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: choijh@dnue.ac.kr
Received 18 January 2015; accepted 9 February 2015; published 13 February 2015
Copyright © 2015 by author and Scientific Research Publishing Inc.
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)( )
(
( )
)
1,(
) (
) (
(
0 ;)
)
1 1 , .
k
k 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 itreduces 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
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)
( )
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
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
( )
( )
(
)
, , , , , , πarg , .
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
( )
21 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 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
z p z p z p
f z f z g z
h z zh z
p g z f z g z
γ
λ µ ν λ µ ν λ µ ν
λ µ ν λ µ ν λ µ ν
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
( )
=zpin 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 2arg 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
( )
(
)
, , , πarg , .
2 z p p f z z z
λ µ ν
α ∆ < ∈
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 πarg 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
( )
21 2
1
h z = +c z+c z + is analytic in , with h
( )
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 πarg 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
z p z p z p
f z f z g z
p
g z f z g z
γ
λ µ ν λ µ ν λ µ ν
σ
λ µ ν λ µ ν λ µ ν
σ σ σ
δ
δ α α
γ σ
+ + + + + +
∆ ∆ ∆
+ − < +
∆ ∆ ∆ +
(2.11)
then
( )( )
( )( )
(
)
, , , , , , πarg , .
2 z p z p f z z g z γ λ µ ν
σ λ µ ν
σ
α
∆
< ∈
∆
(2.12)
( )( )
(
, ,)
(
)
, ,( )
, ,( )( )
, , , .
z p z p z p
z ∆λ µ νσ f z ′= σ+p ∆λ µ νf z − ∆σ λ µ νσ f z (2.13)
If we let
( )
( )( )
( )( )
(
)
, , , , , ,
, 0 ,
z p
z p
f z
h z
g z
γ λ µ ν
σ λ µ ν
σ
γ
∆
= ≠
∆
(2.14)
then
( )
21 2
1
h z = +c z+c z + is analytic in , with h
( )
0 =1 and h′( )
0 ≠0. Differentiating both sides of (2.14) logarithmically, it follows that( )
( )
( )( )
(
)
( )( )
( )( )
(
)
( )( )
, , , ,
, ,
, , , ,
, ,
1
.
z p z p
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.
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