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Abstract and Applied Analysis

VolumeΒ 2012Β (2012), Article IDΒ 837913, 10 pages

http://dx.doi.org/10.1155/2012/837913

## On Certain Sufficiency Criteria for -Valent Meromorphic Spiralike Functions

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan

Received 10 June 2012; Accepted 6 August 2012

Academic Editor: AllanΒ Peterson

Copyright Β© 2012 Muhammad Arif. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider some subclasses of meromorphic multivalent functions and obtain certain simple sufficiency criteria for the functions belonging to these classes. We also study the mapping properties of these classes under an integral operator.

#### 1. Introduction

Let denote the class of functions of the form which are analytic and -valent in the punctured unit disk . Also let and denote the subclasses of consisting of all functions which are defined, respectively, by

We note that for and , the above classes reduce to the well-known subclasses of consisting of meromorphic multivalent functions which are, respectively, starlike and convex of order β. For the detail on the subject of meromorphic spiral-like functions and related topics, we refer the work of Liu and Srivastava [1], Goyal and Prajapat [2], Raina and Srivastava [3], Xu and Yang [4], and Spacek [5] and Robertson [6].

Analogous to the subclass of for meromorphic univalent functions studied by Wang et al. [7] and Nehari and Netanyahu [8], we define a subclass of consisting of functions satisfying

For more details of the above classes see also [9, 10].

Motivated from the work of Frasin [11], we introduce the following integral operator of multivalent meromorphic functions

For , (1.4) reduces to the integral operator introduced and studied by Mohammed and Darus [12, 13]. Similar integral operators for different classes of analytic, univalent, and multivalent functons in the open unit disk are studied by various authors, see [14β19].

In this paper, first, we find sufficient conditions for the classes and and then study some mapping properties of the integral operator given by (1.4).

We will assume throughout our discussion, unless otherwise stated, that is real with , , , for .

To obtain our main results, we need the following Lemmas.

Lemma 1.1 (see [20]). *If with and satisfies the condition
**
then
*

Lemma 1.2 (see [21]). *Let be a set in the complex plane and suppose that is a mapping from to which satisfies for , and for all real such that . If is analytic in and for all , then .*

#### 2. Some Properties of the Classes and

Theorem 2.1. *If satisfies
**
then .*

*Proof. *Let us set a function by
for . Then clearly (2.2) shows that .

Differentiating (2.2) logarithmically, we have
which gives

Thus using (2.1), we have

Hence, using Lemma 1.1, we have .

From (2.3), we can write

Since , it implies that . Therefore, we get
or
and this implies that .

If we take , we obtain the following result.

Corollary 2.2. *If satisfies
**
then .*

Theorem 2.3. *If satisfies
**
then .*

*Proof. *Let us set

Also let

Then clearly and . Now

Differentiating logarithmically and then simple computation gives us

Therefore, by using Lemma 1.1, we have
which implies that . Since
therefore
Since , so
or

It follows that .

Theorem 2.4. *If satisfies
**then , where , and
*

*Proof. *Let us set

Then is analytic in with .

Taking logarithmic differentiation of (2.22) and then by simple computation, we obtain
with

Now for all real and satisfying , we have

Reputing the values of , , , and and then taking real part, we obtain
where , , and are given in (2.21).

Let . Then and , for all real and satisfying , . By using Lemma 1.2, we have , that is .

If we put , we obtain the following result.

Corollary 2.5. *If satisfies
**
then , where , .*

Theorem 2.6. *For , let and satisfy (2.9). If
**
then , where .*

*Proof. *From (1.4), we obtain

Differentiating again logarithmically and then by simple computation, we get
or, equivalently we can write
Now taking real part on both sides, we obtain
This further implies that

Let
Clearly we have
Then by using (2.28) and Corollary 2.2, we obtain
Therefore with .

Making use of (2.27) and Corollary 2.5, one can prove the following result.

Theorem 2.7. *For , let and satisfy (2.27). If
**
then , where .*

#### Acknowledgment

The author would like to thank Prof. Dr. Ihsan Ali, Vice Chancellor Abdul Wali Khan University Mardan for providing excellent research facilities and financial support.

#### References

- J.-L. Liu and H. M. Srivastava, βA linear operator and associated families of meromorphically multivalent functions,β
*Journal of Mathematical Analysis and Applications*, vol. 259, no. 2, pp. 566β581, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. P. Goyal and J. K. Prajapat, βA new class of meromorphic multivalent functions involving certain linear operator,β
*Tamsui Oxford Journal of Mathematical Sciences*, vol. 25, no. 2, pp. 167β176, 2009. View at Zentralblatt MATH - R. K. Raina and H. M. Srivastava, βA new class of meromorphically multivalent functions with applications to generalized hypergeometric functions,β
*Mathematical and Computer Modelling*, vol. 43, no. 3-4, pp. 350β356, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - N. Xu and D. Yang, βOn starlikeness and close-to-convexity of certain meromorphic functions,β
*Journal of the Korea Society of Mathematical Education B*, vol. 10, no. 1, pp. 566β581, 2003. View at Zentralblatt MATH - L. Spacek, βPrispevek k teorii funkei prostych,β
*Časopis Pro Pestování Matematiky A Fysiky*, vol. 62, pp. 12β19, 1933. - M. S. Robertson, βUnivalent functions $f(z)$ for which $zf\u2019(z)$ is spirallike,β
*The Michigan Mathematical Journal*, vol. 16, pp. 97β101, 1969. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z.-G. Wang, Y. Sun, and Z.-H. Zhang, βCertain classes of meromorphic multivalent functions,β
*Computers & Mathematics with Applications*, vol. 58, no. 7, pp. 1408β1417, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Nehari and E. Netanyahu, βOn the coefficients of meromorphic schlicht functions,β
*Proceedings of the American Mathematical Society*, vol. 8, pp. 15β23, 1957. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z.-G. Wang, Z.-H. Liu, and R.-G. Xiang, βSome criteria for meromorphic multivalent starlike functions,β
*Applied Mathematics and Computation*, vol. 218, no. 3, pp. 1107β1111, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z.-G. Wang, Z.-H. Liu, and A. Catas, βOn neighborhoods and partial sums of certain meromorphic multivalent functions,β
*Applied Mathematics Letters*, vol. 24, no. 6, pp. 864β868, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. A. Frasin, βNew general integral operators of $p$-valent functions,β
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 10, no. 4, article 109, 2009. - A. Mohammed and M. Darus, βA new integral operator for meromorphic functions,β
*Acta Universitatis Apulensis*, no. 24, pp. 231β238, 2010. View at Zentralblatt MATH - A. Mohammed and M. Darus, βStarlikeness properties of a new integral operator for meromorphic functions,β
*Journal of Applied Mathematics*, Article ID 804150, 8 pages, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Arif, W. Haq, and M. Ismail, βMapping properties of generalized Robertson functions under certain integral operators,β
*Applied Mathematics*, vol. 3, no. 1, pp. 52β55, 2012. View at Publisher Β· View at Google Scholar - K. I. Noor, M. Arif, and A. Muhammad, βMapping properties of some classes of analytic functions under an integral operator,β
*Journal of Mathematical Inequalities*, vol. 4, no. 4, pp. 593β600, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - N. Breaz, V. Pescar, and D. Breaz, βUnivalence criteria for a new integral operator,β
*Mathematical and Computer Modelling*, vol. 52, no. 1-2, pp. 241β246, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. A. Frasin, βConvexity of integral operators of $p$-valent functions,β
*Mathematical and Computer Modelling*, vol. 51, no. 5-6, pp. 601β605, 2010. View at Publisher Β· View at Google Scholar - B. A. Frasin, βSome sufficient conditions for certain integral operators,β
*Journal of Mathematical Inequalities*, vol. 2, no. 4, pp. 527β535, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - G. Saltik, E. Deniz, and E. Kadioğlu, βTwo new general $p$-valent integral operators,β
*Mathematical and Computer Modelling*, vol. 52, no. 9-10, pp. 1605β1609, 2010. View at Publisher Β· View at Google Scholar - H. Al-Amiri and P. T. Mocanu, βSome simple criteria of starlikeness and convexity for meromorphic functions,β
*Mathematica*, vol. 37(60), no. 1-2, pp. 11β20, 1995. View at Zentralblatt MATH - S. S. Miller and P. T. Mocanu, βDifferential subordinations and inequalities in the complex plane,β
*Journal of Differential Equations*, vol. 67, no. 2, pp. 199β211, 1987. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH