International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 147842, 6 pages

doi:10.1155/2012/147842

Research Article

Bounds of Hankel Determinant for a Class of Univalent Functions

Sarika Verma, Sushma Gupta, and Sukhjit Singh

Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Punjab, Longowal 148106, India

Received 31 March 2012; Revised 1 June 2012; Accepted 1 June 2012

Academic Editor: Teodor Bulboaca

Copyright © 2012 Sarika Verma 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.

The authors study the coefficient condition for the class defined as the family of analytic functions and , which satisfy , where is a real number.

1. Introduction

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. [1] established that, for , functions in satisfy in and so are close to convex in .

In [2], 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 [3] 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 [4]). 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 .

Lemma 2.2 (See [5, 6]). Let . Then

for some such that and .

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

Then,

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. [7].

Corollary 3.2. If an analytic function is such that , , then

The result is sharp.

References

V. Singh, S. Singh, and S. Gupta, “A problem in the theory of univalent functions,” Integral Transforms and Special Functions, vol. 16, no. 2, pp. 179–186, 2005.

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.

M. Fekete and G. Szegö, “Eine Bemerkung uber ungerade schlichte Funktionen,” Journal of the London Mathematical Society, vol. 8, no. 2, pp. 85–89.

U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley, Calif, USA, 1958.

R. J. Libera and E. J. Złotkiewicz, “Early coefficients of the inverse of a regular convex function,” Proceedings of the American Mathematical Society, vol. 85, no. 2, pp. 225–230, 1982.

R. J. Libera and E. J. Złotkiewicz, “Coefficient bounds for the inverse of a function with derivative in ,” Proceedings of the American Mathematical Society, vol. 87, no. 2, pp. 251–257, 1983.

A. Janteng, S. A. Halim, and M. Darus, “Coefficient inequality for a function whose derivative has a positive real part,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 2, article 50, pp. 1–5, 2006.