On certain subclasses of multivalent functions defined by multiplier transformations
© Patel et al.; licensee Springer. 2013
Received: 3 January 2013
Accepted: 1 August 2013
Published: 21 September 2013
The purpose of the present paper is to introduce and investigate various properties of a certain class of multivalent functions in the open unit disk defined by a multiplier transformation. In particular, we obtain some inclusion relationships and integral preserving properties of this class of functions. Relevant connections of the results presented in this paper with various known results are also pointed out.
Dedicated to Professor Hari M Srivastava
which are analytic in the open unit disk . We write .
We note that the series defined by (1.3) converges absolutely for all and hence represents an analytic function in [, Chapter 14].
The operator is related to the multiplier transformation studied in .
By using the operator , we introduce the subclass of as follows.
The object of the present paper is to investigate some inclusion properties of the class . Integral-preserving and convolution properties in connection with the operator are also considered. Relevant connections of the results presented here with those obtained in the earlier works are pointed out.
We shall need the following lemmas to prove our results.
and ψ is the best dominant of (2.1).
Lemma 2 [, p.35]
Lemma 3 
- (i)If , and satisfies
The value of β in (2.2) and the bound in (2.3) are best possible.
Lemma 4 
The value of β in (2.4) and the bound in (2.5) are best possible.
3 Inclusion relationships for the class
where . The bound ρ is best possible.
where q is the best dominant of (3.1) and Q is given by (3.2).
The proof of the remaining part can now be deduced along the same lines as in [, Theorem 1]. The bound ρ in (3.3) is best possible as q is the best dominant of (3.1). This evidently completes the proof of the theorem. □
Setting , and in Theorem 1, we get the following corollary.
The result is best possible.
In the special case when , , () and , Theorem 1 gives the following.
The result is best possible.
Setting , () and in Corollary 1 and in Corollary 2, we get the corresponding result obtained in .
The bound R is best possible.
which is certainly positive if , where R is given by (3.7).
for , we complete the proof of Theorem 2. □
Remark For , , and , Theorem 2 yields the corresponding result contained in .
For convenience, we write , . It readily follows from (3.10) that .
- (i)If , then(3.12)
- (ii)If and , then(3.13)
The bound τ is best possible.
The proof of the remaining part is the same as that of [, Theorem 1], and we choose to omit the details. The result is best possible as q is the best dominant of (3.12). □
where and . The containment relations are best possible, and they improve the corresponding work in  for suitable values of the parameters p, η and δ.
4 Properties involving the operator
The result is best possible.
The proof of the remaining part of the theorem follows by using Lemma 1 and the techniques that proved Theorem 4 in .
where . The bound is best possible. □
and the proof of the theorem is completed similarly to Theorem 5. □
and by following the same lines of proof as in Theorem 5, we get the required result. □
Remark Putting , and in Theorems 5, 6 and 7, respectively, we obtain the corresponding results contained in .
The bound given by (4.13) and the estimate in (4.14) are best possible.
then we obtain (4.14) from part (ii) of Lemma 3.
and the sharpness follows from Lemma 4 (for ). □
Putting , and in Theorem 8, we get the following.
The estimates are best possible.
The estimates are best possible.
we obtain (4.18). That and (4.19) now follow from Theorem 8 and (4.20).
and the sharpness follows from Lemma 4. The fact that is sharp follows from (4.20) and the sharpness of Theorem 8. □
Putting , and in Theorem 9, we have the following.
The result is sharp.
Remark For in Corollary 3 and Corollary 4, we get the corresponding results obtained in .
for some real numbers β and γ such that , , , then .
This contradicts the hypothesis (4.21) and hence for . This proves the theorem. □
Taking , , , , and in Theorem 10, we get the following interesting criterion for starlikeness for multivalent functions, which improves the corresponding work in  for .
for all , then .
Similarly, by setting , , , , and in Theorem 10, we obtain the following sufficient condition for convexity of multivalent functions.
where is defined as in Corollary 5, then .
This work was supported by a Research Grant of Pukyong National University (2013 year) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007037).
- Janowski W: Some extremal problems for certain families of analytic functions I. Ann. Pol. Math. 1973, 28: 297–326.MathSciNetGoogle Scholar
- Whittaker ET, Watson GN: A Course on Modern Analysis. 4th edition. Cambridge University Press, Cambridge; 1927. (reprinted)Google Scholar
- Cho NE, Srivastava HM: Argument estimates of certain analytic functions defined by a class of multiplier transformations. Math. Comput. Model. 2003, 37: 39–49. 10.1016/S0895-7177(03)80004-3MathSciNetView ArticleGoogle Scholar
- Flett TM: The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl. 1972, 38: 746–765. 10.1016/0022-247X(72)90081-9MathSciNetView ArticleGoogle Scholar
- Sǎlǎgean GS: Subclasses of univalent functions. Lecture Notes in Math. 1013. In Complex Analysis - Fifth Romanian-Finnish Seminar, Part I Bucharest, 1981. Springer, Berlin; 1983:362–372.Google Scholar
- Cho NE, Kim TH: Multiplier transformations and strongly close-to-convex functions. Bull. Korean Math. Soc. 2003, 40: 399–410.MathSciNetView ArticleGoogle Scholar
- Cho NE, Kim JA: Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation. Comput. Math. Appl. 2006, 52: 323–330. 10.1016/j.camwa.2006.08.022MathSciNetView ArticleGoogle Scholar
- Uralegaddi BA, Somanatha C: Certain classes of univalent functions. In Current Topics in Analytic Function Theory. World Scientific, River Edge; 1992:371–374.View ArticleGoogle Scholar
- Hallenbeck DJ, Ruscheweyh S: Subordination by convex functions. Proc. Am. Math. Soc. 1975, 52: 191–195. 10.1090/S0002-9939-1975-0374403-3MathSciNetView ArticleGoogle Scholar
- Miller SS, Mocanu PT: Differential Subordinations, Theory and Applications. Dekker, New York; 2000.Google Scholar
- Ponnusamy, S, Singh, V: Convolution properties of some classes of analytic functions. Internal Report, SPIC Science Foundation (1990)Google Scholar
- Miller SS, Mocanu PT: Univalent solutions of Briot-Bouquet differential subordinations. J. Differ. Equ. 1985, 56: 297–309. 10.1016/0022-0396(85)90082-8MathSciNetView ArticleGoogle Scholar
- Patel J, Cho NE, Srivastava HM: Certain classes of multivalent functions associated with a family of linear operators. Math. Comput. Model. 2006, 43: 320–338. 10.1016/j.mcm.2005.06.014MathSciNetView ArticleGoogle Scholar
- Miller SS, Mocanu PT, Reade MO: The order of starlikeness of α -convex functions. Mathematica 1978, 20(43):25–30.MathSciNetGoogle Scholar
- Patel J, Cho NE: Some classes of analytic functions involving Noor integral operator. J. Math. Anal. Appl. 2005, 312: 564–575. 10.1016/j.jmaa.2005.03.047MathSciNetView ArticleGoogle Scholar
- MacGregor TH: Subordination of convex functions of order α . J. Lond. Math. Soc. 1975, 9(2):530–536.MathSciNetView ArticleGoogle Scholar
- Choi JH, Saigo M, Srivastava HM: Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 2002, 276: 432–445. 10.1016/S0022-247X(02)00500-0MathSciNetView ArticleGoogle Scholar
- Fukui S, Kim JA, Srivastava HM: On certain subclasses of univalent functions by some integral operators. Math. Jpn. 1999, 50: 359–370.MathSciNetGoogle Scholar
- Stankiewicz J, Stankiewicz Z: Some applications of the Hadamard convolution in the theory of functions. Ann. Univ. Mariae Curie-Skl̄odowska, Sect. A 1986, 40: 251–256.MathSciNetGoogle Scholar
- Lashin AY: Some convolution properties of analytic functions. Appl. Math. Lett. 2005, 18: 135–138. 10.1016/j.aml.2004.09.003MathSciNetView ArticleGoogle Scholar
- Jack IS: Functions starlike and convex of order α . J. Lond. Math. Soc. 1971, 3(2):469–474.MathSciNetView ArticleGoogle Scholar
- Owa S, Srivastava HM: Univalent and starlike generalized hypergeometric functions. Can. J. Math. 1987, 39: 1057–1077. 10.4153/CJM-1987-054-3MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.