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On certain classes of meromorphic harmonic concave functions defined by al-Oboudi operator

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(1)

ON CERTAIN CLASSES OF MEROMORPHIC HARMONIC CONCAVE FUNCTIONS DEFINED BY AL-OBOUDI OPERATOR

(Berkenaan Kelas Fungsi Cekung Meromorfi Harmonik Tertentu yang Ditakrif oleh Pengoperasi Al-Oboudi)

KHALID CHALLAB & MASLINA DARUS

ABSTRACT

In this work, a class of meromorphic harmonic concave functions defined by Oboudi operator in the punctured unit disc is introduced. Coefficient conditions, distortion inequalities, extreme points, convolution bounds, geometric convolution, integral convolution and convex combinations for functions f belonging to this class are obtained.

Keywords: Meromorphic function; harmonic function; concave function; Al-Oboudi operator

ABSTRAK

Dalam kajian ini, kelas fungsi cekung meromorfi harmonik yang ditakrif oleh pengoperasi Al- Oboudi diperkenalkan dalam cakera unit terpancit. Syarat pekali, ketaksamaan erotan, titik ekstrim, batas konvolusi, konvolusi geometri, konvolusi kamiran dan gabungan cembung untuk fungsi f dalam kelas tersebut diperoleh.

Kata kunci: fungsi meromorfi; fungsi harmonik; fungsi cekung; pengoperasi Al-Oboudi

1. Introduction

Conformal maps of the unit disc onto convex domain are very classic. We note that Avkhadiev and Wirths (2005) discovered the conformal mapping of a unit disc onto concave domains (the complements of convex closed sets). It is quite interesting to see other related results in this direction as not many problems are discussed thoroughly towards this approach.

Let 𝑈 = {𝑧 ∈ ℂ: |𝑧| < 1} denote the open unit disc, where f has the form given by

 

2 n n n

f z z a z

 

(1) that maps U conformally onto a domain whose complement with respect to C is convex and that satisfies the normalisation (1) f . In addtion, they imposed on these functions the condition that the opening angle of ( )f U at infinity is less than or equal to  , (1, 2].

These families of functions are denoted by C0( ) . The class C0( ) is referred to as the class of concave univalent functions.

Chuaqui et al. (2012) defined the concept of meromorphic concave mappings. A conformal mapping of meromorphic function in U, where 𝑈= 𝑈\{0} is said to be a concave mapping if its image is the complement of a compact convex set.

If f has the form

 

0 1 2 2

1 ,

f z a a z a z

 z   

then a necessary and sufficient condition for f to be a concave mapping is

(2)

Khalid Challab & Maslina Darus

'' '

1 ( ) 0, 1.

( ) f z

Re z z

f z

 

 

   

 

 

A continuous function f  u ivis a complex valued harmonic function in a domain

if both uand vare real harmonic in In any simply connected domain, we write where hand gare analytic in A necessary and sufficient condition for f to be

locally univalent and orientation preserving in U is that h  gin U (see Clunie and Sheil- Small 1984). Hengartner and Schober (1987) investigated functions harmonic in the exterior of the unit disc . They showed that complex valued, harmonic, sense preserving, univalent mapping f must admit the representation.

     

,

f zh zg zAlog z where h z

 

and g z( )are defined by

   

1 1

,

n n

n n

n n

h zz a z g zz b z

 

 

for0

and , zU**,where We call h the analytic part and g the co-analytic part of f .

ForzU*, let MHbe the class of functions:

   

1 1

( ) 1 k k k k

k k

f z h z g z a z b z

z

   

(2) which are analytic in the punctured unit discU*, where h z( )and g z( )are analytic in Uand U*, respectively, and h z( )has a simple pole at the origin with residue 1 (see Al-Shaqsi and Darus 2008).

A function fMHis said to be in the subclass MSH of meromorphically harmonic starlike functions in U* if it satisfies the condition

     

'

'( ) *

0, . ( )

zh z zg z

Re z U

h z g z

  

  

  

 

 

Also, a function fMH is said to be in the subclass MCHof meromorphically harmonic convex functions in U* if it satisfies the condition

     

   

'' ''

' '( ) *

0, .

' '( )

zh z h z zg z g z

Re z U

h z g z

    

  

  

 

 

Note that the classes of harmonic meromorphic starlike functions, harmonic meromorphic convex functions and harmonic meromorphic concave functions (MHC0) have been studied by Jahangiri and Silverman (1999), Jahangiri (2000), Jahangiri (1998), Aldawish and Darus (2015) and Challab and Darus (2016). Other works related to concave can be read in Aldawish and Darus (2015), Darus et al. (2015) and Aldawish et al. (2014.)

AL-Oboudi (2004) introduced the operator Dn for fAwhich is the class of function of the form (1) analytic in the unit disc , and defined the following differential operator

   

0 ,

D f zf z

(3)

D f z1

  

 1

 

zf'

 

z D f z

 

,

0, (3) D f zn

 

D

Dn1f z

  

. (4) If f is given by (1), then from (3) and (4) we see that

(5) Al-Oboudi (2004) introduced the operator On for fMHC0 which is the class of functions f  h g that are harmonic univalent and sense-preserving in the unit disc for which { f(0)h(0) fz(0) 1 0  }.

Now, we define Onfor f  h ggiven by (2) as

O f nn ( )O h nn ( )O g nn ( ) ,

n0,1,2, ,... z U *,

where

   

1

( 1) 1 1 ,

n n

n k

k k

O h z k a z

z

  

 

   

 

1

g(z) 1 1 n .

n k

k k

O kb z

 

   

This work is an attempt to give a connection between harmonic function and meromorphic concave functions defined by Oboudi operator by introducing a class O MHCn 0 of meromorphic harmonic concave functions.

Definition 1.1. FornN0 {1, 2,...},

 

1, 2 ,k 1, let 0

O MHCn denote the class of meromorphic harmonic concave functions O f zn ( ) defined by Oboudi differential operator of the form,

1 1

( ) ( 1) / [1 ( 1) ] [1 ( 1) ] ( )

n n n k n k

k k

k k

O f z z ka z kb z

  

  

  (6) such that

   

 

''

1 ' 0.

( )

n n

z O f z Re O f z

 

 

  

 

 

2. Coefficient Conditions

In this section, sufficient condition for a function O f zn ( )O MHCn 0 is derived.

(4)

Khalid Challab & Maslina Darus

Theorem 2.1. LetO f zn ( ) h g be of the form

     

1 1

( 1) 1 1 .

1 1

n n n

n k k

k k

k k

O f z k a z k b z

z

    

 

    

   

If 2

   

1

1 1 n k k 1,

k

k ka b

     

 

then O f z is harmonic univalent, sense preserving in n

 

Proof: First for0 z1z2 1, we have

1 2 1 1 1

1 1

2 2 2

1 1

| ( ) ( ) | | ( 1) / [1 ( 1) ] [1 ( 1) ] ( )

( 1) / [1 ( 1) ] [1 ( 1) ] ( ) |

n n n n k n k

k k

k k

n n k n k

k k

k k

O f z O f z z k a z k b z

z k a z k b z

 

 

        

       

 

 

1 2

1 2 1 2

1 1

1/ | | 1/ | |

[1 ( 1) ] |n k || k k | [1 ( 1) ] |n k || k k |

k k

z z

ka z z kb z z

 

   

  

 

n

 

1 2

1 2 k k

1 2 k 1

z z

z z k 1 k 1 γ a b

z z

  

  

    

2 2

1 2

1

1 2 2

| | / | || | [1 | | [1 ( 1) ] (|n | | |)]

k

k k

z z z z z k ka b

  

  

2

1 2 1 2

1

| | / | || | [1 [1 ( 1) ] (|n k | | k |)].

k

z z z z k ka b

  

  

The last expression is non negative by 2

1

[1 ( 1) ] (|n k | | k|) 1

k

k k

a b

   

and O f zn ( ) is

univalent in U*.

Now we want to show f is sense preserving inU**, we need to show that h z'( )  g z'( ) in U*.

   

1

'

2 1

1 1 1 n k k

k

h z k k a z

z

 

 

   

 

1

2 1

1 1 1 n k k

k

k k a r

r

 

 

   
(5)

 

1

1 1 1 n k

k

k ka

 

 

   

 

2 1

1 1 1 n k

k

k ka

 

 

    2

 

1

1 1 n k

k

k kb

 

   

 

1

 

1 '

 

1 1

1 1 n k k 1 1 n k k .

k k

k k

b r k k

b z g z

    

    

Thus this completes the proof of the Theorem 2.1. 

Theorem 2.2. Let O f zn

 

 h g be of the form,

     

1 1

( 1) 1 1 .

1 1

n n n

n k k

k k

k k

O f z k a z k b z

z

  

    

   

ThenO f zn ( )O MHCn if the inequality

2

1

[1 ( 1) ] (|n k| | k|) 1

k

k ka b

   

(7) holds for coefficient O f zn ( ) h g.

Proof: Using the fact that 1

( ) 0 1

1 Re w w

w

   

 , it suffices to show

1 1

1 w w

 

.

Let

   

 

''

1 '

( )

n n

z O f z

w Re

O f z

 

 

   

 

 

, such that

'( ) ( ) w zg z

g z , where g z

 

z O f z( n

 

)'.

Now,

(6)

Khalid Challab & Maslina Darus

   

     

2 2

1 1

1

2 2

1 1

( ) 1 1 ( ) 1 1

1

1 2 1

( ) 1 1 ( ) 1 1

n k n k

k k

k k

n

n k n k

k k

k k

k k k a z k k k b z

w

w k k k a z k k k b z

z

 

 

   

      

     

  

   

           

 

 

   

   

2 2

1 1

2 2

1 1

( ) 1 1 ( ) 1 1

1 .

1 2 ( ) 1 1 ( ) 1 1

n n

k k

k k

n n

k k

k k

k k k a k k k b

w

w k k k a k k k b

 

 

   

      

     

            

 

 

(8)

The last expression is bounded above by 1 if

   

2 2

1 1

( ) 1 1 n k ( ) 1 1 n k

k k

k k ka k k kb

   

          

 

2 2

1 1

2 ( )[1 ( 1) ] |n k | ( )[1 ( 1) ] |n k |

k k

k k ka k k kb

 

   

  

which is equivalent to our condition by 2

1

[1 ( 1) ] (|n k| | k|) 1.

k

k ka b

   

Theorem 2.3. Let O f zn ( ) h g be of the form

     

1 1

( 1) 1 1 .

1 1

n n n

n k k

k k

k k

O f z k a z k b z

z

    

 

    

   

A necessary and sufficient condition for O f zn

 

to be in O MHC is that n

   

2 1

1 1 n k k 1.

k

k ka b

     

 

Proof: In view of Theorem 2.2, we assume that 2

1

[1 ( 1) ] (|n k| | k|) 1.

k

k ka b

   

SinceO f zn ( )O MHCn 0, then

1Re z O f z{( ( n ( )) ) / ( O f zn ( )) } equivalent to

     

   

2

2 1 2 1

2 1 1

'

1

1 1

1 1 1 1 1

( )

( ) ( 1)

1 1 1 1

n

n k n k

k k

k k

n n k n k

k k

k k

z k k a z k k b z

zg z z

Re Re

g z k k a z k k b z

z

 

 

  

           

 

 

           

 

 

(7)

   

   

1

2 1 2 1

1 1

1

1 1

( 1) 1 1 1 1

( 1) 0

1 1 1 1

n n k n k

k k

k k

n n k n k

k k

k k

k k a z k k b z

Re z

k k a z k k b z

z

 

 

           

 

          

 

 

for z  r 1, the above expression reduce to

   

   

1 2

1 1

1

1 1 1 ( ) ( )

( ) 0.

1 1 1 ( )

n n k

k k

k

n n k

k k

k

k k a b r A r

Re k k a b r B r

          

   

 

          

 

 

As we assumed 2

1

[1 ( 1) ] (|n k| | k|) 1,

k

k k

a b

   

then A(r) and B(r) are positive for r

sufficiently close to 1. Thus there exists a z  r 1 for which the quotient is positive. This contradicts the required condition that ( )

( ) 0 A r

B r  , so the proof is complete.  3. Distortion bounds and extreme points

Bounds and extreme points for functions belonging to the class O MHCn are estimated in this section.

Theorem 3.1. If O fn kO hn kO gn kO MHCn and 0 z  r 1 then

 

1 2

n k

O f z r r

  and

1 2

( ) .

n k

O f z r r

 

Proof: Let O fn kO hn kO gn kO MHCn 0. Taking the absolute value of O fn k, we obtain

       

1 1

1 1 1 1 1

n

n n

n k k

k k k

k k

O f z k a z k b z

z

    

 

    

   

     

1

1 1 1 n

n k

k k k

k

O f z k a b r

r

 

 

    

   

2 1

1 1 1 n k k k

k

k k a b r

r

 

    

   

2 1

1 1 1 n k k

k

k k a b r

r

 

 

    

by applying

2 1

[1 ( 1) ] (|n k| | k|) 1,

k

k k

a b

   

then

(8)

Khalid Challab & Maslina Darus

 

1 1 2.

n k

O f z r r

r r

   

Now,

       

1 1

1 1 1 1 1

n

n n

n k k

k k k

k k

O f z k a z k b z

z

    

 

    

   

   

1

1 1 1 n k k k

k

k a b r

r

 

 

    

   

2 1

1 1 1 n k k .

k

k k a b r

r

 

 

    

By applying 2

1

[1 ( 1) ] (|n k| | k|) 1,

k

k k

a b

   

then

n k

 

1 1 r2.

O f z r

r r

    

Theorem 3.2. Let O fn nO hn nOngn and O f zn n( )O h zn n( )Ong ( )n z That is

       

1 1

1 1 1 1 1 .

n

n n

n k k

n k k

k k

O f z k a z k b z

z

    

 

    

   

Set

 

,0 ,0

g 1

n

n n

n n

O h O

z

   and

 

, 2

1 1

( ) ,

n

n k

O hn k z z

z k

  

Fork1,2,3,... and let

   

, 2

1 1

g ,

n

n k

O n k z z

z k

  

fork1,2,3,... then O fn nO MHCn if and only if O f can be expressed as n n

   

 

, , ,

0

g ,

n n n

n k k n k k n k

k

O f O h zO z

where k0,k0 and

0

( k k) 1

k

 

. In particular the extreme points of O MHC are n

O hn n k,

and {O gn n k, }.
(9)

Proof: For functions O fn nO hn nOngn, where O hn n and Ongn are given by

       

1 1

1 1 1 1 1 ,

n

n n

n k k

n k k

k k

O f z k a z k b z

z

  

    

    we have

   

 

, , ,

0

( ) g

n n n

n k k n k k n k

k

O f z O h zO z

   

 

0 ,0 0 ,0 , ,

1

g g

n n n n

n n k n k k n k

k

O h

O O h z

O z

  

   

0 ,0 0 ,0 2 2

1 1

1 1 1 1

g

n n

n n k k

n n k k

k k

O h O z z

z k z k

     

   

     

   

   

 

 

2

 

0 1

1 1

( ) .

n

k k

k k k k

k k

z z

z k

 

  

Now by Theorem 2.2,

2 2

0 0

2 2

1 1

1 1

1 1,

k k k k

k k

k k

kk  

      

 

We have O fn ( , )( )n k zO MHCn 0.

Conversely, suppose that O fn n k, ( )zO MHCn . Setting

 

2 1 1 n , 1

kk   k  ak k and

2| [1 ( 1) ]n |, 1.

k k k bk k

     

We define

0 0

1 1

1 k k.

k k

  

 

 Therefore O f zn n( ) can be written as

       

1 1

1 1 1 1 1

n

n n

n k k

n k k

k k

O f z k a z k b z

z

    

 

    

   
(10)

Khalid Challab & Maslina Darus

 

2 2

1

1 1 1

n

k k

k k

k

z z

z kk

  

 

,

   

,

   

1 1

1 1 1

g

n n n

n n

n k k n k k

k k

O h z O z

z z z

   

  

   

    

   

   

 

 

,

 

,

 

1 1 1 1

1 1 g

n

n n

k k n k k n k k

k k k k

O h z O z

z

 

   

 

 

0 0

  

,

 

,

 

1 1

1 g

n

n n

n k k n k k

k k

O h z O z

z

   

and finally

  

,

 

,

  

0

g .

n n n

n k n k k n k

k

O f z O h zO z

The proof is completed, therefore ,O hn ( , )( )n k z and O gn ( , )( )n k z are extreme points. .

4. Convolution Properties

In this section, we define and study the convolution, geometric convolution and integral convolution of the class O MHCn .

For harmonic function O f O Fn n, n nis defined as follows:

       

1 1

1 1 1 1 1

n

n n

n k k

n k k

k k

O f z k a z k b z

z

  

    

    (9)

and

1 1

( ) ( 1) / [1 ( 1) ] | | [1 ( 1) ] | | .

n n n k n k

n k k

k k

O F z z kA z kB z

  

  

  (10)

The convolution of O fn n and O Fn n is given by

(O fn n*O Fn n)( )zO f zn n( ) *O F zn n( ),

1 1

( 1) /n [1 ( 1) ] |n k|| k| k [1 ( 1) ] |n k|| k| k.

k k

z ka A z kb B z

  

  

 

The geometric convolution O fn n and O Fn n is given by

   

(O fn nO Fn n)( )zO fn n zO F zn n ,

1 1

( 1) /n [1 ( 1) ]n | k k| k [1 ( 1) ]n | k k| k.

k k

z ka A z kb B z

  

  

 
(11)

The integral convolution of O fn n and O Fn n is given by

1 1

( 1) /n [1 ( 1) ] |n k k| / k [1 ( 1) ] |n k k| / k.

k k

z ka A kz kb B k z

  

  

 

Theorem 4.1. Let O fn nO MHCn 0 and O Fn nO MHCn 0. Then the convolution

* 0.

n n n

n n

O f O FO MHC

Proof: For O fn n and O Fn n given by (9) and (10), then the convolution is given by (10). We show that the coefficients of O fn n*O Fn n satisfy the required condition given in Theorem 2.2.

For O Fn nO MHCn 0, we note that Ak 1and Bk 1. Now for convolution function

* ,

n n

n n

O f O F we obtain

       

2 2

1 1

1 1 n k k 1 1 n k k

k k

k ka A k kb B

        

   

 

   

2 2

1 1

1 1 n k 1 1 n k 1.

k k

k ka k kb

   

    

    

Therefore O fn n* O Fn nO MHCn , this proves the required result.  Theorem 4.2. If O f and n n O F of the form (9) and (10) belong to the class n n O MHC , then n 0 the geometric convolution O fn nO Fn n also belong to O MHC n 0.

Proof: Since O f O Fn n, n nO MHCn 0, it follows that

 

2 1

1 1 n( k k) 1

k

k ka b

     

 

and

 

2 1

1 1 n( k k ) 1.

k

k kA B

     

 

Hence by Cauchy-Schwartz's inequality it is noted that

 

2 1

1 1 n( k k k k ) 1.

k

k k a A b B

     

 

The proof is complete.  Theorem 4.3. If O f and n n O F of the form n n (8) and (9) belong to the class O MHCn then the integral convolution also belong to the O MHCn .

(12)

Khalid Challab & Maslina Darus

Proof: Since O f O Fn n, n nO MHCn 0, it follows that Ak 1and Bk 1.Then because

   

2 2

1 1

1 1 n k k 1 1 n k k

k k

a A b B

k k k k

k k

        

   

 

2

 

2

 

1 1

1 1 n k 1 1 n k

k k

a b

k k k k

k k

   

    

   

   

2 2

1 1

1 1 n k 1 1 n k 1.

k k

k k a k k b

   

    

    

This proves the required result.  5. Convex Combinations

In this section, we show that the class O MHCn is invariant under convex combinations of its members.

Theorem 5.1. The class O MHC is closed under convex combinations. n 0

Proof: Fori1,2,3,... suppose that O f zn i( )O MHCn 0, where O fn i is given by

       

1 1

1 1 1 1 1 ,

n

n n

n k k

i ik ik

k k

O f z k a z k b z

z

    

 

    

   

0, 0,

ik ik

ab  then by Theorem 2.2, 2

1

[1 ( 1) ] (|n i | | i |) 1.

k

k ka k b k

   

For

1

1,0 1,

i i

k

t t

  

the convex combinations of O fn i may be written as

1 1 1 1

( ) ( 1) / [1 ( 1) ] ( ) [1 ( 1) ] ( ) .

i

n n n k n k

i i i ik i ik

i k k

t O f z z kt a z kt b z

       

   

Then by

   

2 1

1 1 n ik ik 1.

k

k k a b

     

 

 

2

1 1 1

1 1 n i ik i ik

k i i

k k t a t b

 

      

   

  

   

2

1 1

1 1 n

i ik ik

i k

t k k a b

 

       

 

 

1 i 1.

i

t

Thus 0

1

( ) .

n n

i i

i

t O f z O MHC

 
(13)

Acknowledgements

The work here is supported by MOHE grant FRGS/1/2016/STG06/UKM/01/1.

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School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia <

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