- Research
- Open access
- Published:
Bounds for the second Hankel determinant of certain univalent functions
Journal of Inequalities and Applications volume 2013, Article number: 281 (2013)
Abstract
The estimates for the second Hankel determinant of the analytic function , for which either or is subordinate to a certain analytic function, are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained.
MSC:30C45, 30C80.
Dedication
Dedicated to Professor Hari M Srivastava
1 Introduction
Let denote the class of all analytic functions
defined on the open unit disk . The Hankel determinants ( , ) of the function f are defined by
Hankel determinants are useful, for example, in showing that a function of bounded characteristic in , i.e., a function which is a ratio of two bounded analytic functions with its Laurent series around the origin having integral coefficients, is rational [1]. For the use of Hankel determinants in the study of meromorphic functions, see [2], and various properties of these determinants can be found in [[3], Chapter 4]. In 1966, Pommerenke [4] investigated the Hankel determinant of areally mean p-valent functions, univalent functions as well as of starlike functions. In [5], he proved that the Hankel determinants of univalent functions satisfy
where and K depends only on q. Later, Hayman [6] proved that ( ; A an absolute constant) for areally mean univalent functions. In [7–9], the estimates for the Hankel determinant of areally mean p-valent functions were investigated. ElHosh obtained bounds for Hankel determinants of univalent functions with a positive Hayman index α [10] and of k-fold symmetric and close-to-convex functions [11]. For bounds on the Hankel determinants of close-to-convex functions, see [12–14]. Noor studied the Hankel determinant of Bazilevic functions in [15] and of functions with bounded boundary rotation in [16–19]. In the recent years, several authors have investigated bounds for the Hankel determinant of functions belonging to various subclasses of univalent and multivalent functions [20–27]. The Hankel determinant is the well-known Fekete-Szegö functional. For results related to this functional, see [28, 29]. The second Hankel determinant is given by .
An analytic function f is subordinate to an analytic function g, written , if there is an analytic function with satisfying . Ma and Minda [30] unified various subclasses of starlike () and convex functions () by requiring that either of the quantity or is subordinate to a function φ with a positive real part in the unit disk , , , φ maps onto a region starlike with respect to 1 and symmetric with respect to the real axis. He obtained distortion, growth and covering estimates as well as bounds for the initial coefficients of the unified classes.
The bounds for the second Hankel determinant are obtained for functions belonging to these subclasses of Ma-Minda starlike and convex functions in Section 2. In Section 3, the problem is investigated for two other related classes defined by subordination. In proving our results, we do not assume the univalence or starlikeness of φ as they were required only in obtaining the distortion, growth estimates and the convolution theorems. The classes introduced by subordination naturally include several well-known classes of univalent functions and the results for some of these special classes are indicated as corollaries.
Let be the class of functions with positive real part consisting of all analytic functions satisfying and . We need the following results about the functions belonging to the class .
Lemma 1 [31]
If the function is given by the series
then the following sharp estimate holds:
Lemma 2 [32]
If the function is given by the series (2), then
for some x, z with and .
2 Second Hankel determinant of Ma-Minda starlike/convex functions
Subclasses of starlike functions are characterized by the quantity lying in some domain in the right half-plane. For example, f is strongly starlike of order β if lies in a sector , while it is starlike of order α if lies in the half-plane . The various subclasses of starlike functions were unified by subordination in [30]. The following definition of the class of Ma-Minda starlike functions is the same as the one in [30] except for the omission of starlikeness assumption of φ.
Definition 1 Let be analytic, and let the Maclaurin series of φ be given by
The class of Ma-Minda starlike functions with respect to φ consists of functions satisfying the subordination
For the function φ given by , , the class is the well-known class of starlike functions of order α. Let
Then the class
is the parabolic starlike functions introduced by Rønning [33]. For a survey of parabolic starlike functions and the related class of uniformly convex functions, see [34]. For , the class
is the familiar class of strongly starlike functions of order β. The class
is the class of lemniscate starlike functions studied in [35].
Theorem 1 Let the function be given by (1).
-
1.
If , and satisfy the conditions
then the second Hankel determinant satisfies
-
2.
If , and satisfy the conditions
or the conditions
then the second Hankel determinant satisfies
-
3.
If , and satisfy the conditions
then the second Hankel determinant satisfies
Proof Since , there exists an analytic function w with and in such that
Define the functions by
or, equivalently,
Then is analytic in with and has a positive real part in . By using (8) together with (6), it is evident that
Since
it follows by (7), (9) and (10) that
Therefore
Let
Then
Since the function () is in the class for any , there is no loss of generality in assuming . Write , . Substituting the values of and respectively from (4) and (5) in (12), we obtain
Replacing by μ and substituting the values of , , and from (11) yield
Note that for , differentiating in (13) partially with respect to μ yields
Then, for and for any fixed c with , it is clear from (14) that , that is, is an increasing function of μ. Hence, for fixed , the maximum of occurs at , and
Also note that
Let
Since
we have
where P, Q, R are given by (15). □
Remark 1 When , Theorem 1 reduces to [[24], Theorem 3.1].
Corollary 1
-
1.
If , then .
-
2.
If , then .
-
3.
If , then .
-
4.
If , then .
Definition 2 Let be analytic, and let be given as in (6). The class of Ma-Minda convex functions with respect to φ consists of functions f satisfying the subordination
Theorem 2 Let the function be given by (1).
-
1.
If , and satisfy the conditions
then the second Hankel determinant satisfies
-
2.
If , and satisfy the conditions
or the conditions
then the second Hankel determinant satisfies
-
3.
If , and satisfy the conditions
then the second Hankel determinant satisfies
Proof Since , there exists an analytic function w with and in such that
Since
equations (9), (17) and (18) yield
Therefore
By writing
we have
Similar as in Theorems 1, it follows from (4) and (5) that
Replacing by μ and then substituting the values of , , and from (19) yield
Again, differentiating in (21) partially with respect to μ yields
It is clear from (22) that . Thus is an increasing function of μ for and for any fixed c with . So, the maximum of occurs at and
Note that
Let
By using (16), we have
where P, Q, R are given in (23). □
Remark 2 For the choice of , Theorem 2 reduces to [[24], Theorem 3.2].
3 Further results on the second Hankel determinant
Definition 3 Let be analytic, and let be as given in (6). Let and . A function is in the class if it satisfies the following subordination:
Theorem 3 Let , , and let the function f as in (1) be in the class . Also, let
-
1.
If , and satisfy the conditions
then the second Hankel determinant satisfies
-
2.
If , and satisfy the conditions
or the conditions
then the second Hankel determinant satisfies
-
3.
If , and satisfy the conditions
then the second Hankel determinant satisfies
Proof For , there exists an analytic function w with and in such that
Since f has the Maclaurin series given by (1), a computation shows that
It follows from (24), (9) and (25) that
Therefore
which yields
where
It can be easily verified that for .
Let
Then (26) becomes
It follows that
Application of the triangle inequality, replacement of by μ and substituting the values of , , and from (27) yield
where , .
Similarly as in the previous proofs, it can be shown that is an increasing function of μ for . So, for fixed , let
which is
Let
Using (16), we have
where P, Q, R are given in (30). □
Remark 3 For the choice with , Theorem 3 reduces to [[36], Theorem 2.1].
Definition 4 Let be analytic, and let be as given in (6). For a fixed real number α, the function is in the class if it satisfies the following subordination:
Al-Amiri and Reade [37] introduced the class and they showed that for . Univalence of the functions in the class was also investigated in [38, 39]. Singh et al. also obtained the bound for the second Hankel determinant of functions in . The following theorem provides a bound for the second Hankel determinant of the functions in the class .
Theorem 4 Let the function f given by (1) be in the class , . Also, let
-
1.
If , and satisfy the conditions
then the second Hankel determinant satisfies
-
2.
If , and satisfy the conditions
or
then the second Hankel determinant satisfies
-
3.
If , and satisfy the conditions
then the second Hankel determinant satisfies
Proof For , a calculation shows that
where
It can be easily verified that for , . Let
Then
Similarly as in earlier theorems, it follows that
and for fixed , with
Let
By using (16), we have
where P, Q, R are given in (35). □
Remark 4 For , Theorem 4 reduces to Theorem 2. For , let . For this function φ, . In this case, Theorem 4 reduces to [[40], Theorem 3.1].
References
Cantor DG: Power series with integral coefficients. Bull. Am. Math. Soc. 1963, 69: 362–366. 10.1090/S0002-9904-1963-10923-4
Wilson R: Determinantal criteria for meromorphic functions. Proc. Lond. Math. Soc. 1954, 4: 357–374.
Vein R, Dale P Applied Mathematical Sciences 134. In Determinants and Their Applications in Mathematical Physics. Springer, New York; 1999.
Pommerenke C: On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. 1966, 41: 111–122.
Pommerenke C: On the Hankel determinants of univalent functions. Mathematika 1967, 14: 108–112. 10.1112/S002557930000807X
Hayman WK: On the second Hankel determinant of mean univalent functions. Proc. Lond. Math. Soc. 1968, 18: 77–94.
Noonan JW, Thomas DK: On the Hankel determinants of areally mean p -valent functions. Proc. Lond. Math. Soc. 1972, 25: 503–524.
Noonan JW: Coefficient differences and Hankel determinants of areally mean p -valent functions. Proc. Am. Math. Soc. 1974, 46: 29–37.
Noonan JW, Thomas DK: On the second Hankel determinant of areally mean p -valent functions. Trans. Am. Math. Soc. 1976, 223: 337–346.
Elhosh MM: On the second Hankel determinant of univalent functions. Bull. Malays. Math. Soc. 1986, 9(1):23–25.
Elhosh MM: On the second Hankel determinant of close-to-convex functions. Bull. Malays. Math. Soc. 1986, 9(2):67–68.
Noor KI: Higher order close-to-convex functions. Math. Jpn. 1992, 37(1):1–8.
Noor KI: On the Hankel determinant problem for strongly close-to-convex functions. J. Nat. Geom. 1997, 11(1):29–34.
Noor KI: On certain analytic functions related with strongly close-to-convex functions. Appl. Math. Comput. 2008, 197(1):149–157. 10.1016/j.amc.2007.07.039
Noor KI, Al-Bany SA: On Bazilevic functions. Int. J. Math. Math. Sci. 1987, 10(1):79–88. 10.1155/S0161171287000103
Noor KI: On analytic functions related with functions of bounded boundary rotation. Comment. Math. Univ. St. Pauli 1981, 30(2):113–118.
Noor KI: On meromorphic functions of bounded boundary rotation. Caribb. J. Math. 1982, 1(3):95–103.
Noor KI: Hankel determinant problem for the class of functions with bounded boundary rotation. Rev. Roum. Math. Pures Appl. 1983, 28(8):731–739.
Noor KI, Al-Naggar IMA: On the Hankel determinant problem. J. Nat. Geom. 1998, 14(2):133–140.
Arif M, Noor KI, Raza M: Hankel determinant problem of a subclass of analytic functions. J. Inequal. Appl. 2012., 2012: Article ID 22
Hayami T, Owa S: Generalized Hankel determinant for certain classes. Int. J. Math. Anal. 2010, 4(49–52):2573–2585.
Hayami T, Owa S: Applications of Hankel determinant for p -valently starlike and convex functions of order α . Far East J. Appl. Math. 2010, 46(1):1–23.
Hayami T, Owa S: Hankel determinant for p -valently starlike and convex functions of order α . Gen. Math. 2009, 17(4):29–44.
Janteng A, Halim SA, Darus M: Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 2007, 1(13–16):619–625.
Mishra AK, Gochhayat P: Second Hankel determinant for a class of analytic functions defined by fractional derivative. Int. J. Math. Math. Sci. 2008., 2008: Article ID 153280
Mohamed N, Mohamad D, Cik Soh S: Second Hankel determinant for certain generalized classes of analytic functions. Int. J. Math. Anal. 2012, 6(17–20):807–812.
Murugusundaramoorthy G, Magesh N: Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant. Bull. Math. Anal. Appl. 2009, 1(3):85–89.
Ali RM, Lee SK, Ravichandran V, Supramaniam S: The Fekete-Szegö coefficient functional for transforms of analytic functions. Bull. Iran. Math. Soc. 2009, 35(2):119–142. 276
Ali RM, Ravichandran V, Seenivasagan N: Coefficient bounds for p -valent functions. Appl. Math. Comput. 2007, 187(1):35–46. 10.1016/j.amc.2006.08.100
Ma WC, Minda D: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis, Tianjin, 1992. Conf. Proc. Lecture Notes Anal., vol. I, pp. 157-169. International Press, Cambridge (1922)
Duren PL Grundlehren der Mathematischen Wissenschaften 259. In Univalent Functions. Springer, New York; 1983.
Grenander U, Szegö G California Monographs in Mathematical Sciences. In Toeplitz Forms and Their Applications. University of California Press, Berkeley; 1958.
Rønning F: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 1993, 118(1):189–196.
Ali RM, Ravichandran V: Uniformly convex and uniformly starlike functions. Math. News Lett. 2011, 21(1):16–30.
Sokół J, Stankiewicz J: Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 1996, 19: 101–105.
Bansal D: Upper bound of second Hankel determinant for a new class of analytic functions. Appl. Math. Lett. 2013, 26(1):103–107. 10.1016/j.aml.2012.04.002
Al-Amiri HS, Reade MO: On a linear combination of some expressions in the theory of the univalent functions. Monatshefte Math. 1975, 80(4):257–264. 10.1007/BF01472573
Singh S, Gupta S, Singh S: On a problem of univalence of functions satisfying a differential inequality. Math. Inequal. Appl. 2007, 10(1):95–98.
Singh V, Singh S, Gupta S: A problem in the theory of univalent functions. Integral Transforms Spec. Funct. 2005, 16(2):179–186. 10.1080/10652460412331270571
Verma S, Gupta S, Singh S: Bounds of Hankel determinant for a class of univalent functions. Int. J. Math. Math. Sci. 2012., 2012: Article ID 147842
Acknowledgements
The work presented here was supported in part by research grants from Universiti Sains Malaysia (FRGS grants) and University of Delhi as well as MyBrain MyPhD programme of the Ministry of Higher Education, Malaysia.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on the results and they read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Lee, S.K., Ravichandran, V. & Supramaniam, S. Bounds for the second Hankel determinant of certain univalent functions. J Inequal Appl 2013, 281 (2013). https://doi.org/10.1186/1029-242X-2013-281
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-281