CERTAIN PROPERTIES FOR ANALYTIC FUNCTIONS DEFINED … · CERTAIN PROPERTIES FOR ANALYTIC FUNCTIONS DEFINED BY A . GENERALISED DERIVATIVE OPERATOR ... Journal of Quality Measurement

36 37

CERTAIN PROPERTIES FOR ANALYTIC FUNCTIONS DEFINED BY A GENERALISED DERIVATIVE OPERATOR

(Sifat Tertentu bagi Fungsi Analisis yang Ditakrif oleh Pengoperasi Terbitan Teritlak)

AISHA AHMED AMER & MASLINA DARUS

ABSTRACT

In this paper, some important properties of analytic functions with negative coefficients defined by a generalised derivative operator are investigated. The properties include the necessary and sufficient conditions, radius of starlikeness, convexity and close-to-convexity.

Keywords: derivative operator; radius of starlikeness; convexity; close-to-convexity

ABSTRAK

Dalam makalah ini dikaji beberapa sifat penting bagi fungsi analisis berpekali negatif yang ditakrif oleh pengoperasi terbitan teritlak. Sifat tersebut termasuklah syarat perlu dan cukup, jejari kebakbintangan, kecembungan dan dekat-dengan-kecembungan.

Kata kunci: pengoperasi terbitan; jejari kebakbintangan; kecembungan; dekat-dengan-kecembungan

1. Introduction

Let A denote the class of functions f in the open unit disc

={ :| |<1},z z∈U C and let T denote the subclass of A consisting of analytic functions of the form

=2( ) = , ( ),k

kk

f z z a z z∞

− ∈∑ U

which are analytic in the unit disc .U

Definition 1.1 Let f A∈ . Then f is said to be convex of order (0 <1)µ µ≤ if and only if

( )1 > , .( )

zf z zf z

µ′′⎧ ⎫

ℜ + ∈⎨ ⎬′⎩ ⎭U

Definition 1.2 Let f A∈ . Then f is said to be starlike of order (0 <1)µ µ≤ if and only if

( ) > , .( )

zf z zf z

µ′⎧ ⎫

ℜ ∈⎨ ⎬⎩ ⎭

U

Amer and Darus (2011; 2012) have recently introduced a new generalised derivative operator

1 2( , , , ) ( )mI l n f zλ λ as follows:

Journal of Quality Measurement and Analysis Jurnal Pengukuran Kualiti dan Analisis

JQMA 8(2) 2012, 37-44

Aisha Ahmed Amer & Maslina Darus

38 39

Definition 1.3 Let ,f A∈ then the generalised derivative operator is given by

1 2=2

( , , , ) ( ) = ,m kk k

kI l n f z z a zλ λ ζ

∞

−∑ (1)

where

1

11

2

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

m

k m m

k l c n kl kλζ

λ

−

−

+ − ++ + −

0, ={0,1,2,...},n m N∈

2 1 0, 0,lλ λ≥ ≥ ≥ and 1

1

( 1)( , ) = .(1)

k

k

nc n k −

−

+

Definition 1.4 Let a function f be in T. Then f is said to be in the class of

1 2( , , , , , )l nτ α β λ λ if,

ℜI m(λ1,λ2 ,l,n) f (z)z[I m(λ1,λ2 ,l,n) f (z) ′]

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪>α

I m(λ1,λ2 ,l,n) f (z)z[I m(λ1,λ2 ,l,n) f (z) ′]

−1 + β ,

(2)

where 0, ={0,1,2,...}, 0,n m N l∈ ≥ 0 <1,α≤ and 0 <1.β≤

The family 1 2( , , , , , )l nτ α β λ λ is a special interest as it contains many well-known classes of analytic univalent functions. This family is studied by (Najafzadeh & Ebadian 2009), and also (Tehranchi & Kulkarni 2006a; 2006b).

2. Necessary and Sufficient Conditions

Theorem 2.1 Let .f T∈ Then 1 2( , , , , , )f l nτ α β λ λ∈ if and only if,

=2

[(1 ) ( )] <1.1 k k

k

k aα α β ζβ

∞ + − +−∑ (3)

Proof: Let us assume that 1 2( , , , , , )f l nτ α β λ λ∈ . So by using the fact that

( ) > | 1|ω α ω βℜ − + if, and only if [ (1 ) ] >i ie θ θω α α βℜ + −

and letting 1 2

1 2

( , , , ) ( )=[ ( , , , ) ( )]

m

m

I l n f zz I l n f z

λ λωλ λ ′

in (2), we obtain [ (1 ) ] > .i ie θ θω α α βℜ + −

So

Certain properties for analytic functions defined by a generalised derivative operator

38 39

=2

1

=2

(1 ) > 0,1

kk k

i ik

kk k

k

z a ze e

z k a z

θ θζ

α α βζ

∞

∞−

⎡ ⎤−⎢ ⎥

⎢ ⎥ℜ + − −⎢ ⎥⎛ ⎞−⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∑

∑

then 1 1

=2 =2

1

=2

1 (1 ) (1 )> 0.

1

k i kk k k k

k k

kk k

k

k a z e k a z

k a z

θβ β ζ α ζ

ζ

∞ ∞− −

∞−

⎡ ⎤− − − − −⎢ ⎥⎢ ⎥ℜ⎢ ⎥−⎢ ⎥⎣ ⎦

∑ ∑

∑

The above inequality must hold for all z in .U Letting = iz re θ− where 0 <1r≤ , we obtain

1

=2

1

=2

1 [(1 ) (1 )]> 0.

1

i kk k

k

kk k

k

k e k a r

k a r

θβ β α ζ

ζ

∞−

∞−

⎡ ⎤− − − + −⎢ ⎥⎢ ⎥ℜ⎢ ⎥−⎢ ⎥⎣ ⎦

∑

∑

By letting 1r → through half line = iz re θ− and by mean value theorem, we have

1

=21 [(1 ) (1 )] > 0,k

k kk

k k a rβ β α ζ∞

−⎡ ⎤ℜ − − − + −⎢ ⎥⎣ ⎦∑

then we get

=2

[(1 ) ( )] <1.1 k k

k

k aα α β ζβ

∞ + − +−∑

Conversely, let (3) holds. We will show that (2) is satisfied and so 1 2( , , , , , ).f l nτ α β λ λ∈ By using the fact that ( ) >ω βℜ if and only if | (1 ) |<| (1 ) |,ω β ω β− + + − it is enough to show that

| (1 | 1| ) |<| (1 | 1| ) |,ω α ω β ω α ω β− + − + + − − + if =| (1 | 1| ) |R ω α ω β+ − − +

=21 2

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

kk km

kz z k a z

z I l n f zβ β α α ζ

λ λ

∞

− − + − + −′ ∑

This implies that

=21 2

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

kk km

k

zR k k k a zz I l n f z

β α α β ζλ λ

∞

− − + + − +′ ∑


40 41

Similarly, if =| (1 | 1| ) |,L ω α ω β− − − + we get

=21 2

| |< [ (1 ) ( ) ] .| [ ( , , , ) ( )] |

kk km

k

zL k k k a zz I l n f z

β α α β ζλ λ

∞

+ − + + − +′ ∑

It is easy to verify that > 0R L− and so the proof is complete.

Corollary 2.1 Let 1 2( , , , , , ),f l nτ α β λ λ∈ then

1< .[(1 ) ( )]k

k

akβ

α α β ζ−

+ − +

Proof: For 0 <1µ≤ , we need to show that ( ) 1 <1 .( )

zf zf z

µ′

− − Now, let us show that

1

=2

1

=2

( 1)( ) ( ) =( ) 1

kk

k

kk

k

k a zzf z f z

f z a z

∞−

∞−

− −′ −

−

∑

∑

1

=2

1

=2

( 1) | |<1

1 | |

kk

k

kk

k

k a z

a zµ

∞−

∞−

−≤ −

−

∑

∑

1

=2| | <1.

1k

kk

ka z µµ

∞− ⎛ ⎞−⇒ ⎜ ⎟−⎝ ⎠

∑

By Theorem 2.1, it is enough to consider

1 (1 )[(1 ) ( )]| | < .( )(1 )

kk

kzk

µ α α β ζµ β

− − + − +− −

�

Theorem 2.2 Let 1 2( , , , , , ).f l nτ α β λ λ∈ Then f is convex of order (0 <1)µ µ≤ in

2 1 2| |< = ( , , , , , , )z r r l nα β λ λ µ where

11

2 1 2(1 )[(1 ) ( )]( , , , , , , ) = .inf ( )(1 )

k

kk

kr l nk kµ α α βα β λ λ µ ζ

µ β−⎡ ⎤− + − +

⎢ ⎥− −⎣ ⎦

Proof: For 0 <1µ≤ , we need to show that ( ) <1 .( )

zf zf z

µ′′

−′

Now again let us show that


40 41

1

=2

1

=2

( 1)

1

kk

k

kk

k

k k a z

ka z

∞−

∞−

− −

−

∑

∑

1

=2

1

=2

( 1) | |<1

1 | |

kk

k

kk

k

k k a z

ka zµ

∞−

∞−

−≤ −

−

∑

∑

1

=2| | <1.

1k

kk

kka z µµ

∞− ⎛ ⎞−⇒ ⎜ ⎟−⎝ ⎠

∑

By Theorem 2.1, it is again enough to consider

1 (1 )[(1 ) ( )]| | < .( )(1 )

kk

kzk kµ α α β ζ

µ β− − + − +

− −

�

Theorem 2.3 Let 1 2( ) ( , , , , , ).f z l nτ α β λ λ∈ Then ( )f z is close-to-convex of order

(0 <1)µ µ≤ in 3 1 2| |< = ( , , , , , , )z r r l nα β λ λ µ where

11

3 1 2(1 )[(1 ) ( )]( , , , , , , ) = .inf (1 )

k

kk

kr l nk

µ α α βα β λ λ µ ζβ

−⎡ ⎤− + − +⎢ ⎥−⎣ ⎦

Proof: For 0 <1µ≤ , we must show that ( ) 1 <1 .f z µ′ − − Similarly we show that

1

=2( ) 1 = k

kk

f z ka z∞

−′ − ∑

1

=2| | 1k

kkka z µ

∞−≤ ≤ −∑

1

=2| | <1.

1k

kk

k a zµ

∞−⇒

−∑

By Theorem 2.1, the above inequality holds true if,

1 (1 )[(1 ) ( )]| | < .(1 )

kk

kzk

µ α α β ζβ

− − + − +−

˜

�

3. Radius of Starlikeness, Convexity and Close-to-convexity

In this section, we will calculate Radius of Starlikeness, Convexity and Close-to-convexity for

the class 1 2( , , , , , )l nτ α β λ λ .


42 43

Theorem 3.1 Let 1 2( , , , , , ).f l nτ α β λ λ∈ Then f is starlike of order (0 <1)µ µ≤ in

1 1 2| |< = ( , , , , , , )z r r l nα β λ λ µ , where

11

1 1 2(1 )[(1 ) ( )]( , , , , , , ) = .inf ( )(1 )

k

kk

kr l nk

µ α α βα β λ λ µ ζµ β

−⎡ ⎤− + − +⎢ ⎥− −⎣ ⎦

Proof: For 0 <1µ≤ , we need to show that ( ) 1 <1 .( )

zf zf z

µ′

− − Now, we have to show that

1

=2

1

=2

( 1)( ) ( ) =( ) 1

kk

k

kk

k

k a zzf z f z

f z a z

∞−

∞−

− −′ −

−

∑

∑

1

=2

1

=2

( 1) | |<1

1 | |

kk

k

kk

k

k a z

a zµ

∞−

∞−

−≤ −

−

∑

∑

1

=2| | <1.

1k

kk

ka z µµ

∞− ⎛ ⎞−⇒ ⎜ ⎟−⎝ ⎠

∑


1 (1 )[(1 ) ( )]| | < .( )(1 )

kk

kzk

µ α α β ζµ β

− − + − +− −

�

Theorem 3.2 Let 1 2( , , , , , ).f l nτ α β λ λ∈ Then f is convex of order (0 <1)µ µ≤ in

2 1 2| |< = ( , , , , , , )z r r l nα β λ λ µ , where

11

2 1 2(1 )[(1 ) ( )]( , , , , , , ) = .inf ( )(1 )

k

kk

kr l nk kµ α α βα β λ λ µ ζ

µ β−⎡ ⎤− + − +

⎢ ⎥− −⎣ ⎦

Proof: For 0 <1µ≤ , we need to show that ( ) <1 .( )

zf zf z

µ′′

−′

We have to show that


42 43

1

=2

1

=2

( 1)

1

kk

k

kk

k

k k a z

ka z

∞−

∞−

− −

−

∑

∑

1

=2

1

=2

( 1) | |<1

1 | |

kk

k

kk

k

k k a z

ka zµ

∞−

∞−

−≤ −

−

∑

∑

1

=2| | <1.

1k

kk

kka z µµ

∞− ⎛ ⎞−⇒ ⎜ ⎟−⎝ ⎠

∑


1 (1 )[(1 ) ( )]| | < .( )(1 )

kk

kzk kµ α α β ζ

µ β− − + − +

− −

Theorem 3.3 Let 1 2( , , , , , ).f l nτ α β λ λ∈ Then f is close-to-convex of order (0 <1)µ µ≤

in 3 1 2| |< = ( , , , , , , )z r r l nα β λ λ µ , where

11

3 1 2(1 )[(1 ) ( )]( , , , , , , ) = .inf (1 )

k

kk

kr l nk

µ α α βα β λ λ µ ζβ

−⎡ ⎤− + − +⎢ ⎥−⎣ ⎦

Proof: For 0 <1µ≤ , we must show that ( ) 1 <1 .f z µ′ − −

We have to show that

1

=2( ) 1 = k

kk

f z ka z∞

−′ − ∑

1

=2| | 1k

kkka z µ

∞−≤ ≤ −∑

1

=2| | <1.

1k

kk

k a zµ

∞−⇒

−∑

By Theorem 2.1, the above inequality holds true if

1 (1 )[(1 ) ( )]| | < .(1 )

kk

kzk

µ α α β ζβ

− − + − +−

�

Acknowledgement

The work presented here was partially supported by UKM-ST-06-FRGS0244-2010.


44 45

References

Amer A. A. & Darus M. 2011. On some properties for new generalized derivative operator. Jordan Journal of Mathematics and Statistics (JJMS) 4(2): 91-101.

Amer A. A. & Darus M. 2012. On preserving the univalence integral operator. Applied Sciences 14: 15-25.Najafzadeh Sh. & Ebadian A. 2009. Neighborhood and partial sum property for univalent holomorphic functions in

terms of Komatu operator. Acta Universitatis Apulensis 19: 81-89. Tehranchi A. & Kulkarni S.R. 2006a. Some integral operators defined on p-valent functions by using hypergeometric

functions. Studia Univ. Babes, Bolyai Mathematica (Cluj) 1: 525-532. Tehranchi. A & Kulkarni S.R. 2006b. Study of the class of univalent functions with negative coefficients defined by

Ruscheweyh derivative. J. Rajasthan academy of Physical Science 5(1): 169-180.

School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia43600 UKM BangiSelangor DE, MALAYSIAE-mail: [email protected], [email protected]*

* Corresponding author

UKM

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CERTAIN PROPERTIES FOR ANALYTIC FUNCTIONS DEFINED … · CERTAIN PROPERTIES FOR ANALYTIC FUNCTIONS DEFINED BY A . GENERALISED DERIVATIVE OPERATOR ... Journal of Quality Measurement