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`Abstract and Applied AnalysisVolume 2014, Article ID 603180, 10 pageshttp://dx.doi.org/10.1155/2014/603180`
Research Article

## Upper Bound of Second Hankel Determinant for Certain Subclasses of Analytic Functions

1School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China

2Department of Mathematics, Wuyi University, Jiangmen, Guangdong 529020, China

Received 19 March 2014; Accepted 25 May 2014; Published 5 June 2014

Copyright © 2014 Ming-Sheng Liu 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.

#### Abstract

In this present investigation, we first give a survey of the work done so far in this area of Hankel determinant for univalent functions. Then the upper bounds of the second Hankel determinant for functions belonging to the subclasses , , , and of analytic functions are studied. Some of the results, presented in this paper, would extend the corresponding results of earlier authors.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the unit disc , and let denote the subclass of that is univalent in . Suppose that and are analytic functions in ; we say that is subordinate to , written , if there exists a Schwarz function , which is analytic in with and for all , such that , . In particular, if is univalent in , then the subordination is equivalent to and .

Let be the family of all functions analytic in for which and for .

It is well known that the following correspondence between the class and the class of Schwarz functions exists [1]:

Let denote the starlike subclass of . It is well known that if and only if Let denote the class of all functions that are convex. Further, is convex if and only if is starlike. Also we know that .

In 1959, Sakaguchi [2] introduced the class of functions starlike with respect to symmetric points, consisting of functions satisfying

In 1977, Das and Singh [3] introduced the class of functions convex with respect to symmetric points, which consists of functions satisfying

It is evident that if and only if .

In 2007, Wang and Jiang [4] introduced the following subclass.

Definition 1 (see [4]). Suppose that and . Let denote the class of functions in satisfying the following inequality:

From [4], one knows that the above condition is equivalent to which implies that

If , then the class reduces to the class . In the similar way, one can easily get the following definitions.

Definition 2. Suppose that and . Let denote the class of functions in satisfying the following inequality:

It is evident that the above condition is equivalent to which implies that

If and , then the class reduces to the class .

Definition 3. Suppose that and . Let denote the class of functions in satisfying the following inequality:

From [5], one knows that the above condition is equivalent to

The function class was introduced and investigated by Sudharsan et al. [6]. If and , then the class reduces to the class .

Definition 4. Suppose that and . Let denote the class of functions in satisfying the following inequality:

It is evident that the above condition is equivalent to

If and , then the class reduces to the class .

In 1966, Pommerenke [7] stated the th Hankel determinant for and as This Hankel determinant is useful and has also been considered by several authors. The growth rate of Hankel determinant as was investigated, respectively, when is a member of certain subclass of analytic functions, such as the class of p-valent functions [7, 8], the class of starlike functions [7], the class of univalent functions [9], the class of close-to-convex functions [10], the class of strong close-to-convex functions [11], a new class [12], and a new class [13]. Similar to the above discussions, we can also refer to [14, 15]. Ehrenborg [16] studied the Hankel determinant of exponential polynomials. The Hankel transform of an integer sequence was defined and some of its properties were discussed by Layman [17]. Pommerenke [9] proved that the Hankel determinants of univalent function satisfy Later, was also proved by Hayman [18]. One can easily observe that the Fekete and Szegö functional is . For results related to the functional, see [19, 20]. Fekete and Szegö further generalized the estimate , where is real and . For results related to the functional, see [21, 22]. In 2010, Hayami and Owa [21, 22] also generalized the estimate for analytic function. Later, in 2012, Krishna and Ramreddy [23] also generalized the estimate for p-valent analytic function; see also [24, 25].

For our discussion in this paper, we consider the second Hankel determinant in the case of and , namely,

Janteng et al. [26] have considered the functional and found a sharp bound, the subclass of denoted by , defined as . In their work, they have shown that if , then . These authors [27, 28] also studied the second Hankel determinant and sharp bound for the classes of starlike and convex functions, close-to-starlike and close-to-convex functions with respect to symmetric points denoted by , , , and and have shown that , , , and , respectively.

Singh [29] established the second Hankel determinant and sharp bound for the classes of close-to-starlike and close-to-convex functions with respect to conjugate and symmetric conjugate points denoted by , , , and and has shown that , , , and , respectively.

Mishra and Gochhayat [30] obtained the sharp bound to for the functions in the class denoted by , and defined as , using the fractional differential operator denoted by and defined by Owa and Srivastava [31]. These authors have shown that if , then .

Mohammed and Darus [32] have obtained a sharp upper bound to for the functions in the class denoted by , and defined as . These authors have proved that if , then .

Similar to the above discussions in a new subclass of analytic function with different operators, we can also refer to [33, 34]. Singh [35] also obtained a sharp upper bound for the functional for the function , where and showed that if , then .

Mehrok and Singh [36] have obtained a sharp upper bound to for the function in the classes denoted by and and defined as, respectively, In their work, they proved that if , then and if , then .

Shanmugam et al. [37] established the sharp upper bound of the second Hankel determinant for the classes of and , defined as, respectively, These authors proved that if , then and if , then

Krishna and Ramreddy [38] obtained a sharp upper bound to the nonlinear functional for a new subclass of analytic functions , , defined by These authors proved that if , then .

Similar to the above discussions defined as different classes of analytic functions, we can also refer to [3949]. Raza and Malik [50] studied the third Hankel determinant of analytic functions related with lemniscate of Bernoulli; see also [51].

Motivated by the above-mentioned results obtained by different authors in this direction, in this present investigation, we determine the upper bounds of the second Hankel determinant for functions belonging to these classes , , , and .

#### 2. Preliminary Results

In order to prove our main results, we need the following lemmas.

Lemma 5 (see [52]). If the function is given by the power series (2), then .

Lemma 6 (see [53, 54]). If the function is given by the power series (2), then for some with and for some with .

#### 3. Main Results

Theorem 7. Let and . Suppose that the function given by (1) is in the class . Then The result is sharp, with the extremal function

Proof. Since , it follows from (8) that there exists a Schwarz function , which is analytic in with and in , such that where Define the function by From (3), we get and In view of (30), (31), and (33), we have Similarly,

Comparing the coefficients of , , and in (34) and (35), we obtain Thus we have

Since the functions and    are members of the class simultaneously, we assume without loss of generality that . For convenience of notation, we take . By substituting the values of and , respectively, from (26) and (27) in (38), we have

Using the triangle inequality and , we have where .

We next maximize the function on the closed square . Differentiating in (40) partially with respect to , we get For and for any fixed with , from (41), we observe that . Consequently, is an increasing function of and hence it cannot have a maximum value at any point in the interior of the closed square . Moreover, for fixed , we have From the relations (40) and (42), upon simplification, we obtain

Next, since we get that for and has real critical point at . Therefore, the maximum of occurs at . Thus, the upper bound of corresponds to and . Hence,

Equality holds for the function By calculating, we have and , , and . So and equality holds. This shows that the result is sharp, and the proof of Theorem 7 is complete.

Setting in Theorem 7, we obtain the following result due to Janteng et al. [27].

Corollary 8. If , then The result is sharp, with the extremal function

By using the similar method as in the proof of Theorem 7, one can similarly prove Theorem 9.

Theorem 9. Let and . Suppose that the function given by (1) is in the class . ThenThe results are sharp, with the extremal function for the case , and there is no extremal function for the case .

Setting in Theorem 9, one obtains the following result due to Janteng et al. [27].

Corollary 10. If , then The result is sharp.

Theorem 11. Let and . Suppose that the function given by (1) is in the class . Then The result is sharp, with the extremal function

Proof. Since , it follows from (14) that there exists a Schwarz function , which is analytic in with and in , such that where was defined by (31).

In view of (31), (33), and (55), we have Similarly, Comparing the coefficients of , , and in (56) and (57), we obtain Thus we have Since the functions and    are members of the class simultaneously, we assume without loss of generality that . For convenience of notation, we take . By substituting the values of and , respectively, from (26) and (27) in (60), we have Using the triangle inequality and , we have where .

We next maximize the function on the closed square . Differentiating in (62) partially with respect to , we get For and for any fixed with , from (63), we observe that . Consequently, is an increasing function of and hence it cannot have a maximum value at any point in the interior of the closed square . Moreover, for fixed , we have

From the relations (62) and (64), upon simplification, we obtain Next, since we get that for and has real critical point at . Therefore, the maximum of occurs at . Thus, the upper bound of corresponds to and . Hence,

Equality holds for the function By calculating, we have and , , and . So and equality holds. This shows that the result is sharp, and the proof of Theorem 11 is complete.

Setting in Theorem 11, we obtain the following result due to Janteng et al. [28].

Corollary 12. If , then The result is sharp, with the extremal function

By using the similar method as in the proof of Theorem 11, one can similarly prove Theorem 13.

Theorem 13. Let and . Suppose that the function given by (1) is in the class . Then The result is sharp, with the extremal function

Setting in Theorem 13, one obtains the following result due to Janteng et al. [28].

Corollary 14. If , then The result is sharp, with the extremal function

#### Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work was financially supported by Foundation for Distinguished Young Talents in Higher Education of Guangdong China (no. 2013LYM0093) and Training plan for the Outstanding Young Teachers in Higher Education of Guangdong (no. Yq 2013159). The authors are also grateful to the anonymous referees and Professor V. Ravichandran for making many valuable suggestions that improved the quality and the readability of this paper.

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