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Axioms 2018, 7(2), 27; doi:10.3390/axioms7020027
Article
Subordination Properties for Multivalent Functions Associated with a Generalized Fractional Differintegral Operator
1
Department of Mathematics, Faculty of Science, Menofia University, Shebin Elkom 32511, Egypt
2
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
*
Author to whom correspondence should be addressed.
Received: 19 January 2018 / Accepted: 17 April 2018 / Published: 24 April 2018
Abstract
:Using of the principle of subordination, we investigate some subordination and convolution properties for classes of multivalent functions under certain assumptions on the parameters involved, which are defined by a generalized fractional differintegral operator under certain assumptions on the parameters involved.
Keywords:
differential subordination; p-valent functions; generalized fractional differintegral operatorJEL Classification:
30C45; 30C501. Introduction and Definitions
Denote by the class of analytic and p-valent functions of the form:
For functions analytic in , f is subordinate to g, written if there exists a function w, analytic in with and such that If g is univalent in then (see [1,2]):
If is analytic in and satisfies:
then is a solution of (2). The univalent function q is called dominant, if for all . A dominant is called the best dominant, if for all dominants q.
Let be the well-known (Gaussian) hypergeometric function defined by:
where:
We will recall some definitions that will be used in our paper.
Definition 1.
Definition 2.
For and in terms of the generalized fractional integral and generalized fractional derivative operators defined by Srivastava et al. [5] (see also [6]) as:
where is an analytic function in a simply-connected region of the complex plane containing the origin with the order when , and the multiplicity of is removed by requiring to be real when .
We note that:
where and are the fractional integral and fractional derivative operators studied by Owa [3].
For , we have:
where “∗” stands for convolution of two power series, and is the well-known generalized hypergeometric function.
Let:
and:
It is easy to verify that:
and:
By using the operator , we introduce the following class.
Definition 3.
For , is in the class if
which is equivalent to:
For convenience, we write , which satisfies the inequality:
In this paper, we investigate some subordination and convolution properties for classes of multivalent functions, which are defined by a generalized fractional differintegral operator. The theory of subordination received great attention, particularly in many subclasses of univalent and multivalent functions (see, for example, [13,15,16,17]).
2. Preliminaries
To prove our main results, we shall need the following lemmas.
Lemma 1.
Denote by the class of functions given by:
which are analytic in and satisfy the following inequality:
Using the well-known growth theorem for the Carathéodory functions (cf., e.g., [19]), we may easily deduce the following result:
Lemma 2.
[19]. If . Then
Lemma 3.
Lemma 4.
[21] Let ϕ be such that and and with
If or and satisfies:
then:
and this is the best dominant.
If and and if satisfies:
then:
and this is the best dominant.
Lemma 5.
[2]. Let , and satisfy for all and all . If is analytic in and if:
then in .
Lemma 6.
[22]. Let be analytic in with and for all If there exist two points such that:
for some and and for all then:
where:
3. Properties Involving
Unless otherwise mentioned, we assume throughout this paper that , and the powers are considered principal ones.
Proof.
Let:
Then, φ is analytic in . After some computations, we get:
Now, by using Lemma 1, we deduce that:
or, equivalently,
and so:
Since:
Now, for defined by (18), we have:
Letting we obtain:
which ends our proof. ☐
Putting and using Lemma 1 for Equation (15) in Theorem 1, we obtain the following example.
Example 1.
Let the function Then, following containment property holds,
Theorem 2.
Let satisfy
Then:
The result is sharp.
Putting in Theorem 2, we obtain the following example.
Example 2.
Let the function Then, following inclusion property holds
For a function the generalized Bernardi–Libera–Livingston integeral operator is defined by (see [23]):
Lemma 7.
If prove that:
- (i)
- (ii)
Proof.
Since
and:
Now, the first part of this lemma follows. Furthermore,
If we replace by and using the first part of this lemma, we get (21). ☐
Theorem 3.
Suppose that and defined by (20). Then:
The result is sharp.
Proof.
Let:
Then, φ is analytic in . After some calculations, we have:
Employing the same technique that was used in proving Theorem 1, the remaining part of the theorem can be proven. ☐
Theorem 4.
Let If each of the functions satisfies:
then:
where:
and:
The result is possible when
Proof.
Thus, by making use of the identity (3) in (29), we get:
which, in view of F given by (27) and (30), yields:
where:
Since it follows from Lemma 3 that:
When we consider the functions , which satisfy (25), are defined by:
Theorem 5.
Let , and let with and Suppose that:
If with for all then:
implies:
where:
is the best dominant.
Proof.
Putting:
Then, is analytic in , and for all Taking the logarithmic derivatives on both sides of (34) and using (3), we have:
Now, the assertions of Theorem 5 follow by Lemma 4. ☐
Theorem 6.
Let If satisfies:
then:
where is the positive root of the equation:
Proof.
Let:
Then, φ is analytic in , and for all Taking the logarithmic derivatives on both sides of (37) and using the identity (3), we have:
and so:
Let:
and:
For and for This proves that Thus, for , , and so, we obtain the required result by an application of Lemma 5. ☐
Theorem 7.
Suppose that If:
then:
where:
Proof.
Let:
Then, from Theorem 1, we have:
Let be the function that maps onto the domain:
with then:
Acknowledgments
The authors would like to thank all referees for their valuable comments which led to the improvement of this paper.
Author Contributions
All the authors read and approved the final manuscript as a consequence of the authors meetings.
Conflicts of Interest
The authors declare no conflict of interest.
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