Advances in Pure Mathematics
Vol.05 No.02(2015), Article ID:54044,4 pages
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
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/
Received 18 January 2015; accepted 9 February 2015; published 13 February 2015
ABSTRACT
The object of the present paper is to investigate various argument results of analytic and multivalent functions which are defined by using a certain fractional derivative operator. Some interesting applications are also considered.
Keywords:
Multivalent Analytic Functions, Argument, Integral Operator, Fractional Derivative Operator
1. Introduction
Let denote the class of functions
of the form
(1.1)
which are analytic in the open unit disk. Also let
denote the class of all analytic functions
with
which are defined on
.
Let a, b and c be complex numbers with. Then the Gaussian hypergeometric function
is defined by
(1.2)
where is the Pochhammer symbol defined, in terms of the Gamma function, by
The hypergeometric function 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 and
. Then the generalized fractional derivative operator
of a function
is defined by
(1.3)
The function is an analytic function in a simply-connected region of the z-plane containing the origin, with the order
for, and the multiplicity of
is removed by requiring that
to be real when
.
Definition 2. Under the hypotheses of Definition 1, the fractional derivative operator of a function
is defined by
(1.4)
With the aid of the above definitions, we define a modification of the fractional derivative operator by
(1.5)
for and
. Then it is observed that
also maps
onto itself as follows:
(1.6)
It is easily verified from (1.6) that
(1.7)
Note that,
and
, where
is the fractional derivative operator defined by Srivastava and Aouf [5] .
In this manuscript, we drive interesting argument results of multivalent functions defined by fractional derivative operator.
2. Main Results
In order to establish our results, we require the following lemma due to Lashin [6] .
Lemma 1 [6] . Let be analytic in
, with
and
. Further suppose that
and
(2.1)
then
(2.2)
We begin by proving the following result.
Theorem 1. Let,
and
, and let
. Suppose that
satisfies the condition
(2.3)
then
(2.4)
Proof. If we define the function by
(2.5)
then is analytic in
, with
and
. Making use of the logarithmic differentiation on both sides of (2.5), we have
(2.6)
By applying the identity (1.7) in (2.6), we observe that
Hence, by using Lemma 1, we conclude that
which completes the proof of Theorem 1.
Remark 1. Putting,
and
in Theorem 1, we obtain the result due to Lashin ([6] , Theorem 2.2).
Taking and
in Theorem 1, we have the following corollary.
Corollary 1. Let,
and
. Suppose that
satisfies the condition
then
Theorem 2. Let,
,
and
. Suppose that
satisfies the condition
(2.7)
then
(2.8)
Proof. If we set
(2.9)
then is analytic in
, with
and
. By using the logarithmic differentiation on both sides of (2.9), we obtain
Thus, in view of Lemma 1, we have
which evidently proves Theorem 2.
Remark 2. Setting and
in Theorem 2, we get the result obtained by Goyal and Goswami ([7] , Corollary 3.6).
Putting in Theorem 2, we obtain the following result.
Corollary 2. Let. Suppose that
satisfies the condition
then
Finally, we consider the generalized Bernardi-Libera-Livingston integral operator
defined by (cf. [8] [9] and [10] )
(2.10)
Theorem 3. Let,
,
and
, and let
. Suppose that
satisfies the condition
(2.11)
then
(2.12)
Proof. From (2.10) we observe that
(2.13)
If we let
(2.14)
then is analytic in
, with
and
. Differentiating both sides of (2.14) logarithmically, it follows that
(2.15)
Hence, by applying the same arguments as in the proof of Theorem 1 with (2.13) and (2.15), we obtain
which proves Theorem 3.
Acknowledgements
This work was supported by Daegu National University of Education Research Grant in 2014.
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