- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Journal of Complex Analysis
Volume 2013 (2013), Article ID 509717, 7 pages
Class of Multivalent Analytic Functions Defined by a Linear Operator
Department of Mathematics, Faculty of Science, Al al-Bayt University, P.O. Box 130095, Mafraq, Jordan
Received 14 August 2012; Accepted 27 September 2012
Academic Editor: Bao Qin Li
Copyright © 2013 B. A. Frasin. 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.
Making use of the linear operator defined in (Prajapat, 2012), we introduce the class of analytic and -valent functions in the open unit disk . Furthermore, we obtain some sufficient conditions for starlikeness and close-to-convexity and some angular properties for functions belonging to this class. Several corollaries and consequences of the main results are also considered.
1. Introduction and Definitions
Let denote the class of functions of the form which are analytic and -valent in the open unit disk and . In particular, we set . A function is said to be in the class of -valently starlike of order in if and only if it satisfies the inequality
Furthermore, a function is said to be in the class of -valently close-to-convex of order in if and only if it satisfies the inequality
In particular, we write and , where and are the usual subclasses of consisting of functions which are starlike and close-to-convex in , respectively.
In , Prajapat define a generalized multiplier transformation operator as follows:
We see that for , we have where and . It is readily verified from (5) that
We observe that the operator generalize several previously studied familiar operators, and we will show some of the interesting particular cases as follows:(i) (see );(ii) (see [3, 4]);(iii) (see [5–7]);(iv) (see [8, 9]);(v) (see );(vi) (see );(vii) (see );(viii) (see ).(For other generalizations of the operator , see ).
Making use of the above operator , we introduce the class of analytic and -valent functions defined as follows.
Definition 1. A function is said to be a member of the class if and only if for some , , , , , and for all .
Note that condition (7) implies that
We note that , the class which has been introduced and studied by the author in . Also, we have , . The class is the class which has been introduced and studied by Frasin and Darus  (see also [16, 17]).
In this paper, we obtain some sufficient conditions and some angular properties for functions belonging to the class . Several corollaries and consequences of the main results are also considered.
In order to derive our main results, we have to recall the following lemmas.
Lemma 2 (see ). Let be analytic in and such that . Then if attains its maximum value on circle at a point , one has where is a real number.
Lemma 3 (see ). 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 the function is analytic in such that for all , then .
Lemma 4 (see ). Let be analytic in with and for all . If there exist two points , such that for , , and for , then we have where
2. Sufficient Conditions for Starlikeness and Close-to-Convexity
Unless otherwise mentioned, we shall assume in the remainder of this paper that Making use of Lemma 2, we first prove the following.
Theorem 5. If satisfies for some , then .
Proof. Define the function by Then is analytic in and . It follows from (15) and the identities (6) and (1.6) that Suppose that there exists such that Then from Lemma 2, we have (9). Therefore, letting , we obtain that Which contradicts our assumption (14). Therefore we have in . Finally, we have that is, . This completes the proof of the theorem.
Putting and in Theorem 5, we obtain the following.
Corollary 6. If satisfies for some , then .
Putting and in Theorem 5, one obtains the following.
Corollary 7. If satisfies for some , then .
Letting in Corollary 7, one has
Corollary 8. If satisfies for some , then . In particular, if satisfies then is close-to-convex in .
Putting , , and in Theorem 5, one obtains the following.
Corollary 9. If satisfies for some , then .
Corollary 10. If satisfies for some , then . In particular, if satisfies then is starlike in .
Next we prove the following.
Theorem 12. If satisfies then , where .
Proof . Define the function by Then, we see that is analytic in . A computation shows that where For all real satisfying , we have Let . Then and for all real and , . By using Lemma 3, we have , that is, .
Putting and in Theorem 12, we have the following.
Corollary 13 (see ). If satisfies then , where . In particular, if satisfies then is close-to-convex in .
Putting , , and in Theorem 12, one has the following.
Corollary 14 (see ). If satisfies then , where . In particular, if satisfies then is starlike in .
3. Argument Properties
Finally, we prove the following.
Theorem 15. Suppose that for and . If satisfies for , then
Proof. Define the function by
Then we see that analytic in , , and for all . It follows from (39) that
Suppose that there exists two points such that the condition (10) is satisfied, then by Lemma 4, we obtain (11) under the constraint (12). Therefore, we have which contradict the assumption of the theorem. This completes the proof.
Putting and in Theorem 15, one has the following.
Corollary 17 (see ). Suppose that for and . If satisfies for , then In particular, if satisfies for , then
Remark 18. Taking different choices of , and in the above theorems, we obtain some sufficient conditions for starlikeness and close-to-convexity and some angular properties for functions belonging to new classes defined by the previously operators mentioned in Section 1.
- J. K. Prajapat, “Subordination and superordination preserving properties for generalized multiplier transformation operator,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1456–1465, 2012.
- A. Catas, “On certain classes of p-valent functions defined by multiplier transformations,” in Proceedings of the International Symposium on Geometric Function Theory and Applications (GFTA '07), S. Owa and Y. Polatoglu, Eds., vol. 91, pp. 241–250, TC Istanbul University Publications, TC Istanbul Kultur University, Istanbul, Turkey, August 2007.
- S. S. Kumar, H. C. Taneja, and V. Ravichandran, “Classes multivalent functions defined by dziok– srivastava linear operator and multiplier transformations,” Kyungpook Mathematical Journal, vol. 46, pp. 97–109, 2006.
- H. M. Srivastava, K. Suchithra, B. A. Stephen, and S. Sivasubramanian, “Inclusion and neighborhood properties of certain subclasses of analytic and multivalent functions of complex order,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 5, article 191, 2006.
- M. K. Aouf and A. O. Mostafa, “On a subclass of n-p-valent prestarlike functions,” Computers and Mathematics with Applications, vol. 55, no. 4, pp. 851–861, 2008.
- M. Kamali and H. Orhan, “On a subclass of certain starlike functions with negative coefficients,” Bulletin of the Korean Mathematical Society, vol. 41, pp. 53–71, 2004.
- H. Orhan and H. Kiziltunç, “A generalization on subfamily of p-valent functions with negative coefficients,” Applied Mathematics and Computation, vol. 155, no. 2, pp. 521–530, 2004.
- N. E. Cho and T. H. Kim, “Multiplier transformations and strongly close-to-convex functions,” Bulletin of the Korean Mathematical Society, vol. 40, pp. 399–410, 2003.
- N. E. Cho and H. M. Srivastava, “Argument estimates of certain analytic functions defined by a class of multiplier transformations,” Mathematical and Computer Modelling, vol. 37, no. 1-2, pp. 39–49, 2003.
- G. S. Sălăgean, “Subclasses of univalent functions,” in Complex Analysis—Fifth Romanian-Finnish Seminar, vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Bucharest, Romania, 1981.
- F. M. Al-Oboudi, “On univalent functions defined by a generalized Sǎlǎgean operator,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 27, pp. 1429–1436, 2004.
- B. A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions,” in Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa, Eds., pp. 371–374, World Scientific Publishing Company, Singapore, 1992.
- M. K. Aouf, A. O. Mostafa, and R. El-Ashwah, “Sandwich theorems for p-valent functions defined by a certain integral operator,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1647–1653, 2011.
- B. A. Frasin, “On certain classes of multivalent analytic functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 4, article 104, 2005.
- B. A. Frasin and M. Darus, “On Certain analytic univalent function,” International Journal of Mathematics and Mathematical Sciences, vol. 25, no. 5, pp. 305–310, 2001.
- B. A. Frasin, “A note on certain analytic and univalent functions,” Southeast Asian Bulletin of Mathematics, vol. 28, no. 5, pp. 829–836, 2004.
- B. A. Frasin and J. M. Jahangiri, “A new and comprehensive class of analytic functions,” Analele Universităţii din Oradea—Fascicola Matematică, vol. 15, pp. 59–62, 2008.
- I. S. Jack, “Functions starlike and convex of order ,” Journal of the London Mathematical Society, vol. 3, no. 2, pp. 469–474, 1971.
- S. S. Miller and P. T. Mocanu, “Differential subordinations and inequalities in the complex plane,” Journal of Differential Equations, vol. 67, no. 2, pp. 199–211, 1987.
- N. Takahashi and M. Nunokawa, “A certain connection between starlike and convex functions,” Applied Mathematics Letters, vol. 16, no. 5, pp. 563–655, 2003.
- S. Owa, “Certain sufficient conditions for starlikeness and convexity of order ,” Chinese Journal of Mathematics, vol. 19, pp. 55–60, 1991.