JJ II

(1)

volume 3, issue 3, article 42, 2002.

Received 5 March, 2002;

accepted 6 April, 2002.

Communicated by:G.V. Milovanovi´c

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

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:[email protected]

Department of Mathematics and Statistics, University of Victoria,

Victoria, British Columbia V8W 3P4, Canada.

EMail:[email protected]

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

c

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

033-02

(2)

Some Generalized Convolution Properties Associated with Certain Subclasses of Analytic

Functions

Shigeyoshi Owa,H.M. Srivastava

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of27

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.

(4)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of27

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.

(5)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of27

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.

(6)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of27

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).

(7)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of27

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),

(8)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of27

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).

(9)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of27

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

.

(10)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of27

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).

(11)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of27

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),

(12)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of27

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).

(13)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of27

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

(14)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of27

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.

(15)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of27

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).

(16)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of27

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),

(17)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of27

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.

(18)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of27

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.

(19)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page19of27

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, . . .).

(20)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page20of27

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).

(21)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page21of27

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

(22)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page22of27

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

.

(23)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page23of27

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).

(24)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page24of27

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

.

(25)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page25of27

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).

(26)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page26of27

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).

(27)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page27of27

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.

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

School of Communications and Informatics,V