Advances in Pure Mathematics
Vol.05 No.02(2015), Article ID:54044,4 pages

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


Copyright © 2015 by author and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

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


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.


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

1. Introduction

Let denote the class of functions of the form


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


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


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


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


for and. Then it is observed that also maps onto itself as follows:


It is easily verified from (1.6) that


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




We begin by proving the following result.

Theorem 1. Let, and, and let. Suppose that satisfies the condition




Proof. If we define the function by


then is analytic in, with and. Making use of the logarithmic differentiation on both sides of (2.5), we have


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


Theorem 2. Let, , and. Suppose that satisfies the condition




Proof. If we set


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


Finally, we consider the generalized Bernardi-Libera-Livingston integral operator defined by (cf. [8] [9] and [10] )


Theorem 3. Let, , and, and let. Suppose that satisfies the condition




Proof. From (2.10) we observe that


If we let


then is analytic in, with and. Differentiating both sides of (2.14) logarithmically, it follows that


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.


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


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