Kyungpook Mathematical Journal 2017; 57(1): 109-124
Published online March 23, 2017
Copyright © Kyungpook Mathematical Journal.
Some New Subclasses of Analytic Functions defined by Srivastava-Owa-Ruscheweyh Fractional Derivative Operator
Khalida Inayat Noor1
Department of Mathematics, COMSATS Institute of Information Technology Park Road, Islamabad, Pakistan1
Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland2
Received: January 5, 2016; Accepted: December 6, 2016
In this article the Srivastava-Owa-Ruscheweyh fractional derivative operator
Keywords: analytic functions, convolution, subordination, Srivastava-Owa-Ruscheweyh  ,fractional derivative operator, multiplier linear fractional differential operator, gamma function, incomplete beta function
Let denote the class of all normalized functions of the form
which are analytic in the open unit disk
Let , , , (
the convolution (Hadamard product) is defined as
Recently, the theory of fractional calculus has found interesting applications in the theory of analytic functions. The classical definitions of fractional operators and their generalizations have fruitfully been applied in obtaining, for example, the characterization properties, coefficient estimates, distortion inequalities and convolution structures for various subclasses of analytic functions.
The fractional derivative of order
where the function
where the incomplete beta function
Mishra and Gochhayat  have studied some properties of the operator Ω
where 0 ≤
Bulut [5, 6, 7, 8] used the Al-Oboudi fractional differential operator of order
Ozkan  defined the convolution of
which is called the Srivastava-Owa-Ruscheweyh fractional derivative operator.
It can easily be seen that
α= 0, λ= 0.
a= n+ 1 > 0, α= 0, λ= 0.
a= n+ 1 > 0, α= 0.
From the definition of
We assume that
Obviously, for the special choices of function
To prove our main results, we need the following Lemmas.
3. Main Results
In this section, we will prove our main results.
We suppose that
Then by (
Thus it follows from the Lemma 2.1 that
As special cases of Theorem 3.1, we have the following results.
Then by (
This means that
This implies by using Lemma 2.1 with
Now by applying Lemma 2.2 and from (
We assume that
Using Lemma 2.2 with (
By using (
Now by using Lemma 2.2, we obtain
we can write
Thus from Lemma 2.2, we have
(i). The case
The proof of part (ii) is similar to (i).
Let Ψ ∈ ,
Note that the above result also holds for the classes
This means that
Therefore using (
Now by using Lemma 2.2, we have
Some of the special cases are given as follows.
k= 0. Then . That is
k= 1, we have
k> 1 and . That is
For the case
k= 2, we note that . This gives us
Now we prove a radius result for the class
It follows that
The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan for providing excellent research and academic environment. This research is supported by the HEC NPRU project No: 20-1966/R&D/11-2553, titled, Research unit of Academic Excellence in Geometric Functions Theory and Applications.
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