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Some properties of an integral operator defined by convolution Journal of Inequalities and Applications 2012, 2012:13 doi:10.1186/1029-242X-2012-13 Muhammad Arif marifmaths@awkum.edu.pk

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Some properties of an integral operator defined by convolution

Journal of Inequalities and Applications 2012, 2012:13 doi:10.1186/1029-242X-2012-13

Muhammad Arif (marifmaths@awkum.edu.pk) Khalida Inayat Noor (khalidanoor@yahoo.com) Fazal Ghani (maxy20052001@yahoo.com)

ISSN 1029-242X

Article type Research

Submission date 15 September 2011

Acceptance date 19 January 2012

Publication date 19 January 2012

Article URL http://www.journalofinequalitiesandapplications.com/content/2012/1/13

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Journal of Inequalities and

Applications

Some properties of an integral operator defined by convolution

Muhammad Arif*1, Khalida Inayat Noor2 and Fazal Ghani1

1

Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan

2Department of Mathematics, COMSATS Institute of Information Technology,

Islamabad, Pakistan

*

Corresponding author: marifmaths@awkum.edu.pk

E-mail addresses:

KIN: khalidanoor@hotmail.com

FG: univalentsfg@yahoo.com

Abstract

In this investigation, motivated from Breaz study, we introduce a new family of integral operator using famous convolution technique We also apply this newly defined operator for investigating some interesting mapping properties of certain subclasses of analytic and univalent functions

2010 Mathematics Subject Classification: 30C45; 30C10

Keywords: close-to-convex functions; convolution; integral operators

1 Introduction

Let A denote the class of analytic function satisfying the condition f ( )0 = f ′( )0 − =1 0 in the open unit disc U={z: z <1 } By * *

, , , ,

S C S C and Kwe means the well-known subclasses of

A which consist of univalent, convex, starlike, quasi-convex, and close-to-convex functions, respectively The well-known Alexander-type relation holds between the classes C and S*, and

* and ,

C K that is,

,

f z ∈C ⇔ zf′ z ∈S

and

f z ∈C ⇔zf′ z ∈K

It was proved in [1] that a locally univalent function f z( ) is close-to-convex, if and only if

( ) ( )

2

1

Re 1 zf z d , z re i , (1.1)

f z

θ

θ θ

′′

+ > − =

′

∫

for each r∈(0,1) and every pair θ θ1, 2 with 0≤θ1 <θ2 ≤2 π

Let P k( )ξ be the class of functions p z( ) analytic in U withp( )0 = and 1

( )

2

0

Re

, , 2

1

i

p z

d k z re k

π

θ

ξ

ξ

−

−

∫

This class was introduced in [2] and for k =2,ξ =0, the class P k( )ξ reduces to the class P of

functions with positive real part We consider the following classes:

( ) ( ) ( )

( ) ( )

zf z

f z

f z

g z

′

′

′

U

U

These classes were studied by Noor [3–5] and Padmanabhan and Parvatham [2] Also it can easily be seen that ( ) *

2 0

R =S and T2( )0 =K, where S* and K are the well-known classes of starlike and close-to-convex functions

Using the same method as that of Kaplan [1], Noor [6] extend the result of Kaplan given in (1.1), and proved that a locally univalent function f z( ) is in the class T k, if and only if

( ) ( )

2

1

Re 1 , , (1.2)

2

i

f z

θ

θ θ

′′

′

∫

for each r∈(0,1) and every pair θ θ1, 2 with 0≤θ1 <θ2 ≤2 π

For any two analytic functions

the convolution (Hadamard product) of f z( ) and g z( ) is defined by

0

g n n n,

n

f z z a b z z

∞

=

Using the techniques from convolution theory many authors generalized Breaz operator in several directions, see [7, 8] for example Here, we introduce a generalized integral operator

( , , )( ): n

I f g h z A →A as follows

( )( ) ( ( ) ( ) ) ( ) ( )

1 0

, , , 1.3

i i

i

i

h t

I f g h z f t g t dt

t

β α

=

=∫ ∏ ∗    

where f i( )z ,g i( )z ,h z i( )∈A with f i( )z ∗ g i( )z ≠0 and α βi, i ≥ for 0 i =1, 2,K, n The operator I n(f g h i, i, i)( )z reduces to many well-known integral operators by varying the parameters α βi, i and by choosing suitable functions instead of f i( )z ,g i( )z For example,

(i) If we take ( )

(1 ) for all 1 ,

i

z

z

− we obtain the integral operator

1 0

, , 1.4

i i

i

i

h t

t

β α

=

′

∏

∫

introduced in [9]

(ii) If we take αi =0 and 1≤ ≤i n, we obtain the integral

1 0

,

i

i

i

h t

t

β

=

=  

∏

∫

introduced and studied by Breaz and Breaz [10]

(iii) If we take ( )

(1 ), 0,

z

g z

z β

− we obtain the integral operator

( )( ) ( ( ) )

1 0

,

i

i

α

=

′

=∫ ∏

introduced and studied by Breaz et al [11]

(iv) If we take n=1,α1 =0 and β1 = in (1.4), we obtain the Alexander integral operator 1

( )( ) 1( ) 1

0

,

z n

h t

t

 

=  

 

∫

introduced in [12]

(v) If we take n=1,α1 =0 and β1 =β , we obtain the integral operator

( )( ) 1( ) 1

0

,

z n

h t

t

β

=  

∫

studied in [13]

In this article, we study the mapping properties of different subclasses of analytic and univalent functions under the integral operator given in (1.3) To prove our main results, we need the following lemmas

Lemma 1.1 [14] Let f z( )∈R k( )ξ for k ≥2, 0≤ξ <1 Then with 0≤θ1 <θ2 ≤2π and

, 1,

i

z =reθ r<

( )

2

1

2

d

f z

θ

θ

′

> − − −

 

∫

Lemma 1.2 [15] If f z( )∈C and g z( )∈K, then f z( )∗g( )z ∈K

2 Main results

Theorem 2.1 Let ( ) * ( ) * ( ) ( )

f z ∈S g z ∈C h z ∈R ξ with 0≤ξ <1, k ≥2 for all

1≤ ≤i n If

1

1 1 1, 2.1 2

n

i

k

=

∑

then integral operator defined by (1.3) belongs to the class of close-to-convex functions

Proof Let ( ) * ( ) *

f z ∈S g z ∈C Then there exists ϕi( )z ∈C such that

( ) ( )

f z = zϕ′ z Now consider

( )* ( ) ( )* ( ) ( )* ( )

f z g z =zϕ′ z g z =ϕ z zg′ z

Since ( ) *

,

i

g z ∈C then by Alexander-type relation zg i′( )z ∈K. So, by Lemma 1.2, we have

( )* ( ) ,

which implies that

( )* ( )

f z g z ∈K

and hence, by using (1.1),

( ) ( )

( ) ( )

2

1

*

Re 1 (2.2)

*

z f z g z

d

f z g z

θ

θ

′

∫

From (1.3), we obtain

1

i i

n

i

i

h z

I f g h z f z g z

z

β α

=

′ =∏ ∗    

Differentiating (2.3) logarithmically, we have

( ) ( )

( ) ( )

( ) ( )

n

, ,

, ,

n

z f z g z

h z

′′

( ) ( )

( ) ( )

( )

n

z f z g z zh z

h z

f z g z

∗

Taking real part and then integrating with respect to θ , we get

( ) ( )

( ) ( )

, ,

, ,

n

i i

z f z g z

I f g h z

=

∑

( )

2

1

zh z

d

h z

θ

θ

Using (2.2) and Lemma 1.1, we have

2

, ,

2 , ,

n

i

d

I f g h z

θ

θ

=

∑

∫

n ( ) ( 2 1)

1

i

=

From (2.1), we can easily write

2

k

+ <  + −  − ≤

This implies that

1

1 ,

n

i

=

+ <

∑

so, minimum is for θ1 =θ2, we obtain

( ) ( )

2

1

, ,

, ,

I f g h z

d

I f g h z

θ

θ

′

∫

and this implies that I n(f g h i, i, i)( )z ∈K

For k =2in Theorem 2.1, we obtain

Corollary 2.3 Let ( ) * ( ) * ( ) *( )

f z ∈S g z ∈C h z ∈S ξ with 0≤ξ <1, for all 1≤ ≤i n If

1

1,

n i i

α

=

≤

∑

then I n(f g h i, i, i)( )z ∈K

Theorem 2.4 Let f i( )z ∈T k and h z i( )∈R k for 1≤ ≤i n If α βi, i ≥ such that 0 αi+βi ≠0 and

1

1, 2.4 2

n

i

k

=

+ − ≤

∑

then I n(f h i, i)( )z defined by (1.4) belongs to the class of close-to-convex functions

Proof From (1.4), we have

1

, 2.5

i i

n

i

i

h z

I f h z f z

z

β α

=

∏

Differentiating (2.5) logarithmically, we have

( ) ( ) ( ) ( )

( ) ( )

( )

,

,

n

i

I f h z z f z z h z

h z

f z

I f h z

Taking real part and then integrating with respect to θ, we get

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

n

,

,

n

h z

f z

I f h z

n ( ) ( 2 1)

1

i

=

n n ( ) ( 2 1)

n

kπ k

where we have used Lemma 1.1 and (1.2)

2

n

k

= −   + −  + − +  −

From (2.4), we can obtain

1

1

n

i

=

+ <

∑

So minimum is for θ1 =θ2, thus we have

( ) ( ) ( ) ( )

2

1

,

,

I f h z

d

I f h z

θ

θ

 +  > −

∫

This implies that I n(f i,h i)( )z ∈K

For k =2 in Theorem 2.4, we obtain the following result

Corollary 2.5 Let ( ) ( ) *

,

f z ∈K h z ∈S for 1≤ ≤i n and

1

1, 2

n

i

k

=

+ − ≤

∑

then I n(f i,h i)( )z defined by (1.4) belongs to the class of close-to-convex functions

Competing interests

The authors declare that they have no competing interests

Authors’ contributions

MA completed the main part of this article, KIN presented the ideas of this article, FG participated in some results of this article MA made the text file and all the communications regarding the manuscript All authors read and approved the final manuscript

Acknowledgments

The authors would like to thank the reviewers and editor for improving the presentation of this article, and they also thank Dr Ihsan Ali, Vice Chancellor AWKUM, for providing excellent

research facilities in AWKUM

References

[1] Kaplan, W: Close-to-convex Schlicht functions Michigan J Math 1, 169–185 (1952)

[2] Padmanabhan, K, Parvatham, R: Properties of a class of functions with bounded

boundary rotation Ann Polon Math 31, 311–323 (1975)

[3] Noor, KI: On radii of convexity and starlikeness of some classes of analytic functions

Int J Math Math Sci 14(4), 741–746 (1991)

[4] Noor, KI: On some integral operators for certain families of analytic function Tamkang

J Math 22, 113–117 (1991)

[5] Noor, KI, Haq W, Arif M, Mustafa S: On bounded boundary and bounded radius

rotations J Inequal Appl 2009 Article ID 813687, 12 (2009)

[6] Noor, KI: On generalization of close-to-convexity Int J Math Math Sci 15(2), 279–

290 (1992)

[7] Bulut, S: A new general integral operator defined by Al-Oboudi differential operator J

Inequal Appl 2009, Article ID 158408, 13 (2009)

[8] Vijayvargy, L, Goswami, P, Malik, B: On some integral operators for certain classes of

P-valent functions Int J Math Math Sci 2011, Article ID 783084, 10 (2011)

[9] Frasin, BA: Order of convexity and univalency of general integral operator J Franklin

Inst 348(6), 1013–1019 (2011)

[10] Breaz, D, Breaz, N: Two integral operator Studia Universitatis Babes-Bolyai,

Mathematica, Clunj-Napoca 3, 13–19 (2002)

[11] Breaz, D, Owa, S, Breaz, N: A new integral univalent operator Acta Univ Apulensis

Math Inf 16, 11–16 (2008)

[12] Alexander, JW: Functions which map the interior of the unit circle upon simple regions

Ann Math 17, 12–22 (1915–1916)

[13] Miller, SS, Mocanu, PT, Read, MO: Starlike integral operators Pacific J Math 79,

157–168 (1978)

[14] Noor, KI: On subclasses of close-to-convex functions of higher order Int J Math

Math Sci 6(2), 327–334 (1983)

[15] Ruscheweyh, S, Sheil-Small, T: Hademard product s of Schlicht functions and the

Polya-Schoenberg conjecture Comment Math Helv 48, 119–135 (1973)