Bounds of Hankel Determinant for a Class of Univalent Functions
The authors study the coefficient condition for the class defined as the family of analytic functions and , which satisfy , where is a real number.
Let be the class of functions of the following form: which are analytic in the unit disc , and let be the subclass of consisting of functions which are univalent in . A function is said to be close to convex in the open unit disc if there exists a convex function (not necessarily normalized) such that For fixed real numbers , let denote the family of functions in which satisfy
In 2005, V. Singh et al.  established that, for , functions in satisfy in and so are close to convex in .
In , Noonan and Thomas defined the Hankel determinant of the function for and by
The determinant has been investigated by several authors with the subject of inquiry ranging from rate of growth of as , to the determination of precise bounds on for specific and for some special classes of functions. In a classical theorem, Fekete and Szeg  considered the Hankel determinant of for and
The well-known result due to them states that if , then where and is a real number. In the present paper, we obtain a sharp bound for when .
2. Preliminary Results
We denote by the family of all functions given by analytic in for which for . It is well known that for , for each .
Lemma 2.1 (See ). The power series for p(z) given in (2.1) converges in to a function in if and only if the Toeplitz determinants and = are all nonnegative. They are strictly positive except for and for ; in this case, for and for .
3. Main Result
Theorem 3.1. Let , , be a real number. If , then where is the root of the equation and
Proof. Since , it follows from (1.3) that there exists a function such that
Equating coefficients in (3.3) yields
Thus, we can easily establish that
Using (2.4), in view of Lemma 2.2, we obtain that
Since , so . Letting , we may assume without restriction that . Thus, applying the triangle inequality on (3.6), with , we obtain
Differentiating , we get the following:
Using elementary calculus, one can show that for . It implies that is an increasing function, and, thus, the upper bound for corresponds to , in which case
Setting , since , we have provided , where is the root of the equation .
Case 1. When , then the maximum value of corresponds to . Therefore, we have
Case 2. When , the maximum value of corresponds to . Therefore, we have where is given by (3.2). This completes the proof of the Theorem.
Setting in above theorem, we get the following result of Janteng et al. .
Corollary 3.2. If an analytic function is such that , , then The result is sharp.
J. W. Noonan and D. K. Thomas, “On the second Hankel determinant of areally mean -valent functions,” Transactions of the American Mathematical Society, vol. 223, pp. 337–346, 1976.View at: Google Scholar
U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley, Calif, USA, 1958.