Argument Estimates of Multivalent Functions Involving a Certain Fractional Derivative Operator

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http://dx.doi.org/10.4236/apm.2015.52011

Argument Estimates of Multivalent

Functions Involving a Certain Fractional

Derivative Operator

Jae Ho Choi

Department of Mathematics Education, Daegu National University of Education, Daegu, South Korea Email: [email protected]

Received 18 January 2015; accepted 9 February 2015; published 13 February 2015

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

The object of the present paper is to investigate various argument results of analytic and multiva-lent functions which are defined by using a certain fractional derivative operator. Some interest-ing applications are also considered.

Keywords

Multivalent Analytic Functions, Argument, Integral Operator, Fractional Derivative Operator

1. Introduction

Let 

( )

p denote the class of functions f z

( )

of the form

( )

(

{

}

)

1

, : 1, 2, 3, ,

p p k

p k k

f z z a z p

∞

+ + =

= +

∑

∈ =  (1.1)

which are analytic in the open unit disk =

{

z z: ∈ and z <1

}

. Also let 

( )

1 = denote the class of all analytic functions p z

( )

with p

( )

0 =1 which are defined on .

Let a, b and c be complex numbers with c≠ − −0, 1, 2,. Then the Gaussian hypergeometric function

(

)

2F a b c z1 , ; ; is defined by

(

)

( ) ( )

_{( )}

2 1

0

, ; ; ,

!

k

k k

k _k

a b z F a b c z

c k

∞

=

∑

(1.2)

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( )

(

_{( )}

)

1,

₍

_{) (}

(

0 ;

)

₎

1 1 , .

k

k k

η η

η η η

η

 =

Γ + 

= _{= }

+ + − ∈

Γ _  

The hypergeometric function 2F a b c z1

(

, ; ;

)

is analytic in  and if a or b is a negative integer, then it

reduces to a polynomial.

There are a number of definitions for fractional calculus operators in the literature (cf., e.g., [1] and [2]). We use here the Saigo type fractional derivative operator defined as follows ([3]; see also [4]):

Definition 1. Let 0≤ <λ 1 and µ ν, ∈. Then the generalized fractional derivative operator , , 0,z

λ µ ν

 of a function f z

( )

is defined by

( )

₍

_{) (}

)

( )

, ,

0, ₀ 2 1

d

,1 ;1 ;1 d .

d 1

z z

z

f z z F f

z z

λ µ

λ

λ µ ν _ζ _{µ λ} _ν _λ ζ _{ζ ζ}

λ

−

 _ _ 

= _ − _ − − − − _ _

Γ −  



∫



 (1.3)

The function f z

( )

is an analytic function in a simply-connected region of the z-plane containing the origin, with the order

( )

,

(

0

)

f z =O z z→

for >max 0,

{

µ ν−

}

−1, and the multiplicity of

(

z−ζ

)

−λ is removed by requiring that log

(

z−ζ

)

to be real when z− >ζ 0.

Definition 2. Under the hypotheses of Definition 1, the fractional derivative operator , , 0,

m m m

z

λ+ µ+ ν+

 of a

function f z

( )

is defined by

( )

(

{ }

)

, , , ,

0, 0, 0

d

, ; : 0

d

m m m m

z f z m z f z z m

z

λ+ µ+ ν+ ₌ λ µ ν _∈ _∈_ _{= ∪}_

   (1.4)

With the aid of the above definitions, we define a modification of the fractional derivative operator , , ,

z p

λ µ ν

∆ by

( )

(

₍

_{) (}

) (

₎

)

( )

, , , ,

, 0,

1 1

z p z

p p

f z z f z

p p

λ µ ν µ λ ν µ λ µ ν

µ ν

Γ + − Γ + − +

∆ =

Γ + Γ + − +  (1.5)

for f z

( )

∈

( )

p and µ ν− − <p 1. Then it is observed that , , ,

z p

λ µ ν

∆ also maps 

( )

p onto itself as follows:

( )

₍

(

) (

_{) (}

)

₎

( )

(

)

, , ,

1

1 1

,

1 1

; 0 1; 1; .

p k k k p

z p k p

k _k _k

p p

f z z a z

p p

z p f p

λ µ ν µ ν

µ λ ν

λ µ ν

∞

+ + =

+ + − +

∆ = +

+ − + − +

∈ ≤ < − − < ∈

∑

 

(1.6)

It is easily verified from (1.6) that

( )

(

, ,

)

(

)

1, 1, 1

( )

, ,

( )

, , , .

z p z p z p

z ∆λ µ ν f z ′= p−µ ∆λ+ µ+ν+ f z + ∆µ λ µ νf z (1.7) Note that 0,0,

,

z p f f

ν

∆ = , 1,1,

,

z p f zf p

ν _′

∆ = and ,, , ( , )

p z p f z f

λ λ λ ν

∆ = Ω , where (z ,p)

λ

Ω is the fractional derivative oper-ator defined by Srivastava and Aouf [5].

In this manuscript, we drive interesting argument results of multivalent functions defined by fractional deriva-tive operator , ,

,

z p

λ µ ν

∆ .

2. Main Results

In order to establish our results, we require the following lemma due to Lashin [6].

Lemma 1 [6]. Let h z

( )

be analytic in , with h

( )

0 =1 and h z

( )

≠0

(

z∈

)

. Further suppose that

(

)

, 0,

α β_∈ + ₌ _∞

 and

( )

(

)

π 2

( ) (

)

arg arctan 0; 0

2 π

h z +βzh z′ < _α+ βα _ α > β>

  (2.1)

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( )

π

(

)

arg , .

2

h z < α z∈ (2.2)

We begin by proving the following result.

Theorem 1. Let λ≥0, µ<min

{

ν+ +p 1,p

}

and α γ δ, , ∈+, and let g z

( )

∈

( )

p . Suppose that

( )

f z ∈ p satisfies the condition

( )

(

)

, , 1, 1, 1 1, 1, 1

, , , , , , , , , , , , arg 1 π 2 arctan , 2 π

z p z p z p

f z f z g z

g z f z g z

p

γ

λ µ ν λ µ ν λ µ ν

λ µ ν δ λ µ ν λ µ ν

δ α α γ µ + + + + + + ___∆ __{ }_ __∆ _∆ ___ _ _{ } ₊ _ ₋ __   __∆ _{ }_ _ _ ∆ ∆ ___     

< _ +  ₋ _

      (2.3) then

( )

(

)

, , , , , , π

arg , .

2 z p z p f z z g z γ λ µ ν

λ µ ν α

∆ 

 _{ <} _∈

_∆ 

 

   (2.4)

Proof. If we define the function h z

( )

by

( )

,, ,

( )

_{( )}

(

)

, , ,

, 0 ,

z p z p f z h z g z γ λ µ ν

λ µ ν γ

∆ 

 

=_∆  ≠

 

  (2.5)

then

( )

2

1 2

1

h z = +c z+c z + is analytic in , with h

( )

0 =1 and h′

( )

0 ≠0. Making use of the logarithmic differentiation on both sides of (2.5), we have

( )

(

)

( )

(

)

( )

, , , , , , , , , , , , 1 .

z p z p

z f z z g z

zh z

h z f z g z

λ µ ν λ µ ν

λ µ ν λ µ ν γ

′ ′

∆ ∆

′

= −

∆ ∆ (2.6)

By applying the identity (1.7) in (2.6), we observe that

( )

,, ,

( )

_{( )}

,1, 1, 1

_{( )}

( )

,1, 1, 1

_{( )}

( )

, , , , , ,

, , ,

1 .

( )

z p z p z p

f z f z g z

h z zh z

p g z f z g z

γ

δ _δ γ µ + + + + + +   ∆  ∆ ∆      ′ + ₋ =_∆   + _ _∆ − _∆ _         

Hence, by using Lemma 1, we conclude that

( )

π

(

)

arg , ,

2

h z < α z∈

which completes the proof of Theorem 1.

Remark 1. Putting λ µ= =0, δ = =p 1 and g z

( )

=z in Theorem 1, we obtain the result due to Lashin ([6], Theorem 2.2).

Taking γ =1 and _{g z}

( )

₌_zp

in Theorem 1, we have the following corollary.

Corollary 1.Let λ≥0, µ<min

{

ν+ +p 1,p

}

and α δ, ∈+. Suppose that f z

( )

∈

( )

p satisfies the condition

(

)

,, ,

( )

,1, 1, 1

( )

π 2

arg 1 arctan ,

2 π

z p z p

p p

f z f z

p

z z

λ µ ν λ µ ν _δα

δ δ α

µ + + +

 ∆ ∆    

 ₋ ₊  _< ₊

     ₋          then

( )

₍

₎

, , , π

arg , .

2 z p p f z z z

λ µ ν

α ∆  < ∈       

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sa-tisfies the condition

( )

(

)

(

)

, ,

, π 2

arg arctan , .

2 π z p p f z z p z

λ µ ν _δ

α α

γ µ

 

∆   

<  +  ∈

   

   __ ₋ __

     (2.7)

then

(

)

( ) ( ) ( )

( )

1 , , , 0 π

arg d .

2

p p p

z

z p p

z t f t t

γ µ γ µ δ

λ µ ν

δ δ γ µ α δ − − − + −  ₋  ∆ <    



∫

 (2.8)

Proof. If we set

( )

(

)

( ) ( ) ( 1) , ,

( )

,

0 d ,

p p p

z

z p p

h z z t f t t

γ µ γ µ δ

λ µ ν

δ δ γ µ δ − − − + − −

=

_∫

∆ (2.9)

then

( )

2

1 2

1

( )

0 =1 and h′

( )

0 ≠0. By using the logarithmic dif-ferentiation on both sides of (2.9), we obtain

( )

₍

_{) ( )}

,, ,

( )

.

z p p

f z h z zh z

p z

λ µ ν δ γ µ ∆ ′ + = −

Thus, in view of Lemma 1, we have

( )

π

(

)

arg , ,

2

h z < α z∈

which evidently proves Theorem 2.

Remark 2. Setting λ µ= =0 and γ δ= = =p 1 in Theorem 2, we get the result obtained by Goyal and Goswami ([7], Corollary 3.6).

Putting λ µ γ δ= = = =1 in Theorem 2, we obtain the following result.

Corollary 2.Let α∈+. Suppose that f z

( )

∈

( ) (

p 1p≠

)

satisfies the condition

( )

1

π 2

arg arctan ,

2 π 1

p f z p pz α α − ′  

  _< ₊  

     ₋        then

( )

₍

₎

1 0 1 π

arg d , .

2 z p f t p t z t

pz − α

′

 − _< _∈

 



∫

 

Finally, we consider the generalized Bernardi-Libera-Livingston integral operator _σ

( )

f

(

σ > −p

)

de-fined by (cf.[8] [9] and [10])

( )

( )( )

1

( )

(

( )

)

0

: p z d , ; .

f f z t f t t f p p z

σ

σ σ σ

σ+ − _σ

≡ =

_∫

∈ > −

   (2.10)

Theorem 3. Let λ≥0, µ<min

{

ν+ +p 1,p

}

, σ > −p and α γ δ, , ∈+, and let g z

( )

∈

( )

p . Suppose that f z

( )

∈

( )

p satisfies the condition

( )( )

( )

( )( )

( )

(

)

, , 1, 1, 1 1, 1, 1

, , ,

, , , , , ,

, , ,

π 2

arg 1 arctan ,

2 π

z p z p z p

f z f z g z

p

g z f z g z

γ

σ

σ σ σ

δ

δ α α

γ σ

+ + + + + +

___∆ _{ }_ _ __∆ _∆ ___ _ _ __

_ _{ } ₊ _ ₋ __ _< _ ₊ _ __

   

∆    ∆ ∆    + 

 



   (2.11)

then

( )( )

(

)

, , , , , , π

arg , .

2 z p z p f z z g z γ λ µ ν

σ λ µ ν

σ

α

∆ 

 _{ <} _∈

 

∆

 

  



 (2.12)

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( )( )

(

, ,

)

(

)

, ,

( )

, ,

( )( )

, , , .

z p z p z p

z ∆λ µ νσ f z ′= σ+p ∆λ µ νf z − ∆σ λ µ νσ f z (2.13)

If we let

( )

( )( )

(

)

, , , , , ,

, 0 ,

z p

f z

h z

g z

γ λ µ ν

σ λ µ ν

σ

γ

∆ 

 

=_ _ ≠

∆

 

 



 (2.14)

then

( )

2

1 2

1

( )

0 =1 and h′

( )

0 ≠0. Differentiating both sides of (2.14) logarithmically, it follows that

( )

( )( )

(

)

( )( )

(

)

( )( )

, , , ,

, ,

, , , ,

, ,

1

.

z p z p

z f z z g z

zh z

h z f z g z

λ µ ν λ µ ν

σ σ

λ µ ν λ µ ν

σ σ

γ

′ ′

∆ ∆

′

= −

∆ ∆

 

  (2.15)

Hence, by applying the same arguments as in the proof of Theorem 1 with (2.13) and (2.15), we obtain

( )

π

(

)

arg , ,

2

h z < α z∈

which proves Theorem 3.

Acknowledgements

This work was supported by Daegu National University of Education Research Grant in 2014.

References

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http://dx.doi.org/10.1016/0898-1221(96)00151-4

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