A subclass of convex functions with respect to conjugate points

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A SUBCLASS OF CONVEX FUNCTIONS WITH RESPECT TO CONJUGATE POINTS

UM WEI CHON

A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR

OF SCIENCE WITH HONOURS

PEEPU3TAlliN UNIVEnSITl MALAYSIA SABAt{

MATHEMATICS WITH ECONOMICS PROGRAMME SCHOOL OF SCIENCE AND TECHNOLOGY

UNIVERSITY MALAYSIA SABAH

2010

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DECLARATION

I affirm that this dissertation is of my own effort, except for the materials referred to as cited in the reference section.

ii

~.

UM WEI CHON (B507110447)

31 March 2010

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CERTIFIED BY

Signature

1. SUPERVISOR

(DR. AINI JANTENG)

2. EXAMINER

(ASSOC. PROF. DR. HO CHONG MUN)

3. DEAN

(PROF. DR. MOHO. HARUN ABDULLAH)

iii

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ACKNOWLEDGEMENT

I wish to acknowledge to my dearest supervisor, Dr. Aini Janteng who abundantly helpful and patiently offered invaluable assistance and guidance. I wish to express my sincere appreciation to my examiner, Assoc. Prof. Dr. Ho Chong Mun, for his comment in this dissertation.

In particular, lowe special thanks to Tan 5er Yuan and Loo Chien Ping for sharing the kindness and invaluable assistance. I also would like to acknowledge the support that

I

received from the entire students under supervision Dr. Aini Janteng for our team work and encouragement through each other so that I can manage to finish my research on time.

Last but not least, I wish to thanks my parents and family members for theirs encouragement and mentally support.

~.

UM WEI

^CHON

(6507110447)

31 March 2010

iv

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ABSTRACT

This research consider U as a class consisting of functions, which are analytic in the open unit disc on the complex plane. Let S be the subclass of U that consisting of univalent functions

f

and normalized. If a function

f

^ES, then

f

has a Maclaurin series expansion. This research introduced the subclass of convex functions with respect to conjugate points, which is denoted as CeCA, B), -1 S B

<

A S 1. By using the mathematical induction, the coefficient estimates was obtained. The distortion theorems and integral operator were also obtained for

f

E CeCA, B).

v

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SUBKELAS FUNGSI CEMBUNG TERHADAP TITK KONJUGAT

ABSTRAK

Kajian ini mempertimbangkan U sebagai kelas yang terdiri daripada fungsi-fungsi yang analisis di dalam cakera unit terbuka pada satah kompleks. Andaikan S merupakan subkelas U, yang terdiri daripada fungsi

f

yang univalen dan ternormal.

Jika suatu fungsi

f

^ES, maka

f

mempunyai kembangan siri Maclaurin. Kajian ini rnernperkenalkan subkelas fungsi cern bung terhadap titik konjugat yang dilambangkan sebagai Ce CA, ^B),^-1^~^B

<

A ~ 1. Dengan mengaplikasikan kaedah aruhan matematik, anggaran pekali telah diperoleh. Teorem herotan dan operator pengkamir juga telah ditentukan bagi

f

E CeCA, B).

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CHAPTER 3 COEFFICIENT ESTIMATES

14

3.1 Introduction 14

3.2 Lemma 15

3.3 Coefficient Estimates 15

CHAPTER 4 DISTORTION THEOREM AND INTEGERAL OPERATOR

26

4.1 Introduction 26

4.2 Distortion Theorem 26

4.3 Integral Operator 31

CHAPTER 5 DISCUSSION, CONCLUSION AND SUGGESTION

35

5.1 Discussion 35

5.2 Conclusion 5.3 Suggestion

REFERENCES

viii

36 37

38

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LIST OF FIGURES

No. Figure

1.1 Starlike domain 1.2 Convex domain

2.1 Starlike with respect to symmetric paints domain

ix

Page

3

4 8

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LIST OF SYMBOLS

= equal

+

^plus

minus dot operator factorial

<

^{less than}

>

greater than

~ less than or equal to

E element of

~ subset of or equal to

< surbodination

~ summation sign

n

product sign Z conjugate z

Izl

^{modulus z}

x

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CHAPTER 1

INTRODUCTION

1.1 Introduction

In this study, we let U be the class of functions w which are analytic in the open unit disc D = {z: \z\

<

l} given by

co

w(z) =

I

^bkz^k

k=l

and satisfying the conditions

w(O) = 0 and \w(z)\

<

l,z E D.

Next, we let S denote the class of functions

f

which are univalent and normalized in D of the form

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00

fez) =

z + L

^anzⁿ ^(1.1)

n=2

where an is a complex number.

In Goodman (1975), a function f(z) is said to be univalent in a domain E, E ~ C if f(zl) = f(Z2), implies thatz₁= ^Z2for Zl,Z2 E D. In other words, a function fez) is said to be univalent in domain E if it matches a one-to-one mapping of E onto its image. Some other terms are used to describe the univalent functions, such as 'schlicht (used in German means simple) and 'odnolistni '(used in Russians means single sheeted). There are many mathematicians studied univalent functions such as Spencer (1947), Oas &. Signh (1977) and Goel

&.

Mehrok (1982).

There are four main subclasses of class S which includes starlike, convex, close to convex and quasi convex.

1.2 Starlike Function

Definition 1.2.1 (Goodman, 1975) A set E, E ~ C in the plane is said to be starlike with respect to ^Woan interior point of E if each ray with Initial point ^Wo intersects the interior of E in a set that is either a line segment or a ray. If a function fez) maps D onto a domain that is starlike with respect to wo, then we say that fez) is starlike with respect to woo In the special case the ^Wo= 0, we say that fez) is a starlike function.

2 UMS UNIVERSITI MALAYSIA SABAH

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The domain shown in Figure 1.1 is starlike represented by its geometrical interpretation.

1m

'"

"

, , , ,

" , "

'"

, , ,

><'"

"

'" '" '"

'"

---~---+~"---~~~--_.

Re

Figure 1.1 Starlike domain

(Source from Goodman, 1975)

The class of starlike function is denoted by S·. The analytic description of the functions

f

^ES· is given below.

Definition 1.2.2 (Goodman, 1975) Let { be analytic in D with {CO) = {'CO) - 1 = O. Then,

f

E S· if and only if

{ z((z)}

Re

fez) >

0, ^ZED. (1.2)

Next, we give geometrical and analytic description for the convex function.

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1.3 Convex Function

Definition 1.3.1 (Goodman, 1975) A set E in plane is called convex if for every pair of points Wi and Wz in the interior of E, the line segment joining WI and Wz is also in the interior of E. If a function

f

maps D onto a convex domain, then

f

is called a convex function.

The domain shown in Figure 1.2 is convex represented by its geometrical interpretation.

1m

"

^"-_"-

"-

"

^"-_"-

"

^"-

Figure 1.2 Convex domain

" " "

^"-

"

^"-

" "

"-

"

(Source from Goodman,1975)

Re

The class of convex function is denoted by C. The analytic description of the functions

fEe

is given below.

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Definition 1.3.2 (Goodman, 1975) Let [be analytic in D with[CO)

=

['CO) -1

=

O. Then, [ E C if and only if

{ ( zt'

^Cz)

Y}

Re ((z)

>

0, zED. C1.3)

1.4 Subordination

In this study, we also consider the concept of differential subordination as mentioned in Miller & Mocanu (2000) as follows:

Definition 1.4.1 (Miller & Mocanu, 2000) let

f,

9 E U. The function [Cz) is said to be subordinate to 9CZ), written as

f

-< g, if there exists a function w E U such that fez)

=

9( wCz)) for all zED. If gCz) is univalent in D, then f

-<

9 if and only if fCO) = 9CO) and fCD) ~ 9CD).

1.5 Objectives of study

In this study, we have four objectives to be obtained. There are

a. to introduce a subclass of convex functions with respect to conjugate points which is denoted by CeCA, 8), -1 ~ 8

<

A ~ 1 ;

b. to determine the coefficients estimates for [ E CeCA, 8) ; c. to determine the distortion theorem for [ E CeCA, B) ; and d. to determine the integral operator for [ E CeCA, B) .

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1.6

Scope of

study

In this paper, we impress on class of convex functions with respect to conjugate points. By combining the idea of Das & Singh (1977) and Goel & Mehrok (1982), we introduce a new subclass of functions which is denoted by CeCA, B), -1 ~ B

<

^A^~

1 and determine the coefficients estimates, distortion theorem and integral operator for the functions

f

^ECeCA, B).

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CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

In this chapter, we discuss the class of starlike with respect to symmetric, conjugate and symmetric conjugate pOints. In year 1959, the class of starlike with respect to symmetric points was introduced by Sakaguchi. Then the class has been quoted and considered by Robertson (1961), Stankiewicz (1965), Wu (1987) and Owa et al.

(1988). EI-Ashwah & Thomas (1987) has introduced two more classes namely starlike with respect to conjugate and symmetric conjugate points.

2.2 Starlike With Respect To Symmetric, Conjugate and Symmetric Conjugate Points

In year 1959, Sakaguchi was the first mathematician who introduced the class of function starlike with respect to symmetric points. He calls a function

I

which is analytic in D starlike with respect to symmetric points if for every r ~ l(T

<

1) and every point Zo , the angular velocity of

I

about the point

I(

-zo) is positive at z = Zo , as z traverses the circle

Izl = r

in the positive direction.

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The starlike with respect to symmetric points can be shown by Figure 2.1.

zo

= re

ⁱ⁸

-Zo

f(-zo)

Figure 2.1.

Starlike with respect to symmetric points domain (Source from Sakaguchi,

1959)

Sakaguchi

(1959)

also proved that the condition is equivalent with the following definition.

Definition 2.2.1

(Sakaguchi,

1959) Let

f be analytic in

D

withf(O) ::;: f

'(0)

-1 ::;:

o . Then f is starlike with respect to symmetric points if and only if

{

zf'(z) }

Re fez) _ fe -z)

>

0, zED. (2.1)

We denote this class as s;.

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Then, EI-Ashwah & Thomas

(1987)

have introduced two more classes which called starlike with respect to conjugate and symmetric conjugate paints.

Definition 2.2.2 (EI-Ashwah & Thomas,

1987)

Let

I

be analytic in D with

ICO) =

I

^{'CO) -} 1 = 0 . Then

I

is starlike with respect to conjugate points if and only if

Re

f_

^{zl'Cz) }}

>

⁰

VCz)

+

^I(z) ^, ^zED. ^(2.2)

We denote this class as

s;.

Definition 2.2.3 (EI-Ashwah & Singh,

1987)

Let

I

be analytic in D with 1(0) =

1'(0) - 1 = O. Then

I

is starlike with respect to symmetric conjugate points if and only if

We denote this class as

S;c.

Ref- zl'(z)

}>O,

V(z) -

I(

-i)

2.3 Class S;CA,B) and S~(A,B)

zED. C2.3)

Goel & Mehrok

(1982)

have introduced the subclass of starlike with respect to symmetric points denoted by S;(A, B). Then, Suci Aida Fitri Mad Dahhar & Aini Janteng

(2009)

introduced another subclass of starlike with respect to conjugate paints denoted by S;CA, B).

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Definition 2.3.1 (Goel & Mehrok, 1982) Let { be analytic in D with

teO)

⁼

f'(o) -1 = O. Then, { E S;(A, B) if and only if

2z( (z) 1

+

Az

~~--:-:---:-<: ,

fez) - {(-z) 1

+

Bz -l~B<A~I,ZED.

By definition of subordination, it follows that { E

s;

^{(A, B)}if and only if

where

2z{'(z)

=

¹

+

^Aw(z)

=

^P(z),

fez) - {( -z) 1

+

Bw(z)

00

P(z) = 1

+ I

^{Pn zn}

n=l

w(z) E U (2.4)

(2.5)

Definition 2.3.2 (Suci Aida Fitri Mad Dahhar & Aini Janteng, 2009) Let { be analytic in D with f(O)

=

^{f'(O) -} ¹

=

O. Then, {E S~(A,B) if and only if

2z{'(z) 1 +Az --:--:---== <: ,

f(z)

+

{(i) 1

+

Bz -1 ~ B

<

A ~ 1, zED.

By definition of subordination, it follows that { E S~ (A, B) if and only if

2z(z) 1

+

^Aw(z)

fez)

+

{(i)

=

¹

+

^Bw(z)

=

^P(z), ^w(z)^E^U

10

(2.6)

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where P(z) of the form (2.5).

2.4 Convex With Respect To Symmetric, Conjugate and Symmetric Conjugate Points

Oas & Singh (1977) using the idea of Sakaguchi and extended the result to other class in D which is convex functions with respect to symmetric paints.

Definition 2.4.1

(Oas & Singh, 1977) Let { be analytic in D with teO)

=

^{'(O)-

1

=

O. Then { is convex with respect to symmetric points if and only if

{

(Z{'(z))' }

Re ,

>

0,

(f(z) - {( -z))

zED. (2.7)

We denote this class as C_{S •}

The class C_scan be extended to C, and Csc.

Definition 2.4.2

(Aini Janteng

et al.,

2006) Let { be analytic in D with teO)

=

{'(O) - 1

=

O. Then { is convex with respect to conjugate points if and only if

{

(Z{'(Z))' }

Re ,

>

0,

(f(z)

+

f(i))

ZED. (2.8)

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We denote this class as C_c•

Definition 2.4.3 (Aini Janteng et aI., 2006) Let { be analytic in D with {CO) = {'CO) -1 =

o.

Then { is convex with respect to symmetric conjugate points if and only if

{

(z{'Cz))' }

Re

, >

0,

([(z) -

[e-z))

^zED. ^(2.9)

We denote this class as C_{sc •}

2.5 Class CsCA,B)

Aini Janteng & Suzeini Abdul Halim (2008) introduced the subclass of convex with respect to symmetric points denoted by Cs(A, B).

Definition 2.5.1 (Aini Janteng & Suzeini Abdul Halim, 2008) Let [ be analytic in D with [(0)

=

^{['(0) -1}⁼O. Then [ E CsCA, B) if and only if

2 (Z['(Z))' 1+Az -'-';"'-""';"---', <: ,

(fCz) - [( -z))

1

+ Bz

^-1^~^B

<

A ~ 1, zED.

We denote this subclass as Cs(A, B).

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By definition of subordination, it follows that

f

^ECs(A, B) if and only if

2 (zf '(z))' 1

+

Aw(z)

-~-..-;..~, = = P(z) w(z) E U

(f(z) - f( -z)) 1

+

Bw(z) , ^(2.10)

where P(z) of the form (2.5).

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REFERENCES

Aini Janteng & Suzeini Abdul Halim. 2008. A subclass of convex functions with respect to symmetric points. Prosiding Simposium Kebangsaan Sains Matematik Ke-16, 3-5 June 2008, Kota Bharu, Kelantan: 2-5.

Aini Janteng, Suzeini Abdul Halim & Maslina Darus. 2006. Functions close-to-convex and quasi-convex with respect to other points. International Journal of Pure and Applied Mathematics, 30(2): 225-236.

AI-Kharsani, H.A. & AI-Khal, R.A. 2003. On the derivatives of a family of analytic functions. Red. Isot. Mat. Univ. Trieste, 35: 1-17.

Das, R.N. & Singh, P. 1977. On subclasses of schlicht function mapping. Indian

J.

Pure Appl. Math.,

8:

864-872.

Duren, P.L. 1983. Univalent functions. Springer-Verlag, New York.

Dziok, J. 2010. Classes of meromorphic function defined by the Hadamard product.

International Journal of Mathematics and Mathematical Sciences, 10: 1-11.

EI-Ashwah, R.M. & Thomas, D.K. 1987. Some subclasses of close-to-convex functions.

J.

Ramanujan Math. Soc.,

2:

85-100.

Ghanim, F. & Darus, M. 2010. Some properties of certain subclass of meromorphically multivalent functions defined by linear operator. Journal of Mathematics and Statistics, 6(1): 34-41.

Goel, R.M. & Mehrok, B.S. 1982. A subclass of starlike functions with respect to symmetric points. Tamkang

J.

Math., 13(1): 11-24.

Goodman, A.W. 1975. Univalent Functions. Volume I. Mariner Publishing Company, Inc., Tampa, Florida.

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Khaimar, S.M. & More, M.

2008.

Certain family of analytic and univalent functions with normalized condition. Acta Mathematica Academice Paedagogicae Nyiregyhaziensis,

24: 333-344.

Miller, 5.5. & Mocanu, P .T.

2000.

Differential Subordinations: Theory and Applications.

Marcel Dekker Inc., Basel, New York.

Noor, K.I.

2005.

On classes of analytic functions defined by convolution with incomplete beta functions. Journal of Mathematical Analysis and Applications,

307: 339-349.

Owa, 5., Wu, Z. & Ren, F.

1988.

A note on certain subclass of Sakaguchi functions.

Bull. de la Royale de Liege, 57:

143-150.

Ravichandran, V.

2004.

Starlike and convex functions with respect to conjugate points. Acta Mathematica Academice Paedagogicae Nyiregyhaziensis, 20:

31- 37.

Robertson, M.I.S.

1961.

Applications of the subordination principle to univalent functions. Pacific

J.

of Math.,

11: 315-324.

Sakaguchi, K.

1959.

On a certain univalent mapping.

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Math. Soc. Japan,

11: 72-75.

Spencer, D.C.

1947.

Some problems in conformal mapping. Bulletin of the American Mathematical SOciety,

53: 417-439.

Srivastava, H.M. & Attiya, A.A.

2004.

Some subordination results associated with certain subclasses of analytic functions.

J.

Inequal. Pure and Appl. Math., 5(4):

1-6.

Stankiewicz,

J. 1965.

Some remarks on functions starlike with respect to symmetric points. Ann. Univ. Marie Curie Sklodowska,

19(7): 53-59.

39

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Sud Aida Fitri Mad Dahhar & Aini Janteng. 2009. A subclass of starlike functions with respect to conjugate paints. International Mathematical Forum, 4(28): 1373- 1377.

Wu, Z. 1987. On classes of Sakaguchi functions and Hadamard products. Sci. Sinica Ser. A.,

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128-135.

A subclass of convex functions with respect to conjugate points