Inclusion results for certain subclasses of p-valent meromorphic functions associated with a new operator
© Mostafa; licensee Springer 2012
Received: 21 May 2012
Accepted: 18 July 2012
Published: 31 July 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.
Now, we define the operator as follows:
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.
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.
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 .
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. □
This completes the proof of Theorem 2. □
in Theorem 1 and Theorem 2, we have the following corollary.
Now, using Lemma 2, we obtain similar inclusion relations for the subclass .
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
Applying Lemma 1, we conclude that , which implies that . □
This completes the proof of Theorem 5. □
From Theorem 4 and Theorem 5, we have the following corollary.
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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