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Inclusion results for certain subclasses of p-valent meromorphic functions associated with a new operator
Journal of Inequalities and Applications volume 2012, Article number: 169 (2012)
In this paper, we introduce new subclasses of p-valent starlike, p-valent convex, p-valent close-to-convex, and p-valent quasi-convex meromorphic functions and investigate some inclusion properties of these subclasses and investigate various inclusion properties and integral-preserving properties for the p-valent meromorphic function classes.
Let denote the class of functions of the form
which are analytic and p-valent in the punctured unit disc . If and are analytic in , we say that is subordinate to , written or (), if there exists a Schwarz function in U with and , such that (). Furthermore, if is univalent in U, then the following equivalence relationship holds true (see  and ):
For functions , given by (1.1) and defined by
the Hadamard product (or convolution) of and is given by
Aqlan et al. defined the operator by:
Now, we define the operator as follows:
and let be defined by
Using (1.5)-(1.7), we have
where is in the form (1.1) and denotes the Pochhammer symbol given by
It is readily verified from (1.8) that
It is noticed that, putting in (1.8), we obtain the operator
Let M be the class of analytic functions with , which are convex and univalent in U and satisfy ().
For , , we denote by , , , and , the subclasses of consisting of all p-valent meromorphic functions which are, respectively, starlike of order η, convex of order η, close-to-convex functions of order γ and type η, and quasi-convex functions of order γ and type η in U.
Making use of the principle of subordination between analytic functions, we introduce the subclasses , , , and (, and ) of the class which are defined by:
Now, by using the linear operator (, , ; ) and for , , , we define new subclasses of meromorphic functions of by:
We also note that
In particular, we set
In this paper, we investigate several inclusion properties of the classes , , , and associated with the operator . Some applications involving integral operators are also considered.
In order to establish our main results, we need the following lemmas.
Let ς and υ be complex constants and letbe convex (univalent) in U withand. If
is analytic in U, then
Letbe convex (univalent) in U andbe analytic in U with. If q is analytic in U and, then
2 Some inclusion results
In this section, we give some inclusion properties for meromorphic function classes, which are associated with the operator . Unless otherwise mentioned, we assume that , , , , and .
Theorem 1 For, with
then we have
Proof We will first show that
Let and put
where is analytic in U with . Applying (1.10) in (2.3), we have
Differentiating (2.4) logarithmically with respect to z, we have
we see that
Applying Lemma 1 to (2.5), it follows that , that is, that . The proof of the second part will follow by using arguments similar to those used in the first part with and using the identity (1.9) instead of (1.10). This completes the proof of Theorem 1. □
Theorem 2 For, with
Proof Applying (1.10) and using Theorem 1, we have
This completes the proof of Theorem 2. □
in Theorem 1 and Theorem 2, we have the following corollary.
Corollary 1 Let and
Then we have
Now, using Lemma 2, we obtain similar inclusion relations for the subclass .
Theorem 3 Let and
Then we have
Proof First, we will prove that
Let . Then, from the definition of the class , there exists a function such that
where is analytic in U with . Applying (1.10) in (2.6), we have
Since, by Theorem 1,
where in U, and . Then, using (2.7) and (2.8), we have
Differentiating both sides of (2.9) with respect to z and multiplying by z, we have
Making use of (2.6), (2.10), and (2.11), we have
Since in U, and
in Eq. (2.12) and applying Lemma 2, we can show that , that is, that . The second part can be proved by using similar arguments and using (1.9). This completes the proof of Theorem 3. □
3 Inclusion properties involving the integral operator
From (3.1), we have
Theorem 4 Let with
Proof Let and put
where h is analytic in U with . Then, by using (3.2) and (3.3), we have
Differentiating (3.4) logarithmically with respect to z, we have
Applying Lemma 1, we conclude that , which implies that . □
Theorem 5 Let with
Proof Applying Theorem 4 and (1.12), we have
This completes the proof of Theorem 5. □
From Theorem 4 and Theorem 5, we have the following corollary.
Corollary 2 Suppose that
Then, for the classesand, the following inclusion relations hold true:
Theorem 6 Let with
Proof Let . Then, from the definition of the class , there exists a function such that
where is analytic in U with . Applying (3.2) in (3.6), we have
Since , then by Theorem 4, we have . Let
where in U. Then, using the same techniques as in the proof of Theorem 3 and using (3.5) and (3.7), we have
Since , then applying Lemma 2, we find that , which yields . This completes the proof of Theorem 6. □
Remark Putting in the above results, we obtain the results corresponding to the operator defined by (1.11).
The author read and approved the final manuscript.
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The author declares that she has no competing interests.
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Cite this article
Mostafa, A. Inclusion results for certain subclasses of p-valent meromorphic functions associated with a new operator. J Inequal Appl 2012, 169 (2012). https://doi.org/10.1186/1029-242X-2012-169
- p-valent meromorphic functions
- Hadamard product
- inclusion properties