1. Introduction
Let the notation
be the family of meromorphic
p-valent functions
f that are holomorphic (analytic) in the region of punctured disk
and obeying the following normalization
In particular
, the familiar set of meromorphic functions. Further, the symbol
represents the set of meromorphic starlike functions which is a subfamily of
and is given by
Two points
p and
are said to be symmetrical with respect to
o if
is the midpoint of the line segment
. This idea was further nourished in [
1,
2] by introducing the family
which is defined in set builder form as;
Now, for
with
is real with
and
we introduce a subfamily of
consisting of all meromorphic
p-valent functions of reciprocal order
, denoted by
, and is defined by
We note that for
and
, the class
reduces to the class
and is represented by
For detail of the related topics, see the work of Al-Amiri and Mocanu [
3], Rosihan and Ravichandran [
4], Aouf and Hossen [
5], Arif [
6], Goyal and Prajapat [
7], Joshi and Srivastava [
8], Liu and Srivastava [
9], Raina and Srivastava [
10], Sun et al. [
11], Shi et al. [
12] and Owa et al. [
13], see also [
14,
15,
16].
For simplicity and ignoring the repetition, we state here the constraints on each parameter as with is real with and
We need to mention the following lemmas which will use in the main results.
Lemma 1. “Let and let be a mapping satisfying H for such that If is regular in and ∀ , then ”
Lemma 2. “Let be regular in and η be regular and starlike univalent in with If then This result is the best possible.”
2. Sufficiency Criterion for the Family
In this section, we investigate the sufficiency criterion for any meromorphic p-valent functions belonging to the introduced family :
Now, we obtain the necessary and sufficient condition for the p-valent function
f to be in the family
as follows:
Theorem 1. Let the function be the member of the family Thenwhere Proof. Suppose that inequality (2) holds. Then, we have
and hence the result follows. □
Next, we investigate the sufficient condition for the p-valent function f to be in the family in the following theorem:
Theorem 2. If belongs to the family of meromorphic p-valent functions and obeyingthen Proof. To prove the required result we only need to show that
Now consider the left hand side of (5), we get
By virtue of inequality (4), we at once get the desired result. □
Also, we obtain another sufficient condition for the p-valent function f to be in the family by using Lemma 1, in the following theorem:
Theorem 3. If satisfiesthen where is defined in Equation (3)
. Proof. Let we choose the function
by
then Equation (6) shows that
is holomorphic in
and also normalized by
From Equation (6), we can easily obtain that
where
Now for all
satisfying
we have
Now, let us define a set as
then, we see that
and
Therefore, by using Lemma 1, we obtain that ℜ
□
Further, in the next theorem, we obtain the sufficient condition for the p-valent function f to be in the family by using Lemma 2.
Theorem 4. If is a member of the family of meromorphic p-valent functions and satisfiesthen where is given by Equation (3)
. Proof. In order to prove the required result, we need to define the following function
then, Equation (6) shows that th function
is holomorphic in
and also normalized by
Now, by routine computations, we get
Now, let us consider
and then by using inequality (7), we have
therefore
Using Lemma 2, we have
equivalently
After simplifications, we get
The region
shows that it is symmetric about the real axis and also
is convex. Hence
or
or
□
Finally, we investigate the sufficient condition for the p-valent function f to be in the family in the following theorem:
Theorem 5. If satisfiesthen where is defined in Equation (3)
. Proof. Then
and
is analytic in
. Using inequality (9), we can write
Now,
and this implies that
Now by using Theorem 1, we get the result which we needed. □
3. Conclusions
In our results, a new subfamily of meromorphic p-valent (multivalent) functions were introduced. Further, various sufficient conditions for meromorphic p-valent functions belonging to these subfamilies were obtained and investigated.
Author Contributions
Conceptualization, H.M.S. and M.A.; Formal analysis, H.M.S. and S.M.; Funding acquisition, S.M. and G.S.; Investigation, E.S.A.A. and S.M.; Methodology, M.A. and F.G.; Supervision, H.M.S. and M.A.; Validation, M.A. and S.M.; Visualization, G.S. and E.S.A.A.; Writing original draft, M.A., S.M. and F.G.; Writing review and editing, M.A., F.G. and S.M.
Funding
This research received no external funding.
Acknowledgments
The authors would like to thank the reviewers of this paper for their valuable comments on the earlier version of the paper. They would also like to acknowledge Salim ur Rehman, the Vice Chancellor, Sarhad University of Science & I.T, for providing excellent research environment and his financial support.
Conflicts of Interest
All the authors declare that they have no conflict of interest.
References
- Srivastava, H.M.; Yang, D.-G.; Xu, N.-E. Some subclasses of meromorphically multivalent functions associated with a linear operator. Appl. Math. Comput. 2008, 195, 11–23. [Google Scholar] [CrossRef]
- Wang, Z.-G.; Jiang, Y.-P.; Srivastava, H.M. Some subclasses of meromorphically multivalent functions associated with the generalized hypergeometric function. Comput. Math. Appl. 2009, 57, 571–586. [Google Scholar] [CrossRef] [Green Version]
- Al-Amiri, H.; Mocanu, P.T. Some simple criteria of starlikeness and convexity for meromorphic functions. Mathematica (Cluj) 1995, 37, 11–21. [Google Scholar]
- Ali, R.M.; Ravichandran, V. Classes of meromorphic α-convex functions. Taiwanese J. Math. 2010, 14, 1479–1490. [Google Scholar] [CrossRef]
- Aouf, M.K.; Hossen, H.M. New criteria for meromorphic p-valent starlike functions. Tsukuba J. Math. 1993, 17, 481–486. [Google Scholar] [CrossRef]
- Arif, M. On certain sufficiency criteria for p-valent meromorphic spiralike functions. In Abstract and Applied Analysis; Hindawi: London, UK, 2012. [Google Scholar]
- Goyal, S.P.; Prajapat, J.K. A new class of meromorphic multivalent functions involving certain linear operator. Tamsui Oxf. J. Math. Sci. 2009, 25, 167–176. [Google Scholar]
- Joshi, S.B.; Srivastava, H.M. A certain family of meromorphically multivalent functions. Comput. Math. Appl. 1999, 38, 201–211. [Google Scholar] [CrossRef] [Green Version]
- Liu, J.-L.; Srivastava, H.M. A linear operator and associated families of meromorphically multivalent functions. J. Math. Anal. Appl. 2001, 259, 566–581. [Google Scholar] [CrossRef]
- Raina, R.K.; Srivastava, H.M. A new class of mermorphically multivalent functions with applications of generalized hypergeometric functions. Math. Comput. Model. 2006, 43, 350–356. [Google Scholar] [CrossRef]
- Sun, Y.; Kuang, W.-P.; Wang, Z.-G. On meromorphic starlike functions of reciprocal order α. Bull. Malays. Math. Sci. Soc. 2012, 35, 469–477. [Google Scholar]
- Shi, L.; Wang, Z.-G.; Yi, J.-P. A new class of meromorphic functions associated with spirallike functions. J. Appl. Math. 2012, 2012, 1–12. [Google Scholar] [CrossRef]
- Owa, S.; Darwish, H.E.; Aouf, M.A. Meromorphically multivalent functions with positive and fixed second coefficients. Math. Japon. 1997, 46, 231–236. [Google Scholar]
- Arif, M.; Ahmad, B. New subfamily of meromorphic starlike functions in circular domain involving q-differential operator. Math. Slovaca 2018, 68, 1049–1056. [Google Scholar] [CrossRef]
- Arif, M.; Raza, M.; Ahmad, B. A new subclass of meromorphic multivalent close-to-convex functions. Filomat 2016, 30, 2389–2395. [Google Scholar] [CrossRef]
- Arif, M.; Sokół, J.; Ayaz, M. Sufficient condition for functions to be in a class of meromorphic multivalent Sakaguchi type spiral-like functions. Acta Math. Sci. 2014, 34, 1–4. [Google Scholar] [CrossRef]
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).