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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.

Let

which are analytic in the open unit disk

Let a, b and c be complex numbers with

where

The hypergeometric function

There are a number of definitions for fractional calculus operators in the literature (cf., e.g., [

Definition 1. Let

The function

for

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

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

for

It is easily verified from (1.6) that

Note that

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

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

Lemma 1 [

then

We begin by proving the following result.

Theorem 1. Let

then

Proof. If we define the function

then

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

Taking

Corollary 1. Let

then

Theorem 2. Let

then

Proof. If we set

then

Thus, in view of Lemma 1, we have

which evidently proves Theorem 2.

Remark 2. Setting

Putting

Corollary 2. Let

then

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

Theorem 3. Let

then

Proof. From (2.10) we observe that

If we let

then

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.