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 pvalent 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 [79], the estimates for the Hankel determinant of areally mean pvalent functions were investigated. ElHosh obtained bounds for Hankel determinants of univalent functions with a positive Hayman index α[10] and of kfold symmetric and closetoconvex functions [11]. For bounds on the Hankel determinants of closetoconvex functions, see [1214]. Noor studied the Hankel determinant of Bazilevic functions in [15] and of functions with bounded boundary rotation in [1619]. In the recent years, several authors have investigated bounds for the Hankel determinant of functions belonging to various subclasses of univalent and multivalent functions [2027]. The Hankel determinant is the wellknown FeketeSzegö 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 MaMinda 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 wellknown 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 functionis given by the series
then the following sharp estimate holds:
Lemma 2[32]
If the functionis given by the series (2), then
2 Second Hankel determinant of MaMinda starlike/convex functions
Subclasses of starlike functions are characterized by the quantity lying in some domain in the right halfplane. 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 halfplane . The various subclasses of starlike functions were unified by subordination in [30]. The following definition of the class of MaMinda 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 MaMinda starlike functions with respect toφ consists of functions satisfying the subordination
For the function φ given by , , the class is the wellknown 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 orderβ. The class
is the class of lemniscate starlike functions studied in [35].
Theorem 1Let the functionbe given by (1).
1. If, andsatisfy the conditions
then the second Hankel determinant satisfies
2. If, andsatisfy the conditions
or the conditions
then the second Hankel determinant satisfies
3. If, andsatisfy the conditions
then the second Hankel determinant satisfies
Proof Since , there exists an analytic function w with and in such that
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
Definition 2 Let be analytic, and let be given as in (6). The class of MaMinda convex functions with respect toφ consists of functions f satisfying the subordination
Theorem 2Let the functionbe given by (1).
1. If, andsatisfy the conditions
then the second Hankel determinant satisfies
2. If, andsatisfy the conditions
or the conditions
then the second Hankel determinant satisfies
3. If, andsatisfy 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 3Let, , and let the functionfas in (1) be in the class. Also, let
1. If, andsatisfy the conditions
then the second Hankel determinant satisfies
2. If, andsatisfy the conditions
or the conditions
then the second Hankel determinant satisfies
3. If, andsatisfy 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
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:
AlAmiri 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 4Let the functionfgiven by (1) be in the class, . Also, let
1. If, andsatisfy the conditions
then the second Hankel determinant satisfies
2. If, andsatisfy the conditions
or
then the second Hankel determinant satisfies
3. If, andsatisfy 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
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].
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.
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.
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