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volume 5, issue 4, article 82, 2004.

Received 01 June, 2004;

accepted 21 July, 2004.

Communicated by:G.V. Milovanovi´c

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Journal of Inequalities in Pure and Applied Mathematics

SOME SUBORDINATION RESULTS ASSOCIATED WITH CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS

H.M. SRIVASTAVA AND A.A. ATTIYA

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/

Department of Mathematics Faculty of Science University of Mansoura Mansoura 35516, Egypt EMail:aattiy@mans.edu.eg

c

2000Victoria University ISSN (electronic): 1443-5756 113-04

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Some Subordination Results Associated with Certain

Subclasses of Analytic Functions

H.M. Srivastava and A.A. Attiya

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J. Ineq. Pure and Appl. Math. 5(4) Art. 82, 2004

Abstract

For functions belonging to each of the subclassesM^∗(α)andN^∗(α)of normalized analytic functions in the open unit diskU, which are investigated in this paper whenα > 1,the authors derive several subordination results involving the Hadamard product (or convolution) of the associated functions. A number of interesting consequences of some of these subordination results are also discussed.

2000 Mathematics Subject Classification: Primary 30C45; Secondary 30A10, 30C80.

Key words: Analytic functions, Univalent functions, Convex functions, Subordination principle, Hadamard product (or convolution), Subordinating factor sequence.

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

1. Introduction, Definitions and Preliminaries

LetAdenote the class of functionsf normalized by

(1.1) f(z) = z+

∞

X

n=2

a_nzⁿ,

which are analytic in the open unit disk

U={z :z ∈C and |z|<1}.

We denote byM(α)andN (α)two interesting subclasses of the classA, which are defined (forα >1) as follows:

(1.2) M(α) :=

f :f ∈ A and R

zf⁰(z) f(z)

< α (z ∈U; α >1)

and

(1.3) N (α) :=

f :f ∈ Aand R

1 + zf⁰⁰(z) f⁰(z)

< α (z ∈U; α >1)

.

The classes M(α) and N(α) were introduced and studied by Owa et al.

([1] and [2]). In fact, for1 < α 5 ⁴₃,these classes were investigated earlier by Uralegaddi et al. (cf. [5]; see also [3] and [4]).

It follows from the definitions (1.2) and (1.3) that (1.4) f(z)∈ N(α)⇐⇒zf⁰(z)∈ M(α).

We begin by recalling each of the following coefficient inequalities associated with the function classesM(α)andN (α).

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Theorem A (Nishiwaki and Owa [1, p. 2, Theorem 2.1]). Iff ∈ A, given by (1.1), satisfies the coefficient inequality:

∞

X

n=2

[(n−λ) +|n+λ−2α|]|a_n|52 (α−1) (1.5)

(α >1; 05λ51), thenf ∈ M(α).

Theorem B (Nishiwaki and Owa [1, p. 3, Theorem 2.3]). Iff ∈ A, given by (1.1), satisfies the coefficient inequality:

∞

X

n=2

n[(n−λ) +|n+λ−2α|]|an|52 (α−1) (1.6)

(α >1; 05λ51), thenf ∈ N (α).

In view of TheoremAand TheoremB, we now introduce the subclasses (1.7) M^∗(α)⊂ M(α) and N^∗(α)⊂ N(α) (α >1), which consist of functions f ∈ Awhose Taylor-Maclaurin coefficientsa_n sat- isfy the inequalities (1.5) and (1.6), respectively. In our proposed investigation of functions in the classes M^∗(α)and N^∗(α), we shall also make use of the following definitions and results.

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Defintition 1 (Hadamard Product or Convolution). Given two functionsf, g ∈ A, wheref(z)is given by(1.1)andg(z)is defined by

g(z) =z+

∞

X

n=2

b_nzⁿ,

the Hadamard product(or convolution)f ∗gis defined(as usual)by

(f ∗g) (z) := z+

∞

X

n=2

a_nb_nzⁿ=: (g∗f) (z).

Defintition 2 (Subordination Principle). For two functionsf andg, analytic inU, we say that the functionf(z)is subordinate tog(z)inU, and write

f ≺g or f(z)≺g(z) (z ∈U), if there exists a Schwarz functionw(z), analytic inUwith

w(0) = 0 and |w(z)|<1 (z ∈U), such that

f(z) =g w(z)

(z ∈U).

In particular, if the function g is univalent in U, the above subordination is equivalent to

f(0) =g(0) and f(U)⊂g(U).

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Defintition 3 (Subordinating Factor Sequence). A sequence{b_n}^∞_n=1of com- plex numbers is said to be a subordinating factor sequence if, wheneverf(z)of the form(1.1)is analytic, univalent and convex inU, we have the subordination given by

(1.8)

∞

X

n=1

a_nb_nzⁿ ≺f(z) (z ∈U; a₁ := 1).

Theorem C (cf. Wilf [6]). The sequence {b_n}^∞_n=1 is a subordinating factor sequence if and only if

(1.9) R 1 + 2

∞

X

n=1

b_nzⁿ

!

>0 (z∈U).

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2. Subordination Results for the Classes M

^∗

(α) and M(α)

Our first main result (Theorem 1below) provides a sharp subordination result involving the function classM^∗(α).

Theorem 1. Let the functionf(z)defined by(1.1)be in the classM^∗(α). Also letKdenote the familiar class of functionsf ∈ Awhich are also univalent and convex inU. Then

(2−λ) +|2 +λ−2α|

2 [(2α−λ) +|2 +λ−2α|](f ∗g)(z) ≺g(z) (2.1)

(z ∈U; 05λ51; α >1; g ∈ K) and

(2.2) R f(z)

>− (2α−λ) +|2 +λ−2α|

(2−λ) +|2 +λ−2α| (z ∈U). The following constant factor in the subordination result(2.1):

(2−λ) +|2 +λ−2α|

2 [(2α−λ) +|2 +λ−2α|]

cannot be replaced by a larger one.

Proof. Letf(z)∈ M^∗(α)and suppose that g(z) =z+

∞

X

n=2

cnzⁿ ∈ K.

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Then we readily have

(2.3) (2−λ) +|2 +λ−2α|

2 [(2α−λ) +|2 +λ−2α|](f∗g)(z)

= (2−λ) +|2 +λ−2α|

2 [(2α−λ) +|2 +λ−2α|] z+

∞

X

n=2

c_na_nzⁿ

! .

Thus, by Definition3, the subordination result (2.1) will hold true if (2.4)

(2−λ) +|2 +λ−2α|

2 [(2α−λ) +|2 +λ−2α|] a_n ^∞

n=1

is a subordinating factor sequence (with, of course,a1 = 1). In view of Theorem C, this is equivalent to the following inequality:

(2.5) R 1 +

∞

X

n=1

(2−λ) +|2 +λ−2α|

(2α−λ) +|2 +λ−2α| a_nzⁿ

!

>0 (z ∈U).

Now, since

(n−λ) +|n+λ−2α| (05λ51; α >1) is an increasing function ofn, we have

R 1 +

∞

X

n=1

(2−λ) +|2 +λ−2α|

(2α−λ) +|2 +λ−2α| anzⁿ

!

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=R

1 + (2−λ) +|2 +λ−2α|

(2α−λ) +|2 +λ−2α| z

+ 1

(2α−λ) +|2 +λ−2α|

∞

X

n=2

[(2−λ) +|2 +k−2α|]a_nzⁿ

!

=1− (2−λ) +|2 +λ−2α|

(2α−λ) +|2 +λ−2α| r

− 1

(2α−λ) +|2 +λ−2α|

∞

X

n=2

[(n−λ) +|n+λ−2α|]|a_n| rⁿ

>1− (2−λ) +|2 +λ−2α|

(2α−λ) +|2 +λ−2α| r− 2(α−1)

[(2α−λ) +|2 +λ−2α|] r

>0 (|z|=r <1), (2.6)

where we have also made use of the assertion (1.5) of Theorem A. This evi- dently proves the inequality (2.5), and hence also the subordination result (2.1) asserted by Theorem1.

The inequality (2.2) follows from (2.1) upon setting

(2.7) g(z) = z

1−z =z+

∞

X

n=2

zⁿ ∈ K.

Next we consider the function:

(2.8) q(z) := z− 2(α−1)

(2−λ) +|2 +λ−2α| z² (05λ 51; α >1),

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which is a member of the classM^∗(α). Then, by using (2.1), we have (2.9) (2−λ) +|2 +λ−2α|

2 [(2α−λ) +|2 +λ−2α|]q(z)≺ z

1−z (z ∈U). It is also easily verified for the functionq(z)defined by (2.7) that (2.10) min

R

(2−λ) +|2 +λ−2α|

2 [(2α−λ) +|2 +λ−2α|]q(z)

=−1

2 (z ∈U), which completes the proof of Theorem1.

Corollary 1. Let the function f(z) defined by (1.1) be in the class M(α).

Then the assertions (2.1)and(2.2) of Theorem 1hold true. Furthermore, the following constant factor:

(2−λ) +|2 +λ−2α|

2 [(2α−λ) +|2 +λ−2α|]

By takingλ= 1 and1< α5 3

2 in Corollary1, we obtain

Corollary 2. Let the function f(z) defined by (1.1) be in the class M(α).

Then

1− 1

2α

(f∗g)(z) ≺g(z) (2.11)

z ∈U; 1< α5 3

2; g ∈ K

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and

(2.12) R f(z)

>− 1

2−α (z ∈U).

The constant factor1−¹₂αin the subordination result(2.11)cannot be replaced by a larger one.

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3. Subordination Results for the Classes N

^∗

(α) and N (α)

Our proof of Theorem 2 below is much akin to that of Theorem 1. Here we make use of TheoremBin place of TheoremA.

Theorem 2. Let the function f(z) defined by(1.1)be in the classN^∗(α). Then (2−λ) +|2 +λ−2α|

2 [(α+ 1−λ) +|2 +λ−2α|](f∗g)(z) ≺g(z) (3.1)

(z ∈U; 05λ51; α >1; g ∈ K) and

(3.2) R f(z)

>− (α+ 1−λ) +|2 +λ−2α|

(2−λ) +|2 +λ−2α| (z ∈U). The following constant factor in the subordination result(3.1):

(2−λ) +|2 +λ−2α|

2 [(α+ 1−λ) +|2 +λ−2α|]

Corollary 3. Let the function f(z) defined by (1.1) be in the class N(α).

Then the assertions (3.1) and(3.2) of Theorem2 hold true. Furthermore, the following constant factor:

(2−λ) +|2 +λ−2α|

2 [(α+ 1−λ) +|2 +λ−2α|]

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By lettingλ = 1 and1 < α 5 3

2 in Corollary3, we obtain the following further consequence of Theorem2.

Corollary 4. Let the function f(z) defined by(1.1)be in the classN(α). Then 2−α

2(3−α) (f∗g)(z) ≺g(z) (3.3)

z ∈U; 1< α5 3

2; g ∈ K

.

and

(3.4) R f(z)

>− 3−α

2−α (z ∈U). The following constant factor in the subordination result(3.3):

2−α 2 (3−α) cannot be replaced by a larger one.

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References

[1] S. OWA AND J. NISHIWAKI, Coefficient estimates for certain classes of analytic functions, J. Inequal. Pure Appl. Math., 3(5) (2002), Article 72, 1–5 (electronic). [ONLINE http://jipam.vu.edu.au/article.

php?sid=224]

[2] S. OWA AND H.M. SRIVASTAVA, Some generalized convolution prop- erties associated with certain subclasses of analytic functions, J. Inequal.

Pure Appl. Math., 3(3) (2002), Article 42, 1–13 (electronic). [ONLINE http://jipam.vu.edu.au/article.php?sid=194]

[3] 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.

[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.S. WILF, Subordinating factor sequences for convex maps of the unit circle, Proc. Amer. Math. Soc., 12 (1961), 689–693.