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Article

# Sufficiency Criterion for A Subfamily of Meromorphic Multivalent Functions of Reciprocal Order with Respect to Symmetric Points

1
2
Department of Mathematics and Computer Science, Brandon University, 270 18th Street, Brandon, MB R7A 6A9, Canada
3
Research Center for Interneural Computing, China Medical University, Taichung 40402, Taiwan
4
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
5
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
6
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
7
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(6), 764; https://doi.org/10.3390/sym11060764
Received: 5 May 2019 / Revised: 26 May 2019 / Accepted: 30 May 2019 / Published: 5 June 2019
(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

## Abstract

:
In the present research paper, our aim is to introduce a new subfamily of meromorphic p-valent (multivalent) functions. Moreover, we investigate sufficiency criterion for such defined family.

## 1. Introduction

Let the notation $Ω p$ 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 $Ω 1 = Ω$, the familiar set of meromorphic functions. Further, the symbol $MS *$ represents the set of meromorphic starlike functions which is a subfamily of $Ω$ and is given by
Two points p and $p ′$ are said to be symmetrical with respect to o if $o ′$ is the midpoint of the line segment $p p ′$. This idea was further nourished in [1,2] by introducing the family $MS s *$ which is defined in set builder form as;
Now, for $− 1 ≤ t < s ≤ 1$ with $s ≠ 0 ≠ t ,$ $0 < ξ < 1 ,$ $λ$ is real with $λ < π 2$ and $p ∈ N ,$ we introduce a subfamily of $Ω p$ consisting of all meromorphic p-valent functions of reciprocal order $ξ$, denoted by , and is defined by
We note that for $p = s = 1$ and $t = − 1$, the class reduces to the class and is represented by
For detail of the related topics, see the work of Al-Amiri and Mocanu , Rosihan and Ravichandran , Aouf and Hossen , Arif , Goyal and Prajapat , Joshi and Srivastava , Liu and Srivastava , Raina and Srivastava , Sun et al. , Shi et al.  and Owa et al. , see also [14,15,16].
For simplicity and ignoring the repetition, we state here the constraints on each parameter as $0 < ξ < 1 ,$ $− 1 ≤ t < s ≤ 1$ with $s ≠ 0 ≠ t ,$ $λ$ is real with $λ < π 2$ and $p ∈ N .$
We need to mention the following lemmas which will use in the main results.
Lemma 1.
“Let $H ⊂ C$ and let $Φ : C 2 × E * → C$ be a mapping satisfying H for $a , b ∈ R$ such that $b ≤ − n 1 + a 2 2 .$ If $p z = 1 + c n z n + ⋯$ is regular in $E *$ and $z ∈ E *$, then
Lemma 2.
“Let $p ( z ) = 1 + c 1 z + ⋯$ be regular in $E *$ and η be regular and starlike univalent in $E *$ with $η ( 0 ) = 0 .$ If $z p ′ ( z ) ≺ η ( z ) ,$ then
$p ( z ) ≺ 1 + ∫ 0 z η ( t ) t d t .$
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 $f ( z )$ be the member of the family $Ω p .$ Then
where
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 $f ( z )$ belongs to the family $Ω p$ of meromorphic p-valent functions and obeying
then
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 $f ( z ) ∈ Ω p$ satisfies
then where $G z$ is defined in Equation (3).
Proof.
Let we choose the function $q z$ by
then Equation (6) shows that $q z$ is holomorphic in $E$ and also normalized by $q 0 = 1 .$
From Equation (6), we can easily obtain that
where
Now for all $a , b ∈ R$ 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 $f ( z )$ is a member of the family $Ω p$ of meromorphic p-valent functions and satisfies
then where $G z$ is given by Equation (3).
Proof.
In order to prove the required result, we need to define the following function
$q z cos λ = e − i λ G z + i sin λ ,$
then, Equation (6) shows that th function $q z$ is holomorphic in $E$ and also normalized by $q 0 = 1 .$
Now, by routine computations, we get
$z q ′ ( z ) q z − i tan λ = z G ′ z G z .$
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 $H E$ shows that it is symmetric about the real axis and also $H z$ 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 $f ( z ) ∈ Ω p$ satisfies
then where $G z$ is defined in Equation (3).
Proof.
Let us put
Then $G 0 = 0$ and $G z$ is analytic in $E$. 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.

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## Share and Cite

MDPI and ACS Style

Mahmood, S.; Srivastava, G.; Srivastava, H.M.; Abujarad, E.S.A.; Arif, M.; Ghani, F. Sufficiency Criterion for A Subfamily of Meromorphic Multivalent Functions of Reciprocal Order with Respect to Symmetric Points. Symmetry 2019, 11, 764. https://doi.org/10.3390/sym11060764

AMA Style

Mahmood S, Srivastava G, Srivastava HM, Abujarad ESA, Arif M, Ghani F. Sufficiency Criterion for A Subfamily of Meromorphic Multivalent Functions of Reciprocal Order with Respect to Symmetric Points. Symmetry. 2019; 11(6):764. https://doi.org/10.3390/sym11060764

Chicago/Turabian Style

Mahmood, Shahid, Gautam Srivastava, Hari Mohan Srivastava, Eman S.A. Abujarad, Muhammad Arif, and Fazal Ghani. 2019. "Sufficiency Criterion for A Subfamily of Meromorphic Multivalent Functions of Reciprocal Order with Respect to Symmetric Points" Symmetry 11, no. 6: 764. https://doi.org/10.3390/sym11060764

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