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Some properties of a subclass of analytic functions

Presented by

Dr. Wasim Ul-Haq

Department of Mathematics

College of Science in Al-Zulfi, Majmaah University

KSA

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Presentation Layout

• Introduction• Basic Conceps• Preliminary Results• Main Results

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Introduction

Geometric Function Theory is the branch of Complex Analysiswhich deals with the geometric properties of analytic functions.The famous Riemann mapping theorem about the replacement of an arbitrary domain (of analytic function) with the open unit disk is the founding stone of the geometric function theory. Later, Koebe (1907) and Bieberbach (1916) studied analytic univalent functions which map E onto the domain with some nice geometric properties. Such functions and their generalizations serve a key role in signal theory,

constructing quadrature formulae and moment problems.

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:| | 1 ,E z z C

5

Functions with bounded turning, that is, functions whose derivative has positive real part and their generalizations have very close connection to various classes of analytic univalent functions. These classes have been considered by many mathematicians such as Noshiro and Warchawski (1935), Chichra (1977), Goodman (1983) and Noor (2009).

In this seminar, we define and discuss a certain subclass of analytic functions related with the functions with bounded turning. An inclusion result, a radius problem, invariance under certain integral operators and some other interesting properties for this class will be discussed.

Basic Concepts

The class A (Goodman, vol.1)[2]

The class S of univalent functions[2]

6

2

Let be the class of function :

( ) , (1)

which are analytic in the open unit disk :| | 1 .

nn

n

A f

f z z a z

E z z

C

1 2 1 2: ( ) ( ) for (2)S f A f z f z z z

The class P (Caratheodory functions) [2]

The class 19()

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Let be the class of functions which are analytic

in the open unit disk with (0) 1 and Re ( ) 0, for in .

P p

E p p z z E

1 2 1 2

1 1: ( ) ( ) ( ) ( ) ( ) , , , 2 .4 24 2k

k kP p p z p z p z p p P k

(Pinchuk,1971)kP

Some related classes to the class P (Noor, 2007)[6]

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We define the following.

is analytic in with and

Note that (1) , the class of functions with positive real part.

(ii). Let be the class of functions

(i). ( ) : ( ) (0) 1 arg ( ) , 0 1 .2

( )k

P

P p p z E p p z

P

P

1 2

2

in atisfying and

where 0 1, . Note that

analytic s (0) 1

1 1( ) ( ) ( ) , (3)

4 2 4 2

2, ( ), 1, 2 ( ) ( ).i

p E p

k kp z p z p z

k p P i P P

Examples

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(i). For

For

, the function

1 ( ) ( ).

1

(ii). , the function

1 1 1 1( ) ,

4 2 1 4 2 1

belongs to the class ( ).k

z E

zL z P

z

z E

k z k zp z

z z

P

Special classes of univalent functions [2]

Starlike functions (Nevanilinna, 1913)

Convex functions (Study, 1913)

Alexander relation (1915)

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* ( ): , .

( )

zf zS f A P z E

f z

( ( )): , .

( )

zf zC f A P z E

f z

*if and only if .f C zf S

Functions with bounded turning and related classes

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1 1 1

*

: ( ) , . ([2, vol.1, p.101])

( ) : ( ) ( ) , , (Chichra [1]).

It is well-known that functions in the class are univalent.

J. Krzyz [3] disproved that .We observe that

R f A f z P z E

R f A f z zf z P z E

R

R S

R

1 1( ) (0) , for >0.R R R

Convolution (or Hadamard Product)

Lemma1 (Singh and Singh)[9]

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2

2

Hadamard product (or convolution) of two analytic functions given by (1)

and ( ) , is defined by

( ) , .

nn

nn

nn

n nn

f

g z z b z

f g z z a b z z E

1If is analytic in , (0) 1 and Re ( )> , ,

2then for any function analytic in ,the function

takes the values in the convex hull of ( ).

p E p p z z E

F E p F

F E

A class of analytic functions [Noor and Haq, 7]

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2

2

( , , ) : ( ) ( ) ( ), ,

where , 0, 0,0 1and 2.

Special Cases

(i). ( ,0,1)

(ii). ( , ,1)

k kR f A f z zf z P z E

k

R R

R R

Preliminary Results

Lemma 2 (Lashin, 2005)[4]

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11 1 1 1 1 1

1

Let be analytic in with (0) 1, ( ) 0 in and let

2arg ( ) '( ) tan , 0, 0. (4)

2

Then

arg ( ) for .2

p E p p z E

p z zp z

p z z E

Inclusion result

Theorem 1

Main Results

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1

11 1 1 1

For , 0, 0, 2, and 0 1, ( , , ) ( ,0, ),

where

2 = tan , (5)

and

( , , ) : ( ) ( ) ( ), .

k k

k k

k R R

R f A f z zf z P z E

Proof

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1 2

Set

1 1 ( ) ( ) ( ) ( ) , ,

4 2 4 2

where ( ) is analytic in with (0) 1. Then

( ) ( ) .

This implies that

( ) ( ) , 1, 2.

Now with =

k

i i

k kf z p z p z p z z E

p z E p

p z zp z P

p z zp z P i

11 1 1 1

1 1

1

2tan and , we apply Lemma 2

to have , 1, 2, . Consequently, and

hence, ( ,0, ).

ki

k

p P i z E p P

f R

Applications of Theorem 1

Theorem 2

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-11

Let and let for 0, 0 and ,

( ) 2 ( ) + tan . (6)

Then

( ) .

k

k

f A z E

f zf z P

z

f zP

z

Proof

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1

1

11

( )Let for >0, ( ). We note that is analytic

in with (0) 1.Then

1 ( )( ) ( ) ( ) ,

2where tan .This implies that, for and 1, 2,

( ) ( )

k

i i

f zp z p

z

E p

f zp z zp z f z P

z

z E i

p z zp z

1

1

,

and, using Lemma 2, we have , 1, 2, .

Therefore, by (2), and this proves our result.

i

k

P

p P i z E

p P

Integral preserving property

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,

1

1,

0

1,

For a function given by (1), consider the integral operator ,

discussed in [4] and defined by

( )= [ ]( )= ( ) ( >0, 1), . (7)

The operator with =1,2,...

c

zc

c c

c

f A I

cH z I f z t f t dt c z E

z

I c

1,1

, was introduced by Bernardi. In particular,

the operator was studied by Libera. I

Theorem 3

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-11

-1

Let and ( ) be the integral operator defined by (7).

Let for >0, -1

( ) 2 ( ) + tan .

Then

( ) ( ) .

k

k

f A H z

c

f zf z P

z c

H zH z P

z

Convolution properties

Theorem 4

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1 2

-11 1

Let ( , , ), 1, 2 and let ( )= ( ). Then

( )( ), for ,

( )

2where = + and = + tan .

ki

k

f R i z f f z

z zP z E

z

Proof.

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2 2 1

1 1 1

2 1 2

1 1 2 1

Since ( , , ), it follows from Theorem 1, ,

where is given by (5). Similarly . Let

1 1 ( ) ( ) ( ) ( ) ,

4 2 4 2

( ) ( ), , , .

Now

(

k kf R f P

f P

k kf z p z p z p z

f z h z h p p P

z

1 2 1 2

1 1) ( ) ( * )( ) ( * )( ) ( * )( ).

4 2 4 2

k kz z f f z h p z h p z

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11

1

Applying Lemma 2, we have , where

2tan .

( ( ))Let ( ) . Then

( )

arg ( ) arg( ( )) arg ( )

( ) ( ) .2 2 2 2 4

This implies that .

k

k

P

z zH z

z

H z z z z

k k k

H P

Proof (Cont…..)

Theorem 5

Proof.

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Let ( , , ) and be a convex univalent function.

Then ( , , ) in .

k

k

f R

f R E

1 2

Let

(( * )( )) ( ( * )( )) )+ +

( )* ( ) ( ( ))

+ +

( )* ( ), ( ).

We can write

( ) 1 ( ) 1 ( )* ( ) * ( ) * ( ) .

4 2 4 2

( )Since is convex, Re

k

f z z f z

zf z zf z

z

zF z F P

z

z k z k zF z p z p z

z z z

z

z

1 in , and ( ), 1, 2.

2 iE p P i

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( )Therefore, by Lemma 1, * ( ) lies in the

connvex hull of ( ). Since is analytic in and

( )( ) : arg ( ) . It follows * ( )

2

( )lies in the . It implies that * ( )

i

i i

i i i i

zp z

zp E p E

zp E w w z p z

z

zF z

z

( ) and

conseqently ( , , ).

k

k

P

f R

Proof (Cont…..)

Applications of Theorem 5

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1

0

2

0

3

0

4

Let ( , , ) and , 1, 2,3,4, also belongs to ( , , ), where

( )( ) , (Alexander integral operator),

2( ) ( ) , (Libera integral operator),

( ) ( )( ) , ( x 1, x 1),

(

k ki

z

z

z

f R f i R

f tf z dt

t

f z f t dtz

f t f txf z dt

t tx

f z

1

0

1) ( ) , (Bernardi integral operator).

zc

c

ct f t dt

z

1( ) log(1 )z z

2

2( ) [ log(1 )]z z z

z

3

1 1( ) log

1 1

xzz

x x

411

1) n

n

cz z

n c

Radius problem (Inverse inclusion)

Theorem 6

Corollary 1

Miller and Mocanu [5] proved this result with a different technique.

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1

1 121 1 1

Let ( ,0, ). Then ( , , ) for , where

1, . (8)

2 4 2 1

For 2, 1, we have

k kf R f R z r

r

k

1

1

1

1

Let . Then ( ) for , where

and are given by (8).

f R f R z r

r

( , , ) ( ) : ( ) ( ) ( ), .k kR f z A f z zf z P z E

ConclusionThe arrow heads show the

inclusion relations.

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S

C

A

R*S

1( )R ( , , )kR

1 0

2k

11 1 1 1

2 = tan ,

1( ,0, )kR ( , , )kR

1 121 1 1

1,

2 4 2 1r

*

A number of analogous results, in finding connections of the class ( , , )

to the classes and , can be developed by using the techniques given in the book,

kR

C S

Differential Subordination by Miller and Mocanu [5].

References

[1] P.N. Chichra, New subclasses of the class of close-to-convex functions, Proc. Amer. Math. Soc., 62(1977) 37-43.

[2] A.W. Goodman, Univalent functions, Vol. I, II, Mariner Publishing Company, Tempa Florida, U.S.A 1983.

[3] J. Krzyz, A counter example concerning univalent functions, Folia Soc. Scient.. Lubliniensis 2(1962) 57-58.

[4] A.Y. Lashin, Applications of Nunokawa's theorem, J. Ineq. Pure Appl. Math., 5(2004), 1-5, Article 111.

[5] S. S. Miller and P. T. Mocanu, Differential subordination theory and applications, Marcel Dekker Inc., New York, Basel, 2000.

[6] K.I. Noor , On a generalization of alpha convexity, J. Ineq. Pure Appl. Math., 8(2007), 1-4, Article 16.

[7] K.I. Noor and W. Ul-Haq, Some properties of a subclass of analytic functions, Nonlinear Func. Anal. Appl, 13(2008)265-270.

[8] B. Pinchuk, Functions of bounded boundary rotations, Isr. J. Math., 10(1971),6-16.

[9] S. Singh and R. Singh, Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc., 106(1989), 145-152.

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• THANK YOU

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