decreasing

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fork =n.Consequently, we have

δ= 1 + (n−1)Q2

j=1(αj −1)

1 pj

Q2

j=1(n−α_j)

1 pj +Q2

j=1(α_j −1)

1 pj

. Thus the assertion of Theorem3.1holds true whenm= 2.

Next we suppose that

(f₁• · · · •f_m) (z)∈ M^∗_n(γ), where

γ = 1 + (n−1)Qm

j=1(α_j−1)

1 pj

Qm

j=1(n−α_j)

1

pj +Qm

j=1(α_j−1)

1 pj

. Then, clearly, the first half of the above proof implies that

(f₁• · · · •f_m+1) (z)∈ M^∗_n(β) with

β = 1 + (n−1) (γ −1)¹⁻

1

pm+1 (α_m+1−1)

1 pm+1

(n−γ)¹⁻

1

pm+1 (n−α_m+1)

1

pm+1 + (γ−1)¹⁻

1

pm+1 (α_m+1−1)

1 pm+1

. It is easy to verify that

β = 1 + (n−1)Qm+1

j=1 (α_j−1)

1 pj

Qm+1

j=1 (n−α_j)

1

pj +Qm+1

j=1 (α_j−1)

1 pj

.

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Thus, by the principle of mathematical induction, we conclude that (f₁ • · · · •f_m) (z)∈ M^∗_n(β),

whereβis given by (3.2).

Finally, by taking the functionsf_j(z) (j = 1, . . . , m)given by (3.3), we have (f1• · · · •fm) (z) = z+

m

Y

j=1

α_j −1 n−α_j

! zⁿ, which shows that

n−β β−1

^m Y

j=1

αj −1 n−α_j

_pj¹ !

= 1.

Therefore, Theorem 3.1 is sharp for the functionsf_j(z) (j = 1, . . . , m)given by (3.3). This completes the proof of Theorem3.1.

By puttingα_j =α(j = 1, . . . , m)in Theorem3.1, we obtain Corollary 3.2. Iff_j(z)∈ M^∗_n(α) (j = 1, . . . , m), then

(f₁• · · · •f_m) (z)∈ M^∗_n(α).

The result is sharp for the functionsfj(z) (j = 1, . . . , m)given by fj(z) = z+

α−1 n−α

zⁿ (j = 1, . . . , m).

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Similarly, for the generalized convolution defined by (3.1) for functions in the classN_n^∗(α), we derive

Theorem 3.3. Iffj(z)∈ N_n^∗(αj) (j = 1, . . . , m), then (f₁• · · · •f_m) (z)∈ N_n^∗(β), where

(3.6) β = 1 + (n−1)Qm

j=1(α_j −1)

1 pj

Qm

j=1(n−αj)

1

pj +Qm

j=1(αj −1)

1 pj

.

The result is sharp for the functionsf_j(z) (j = 1, . . . , m)given by (3.7) f_j(z) = z+

α_j −1 n(n−α_j)

zⁿ (j = 1, . . . , m).

Proof. By applying the same technique as in the proof of Theorem3.1, we find that(f₁•f₂) (z)∈ N_n^∗(δ), where

δ =1 + (k−1)Q2

j=1(α_j−1)

1 pj

Q2

j=1(k−α_j)

1

pj +Q2

j=1(α_j−1)

1 pj

(k =n, n+ 1, n+ 2, . . .), forf₁(z)∈ N_n^∗(α₁)andf₂(z)∈ N_n^∗(α₂). Therefore, we have

(3.8) δ = 1 + (n−1)Q2

j=1(α_j −1)

1 pj

Q2

j=1(k−α_j)

1

pj +Q2

j=1(α_j−1)

1 pj

.

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Furthermore, by assuming that

(f₁• · · · •f_m) (z)∈ N_n^∗(γ), where

γ = 1 + (n−1)Qm

j=1(αj−1)

1 pj

Qm

j=1(k−α_j)

1

pj +Qm

j=1(α_j−1)

1 pj

, we can show that

(f₁• · · · •f_m+1) (z)∈ N_n^∗(β), where

β = 1 + (n−1)Qm+1

j=1 (αj−1)

1 pj

Qm+1

j=1 (k−α_j)

1

pj +Qm+1

j=1 (α_j −1)

1 pj

.

Therefore, using the principle of mathematical induction once again, we conclude that

(f₁• · · · •f_m) (z)∈ N_n^∗(β) withβgiven by (3.6).

It is clear that the result of Theorem 3.3 is sharp for the functions f_j(z) (j = 1, . . . , m)given by (3.7).

Finally, by lettingα_j =α(j = 1, . . . , m)in Theorem3.3, we deduce Corollary 3.4. Iff_j(z)∈ N_n^∗(α) (j = 1, . . . , m), then

(f₁• · · · •f_m) (z)∈ N_n^∗(α).

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The result is sharp for the functionsf_j(z) (j = 1, . . . , m)given by (3.9) f_j(z) =z+

α−1 n(n−α)

zⁿ (j = 1, . . . , m).

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References

[1] J. NISHIWAKI AND S. OWA, Coefficient inequalities for certain analytic functions, Internat. J. Math. and Math. Sci., 29 (2002), 285–290.

[2] S. OWA, The quasi-Hadamard products of certain analytic functions, in Current Topics in Analytic Function Theory (H.M. Srivastava and S. Owa, Editors), World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992, pp. 234–251.

[3] S. SAITA ANDS. OWA, Convolutions of certain analytic functions, Alge- bras Groups Geom., 18 (2001), 375–384.

[4] B.A. URALEGADDI AND A.R. DESAI, Convolutions of univalent func- tions with positive coefficients, Tamkang J. Math., 29 (1998), 279–285.

[5] B.A. URALEGADDI, M.D. GANIGI AND S.M. SARANGI, Univalent functions with positive coefficients, Tamkang J. Math., 25 (1994), 225–

230.

[6] H.M. SRIVASTAVA AND S. OWA (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992.