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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 396484, 6 pages
New Criteria for Meromorphic Multivalent Alpha-Convex Functions
1Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
Received 9 March 2013; Revised 14 July 2013; Accepted 15 July 2013
Academic Editor: Zhihua Zhang
Copyright © 2013 Muhammad Arif et al. 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.
The aim of the present paper is to obtain sufficient condition for the class of meromorphic -valent alpha convex functions of order and then to study mapping properties of the newly defined integral operators. Many known results appeared as special consequences of our work.
Let denote the class of meromorphic functions normalized by which are analytic and -valent in the punctured unit disk . In particular, , , and . For which is real with , , , and , , we denote by , , and , the subclasses of consisting of all meromorphic -valent functions of the form (1) which are defined, respectively, by Making , in (2), we get the well-known subclasses of consisting of meromorphic-valent functions which are starlike, convex, and alpha convex of order , respectively. For details of the classes defined by (2) and related topics, we refer the raeder to the work of Aouf and Hossen , Aouf and Srivastava , Ali and Ravichandran , Goyal and Prajapat , Joshi and Srivastava , Liu and Srivastava , Raina and Srivastava , Xu and Yang , and Owa et al. .
For , Wang et al.  and Nehari and Netanyahu  introduced and studied the subclass of consisting of functions satisfying We now extended this concept to define a subclass of consisting of functions of the form (1) satisfying For and in (4), we obtain the classes and of , respectively, studied by Arif ; also see [13, 14].
Integral operators for different classes of analytic, univalent, and multivalent functions in the open unit disk are studied by various authors; see [15–21]. We now define the following general integral operator of meromorphic-valent functions: For , we obtain the integral operator studied recently in [22, 23], and, further for , we obtain the integral operator introduced and studied by Mohammed and Darus .
Sufficient conditions were studied by various authors for different subclasses of analytic and multivalent functions; for some of the related work see [25–27]. The object of the present paper is to obtain sufficient conditions for the class and then study mapping properties of the integral operator given by (5). We also consider some special cases of our results which lead to various interesting corollaries and relevance of some of these results with other known results which are also mentioned.
We will assume throughout our discussion, unless otherwise stated, that is real with , , ,, , for , , , and
To obtain our main results, we need the following lemmas.
Lemma 1 (see ). If with and satisfies the condition then
Lemma 2 (see ). Let satisfy the following condition:
If the function is analytic in and then
2. Sufficiency Criteria for the Class
In this section we establish a new sufficiency criteria for the subclass of .
Theorem 3. If satisfies then , where is given by (6).
Proof. Let us set a function by
for . Then clearly (13) shows that .
Logarithmic differentiating of (13) gives which further implies Thus using (12), we get Therefore by Lemma 1, we have .
From (14), we can write Since , it implies that . Therefore, we get or And therefore .
By taking and in Theorem 3, we obtain Corollaries 4 and 5, respectively, proved by Arif .
Corollary 4. If satisfies then .
Corollary 5. If satisfies then .
Remarks. We note that by simple computation (13) gives By taking suitable meromorphic starlike function for in (22) such as , which satisfies the inequality of Lemma 1, we can conclude that the functionof (22) is the subclass of a meromorphic function.
3. Some Properties of the Integral Operator
In this section, we discuss some mapping properties of the integral operator .
Proof. From (5), we obtain
Dividing both sides by , we have
Differentiating again logarithmically, we have
Now by simple computation, we get
or equivalently we have
By taking real part on both sides, we obtain
which further implies that
Clearly we have
Then by using (23) and Theorem 3 with , we obtain
Therefore with .
Making in Theorem 6, we have the following.
Corollary 7. For , let and satisfy (20). If then with .
Theorem 8. For , let . If then , where with .
Proof. From (26), we obtain Let such that is analytic in with . Then (36) can be written as Taking real part on both sides, we have where we have used (35) and the assumption that . Let us put Then, for such that , we have Thus using Lemma 2, we conclude that , which is equivalent to That is, .
The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02.
- M. K. Aouf and H. M. Hossen, “New criteria for meromorphic -valent starlike functions,” Tsukuba Journal of Mathematics, vol. 17, no. 2, pp. 481–486, 1993.
- M. K. Aouf and M. H. Srivastava, “A new criteria for meromorphically p-valent convex functions of order alpha,” Mathematical Sciences Research Hot-Line, vol. 1, no. 8, pp. 7–12, 1997.
- R. M. Ali and V. Ravichandran, “Classes of meromorphic -convex functions,” Taiwanese Journal of Mathematics, vol. 14, no. 4, pp. 1479–1490, 2010.
- S. P. Goyal and J. K. Prajapat, “A new class of meromorphic multivalent functions involving certain linear operator,” Tamsui Oxford Journal of Mathematical Sciences, vol. 25, no. 2, pp. 167–176, 2009.
- S. B. Joshi and H. M. Srivastava, “A certain family of meromorphically multivalent functions,” Computers & Mathematics with Applications, vol. 38, no. 3-4, pp. 201–211, 1999.
- J. L. Liu and H. M. Srivastava, “A linear operator and associated families of meromorphically multivalent functions,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 566–581, 2001.
- R. K. Raina and H. M. Srivastava, “A new class of meromorphically multivalent functions with applications to generalized hypergeometric functions,” Mathematical and Computer Modelling, vol. 43, no. 3-4, pp. 350–356, 2006.
- N. Xu and D. Yang, “On starlikeness and close-to-convexity of certain meromorphic functions,” Journal of the Korea Society of Mathematical Education B, vol. 10, no. 1, pp. 566–581, 2003.
- S. Owa, H. E. Darwish, and M. K. Aouf, “Meromorphic multivalent functions with positive and fixed second coefficients,” Mathematica Japonica, vol. 46, no. 2, pp. 231–236, 1997.
- Z. G. Wang, Y. Sun, and Z. H. Zhang, “Certain classes of meromorphic multivalent functions,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1408–1417, 2009.
- Z. Nehari and E. Netanyahu, “On the coefficients of meromorphic schlicht functions,” Proceedings of the American Mathematical Society, vol. 8, pp. 15–23, 1957.
- M. Arif, “On certain sufficiency criteria for -valent meromorphic spiralike functions,” Abstract and Applied Analysis, vol. 2012, Article ID 837913, 10 pages, 2012.
- Z. G. Wang, Z. H. Liu, and R. G. Xiang, “Some criteria for meromorphic multivalent starlike functions,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1107–1111, 2011.
- Z. G. Wang, Z. H. Liu, and A. Cǎtaş, “On neighborhoods and partial sums of certain meromorphic multivalent functions,” Applied Mathematics Letters, vol. 24, no. 6, pp. 864–868, 2011.
- B. A. Frasin, “New general integral operators of -valent functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 4, article 109, 2009.
- M. Arif, K. I. Noor, and F. Ghani, “Some properties of an integral operator defined by convolution,” Journal of Inequalities and Applications, vol. 2012, article 13, 6 pages, 2012.
- K. I. Noor and M. Arif, “Mapping properties of an integral operator,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1826–1829, 2012.
- K. I. Noor, M. Arif, and A. Muhammad, “Mapping properties of some classes of analytic functions under an integral operator,” Journal of Mathematical Inequalities, vol. 4, no. 4, pp. 593–600, 2010.
- N. Breaz, V. Pescar, and D. Breaz, “Univalence criteria for a new integral operator,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 241–246, 2010.
- B. A. Frasin, “Convexity of integral operators of -valent functions,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 601–605, 2010.
- B. A. Frasin, “Some sufficient conditions for certain integral operators,” Journal of Mathematical Inequalities, vol. 2, no. 4, pp. 527–535, 2008.
- A. Mohammed and M. Darus, “Some properties of certain integral operators on new subclasses of analytic functions with complex order,” Journal of Applied Mathematics, vol. 2012, Article ID 161436, 9 pages, 2012.
- A. Mohammed and M. Darus, “The order of starlikeness of new -valent meromorphic functions,” International Journal of Mathematical Analysis, vol. 6, no. 27, pp. 1329–1340, 2012.
- A. Mohammed and M. Darus, “Starlikeness properties of a new integral operator for meromorphic functions,” Journal of Applied Mathematics, vol. 2011, Article ID 804150, 8 pages, 2011.
- M. Arif, I. Ahmad, M. Raza, and K. Khan, “Sufficient condition of a subclass of analytic functions defined by Hadamard product,” Life Science Journal, vol. 9, no. 4, pp. 2487–2489, 2012.
- M. Arif, M. Raza, S. Islam, J. Iqbal, and F. Faizullah, “Some sufficient conditions for spirallike functions with argument properties,” Life Science Journal, vol. 9, no. 4, pp. 3770–3773, 2012.
- H. Al-Amiri and P. T. Mocanu, “Some simple criteria of starlikeness and convexity for meromorphic functions,” Mathematica, vol. 37, no. 60, pp. 11–21, 1995.
- S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.