Toeplitz Matrices Whose Elements Are the Coefficients of Functions with Bounded Boundary Rotation

Abstract

Let denote the family of functions of bounded boundary rotation so that in the open unit disk . We obtain sharp bounds for Toeplitz determinants whose elements are the coefficients of functions .

1. Introduction

Let denote the class of all functions of the form which are analytic in the open unit disk and let denote the subclass of consisting of univalent functions. Obviously, for functions , we must have in . For we consider the family of functions of bounded boundary rotation so that in . The family is properly contained in the class of close-to-convex functions (e.g., see Brannan [1], Pinchuk [2], or Duren [3] pp. 269–271.)

Toeplitz matrices are one of the well-studied classes of structured matrices. They arise in all branches of pure and applied mathematics, statistics and probability, image processing, quantum mechanics, queueing networks, signal processing, and time series analysis, to name a few (e.g., see Ye and Lim [4]). Toeplitz matrices have some of the most attractive computational properties and are amenable to a wide range of disparate algorithms and determinant computations. Here we consider the symmetric Toeplitz determinantand obtain sharp bounds for the coefficient body ; ; , where the entries of are the coefficients of functions of form (1) that are in the family of functions of bounded boundary rotation. As far as we are concerned, the results presented here are new and noble and the only prior compatible result is published by Thomas and Halim [5] for the classes of starlike and close-to-convex functions. It is worth noticing that the bounds presented here are much finer than those presented in [5].

2. Main Results

We note that, for the functions of form (1) that are in the family of functions of bounded boundary rotation, we can write , where , the class of positive real part function satisfying for and is of the form

We shall state the following result [6], to prove our main theorems.

Lemma 1. Let . Then for some complex valued with and some complex valued with we have

In our first theorem we obtain a sharp bound for the coefficient body .

Theorem 2. Let be given by (1). Then we have the sharp bound

Proof. First note that by equating the corresponding coefficients of , we obtainNow by (2), (6), and (7) we have Making use of Lemma 1 to express in terms of , we obtain Without loss of generality, let . Applying triangle inequality, we get Differentiating with respect to we obtain Setting leads to either or But for . Therefore the maximum of occurs either at or .
For we obtain and which implies .
For we obtain and which implies .
This bound is sharp and the extremal function is given by .

We remark that the sharp bound given by Theorem 2 is much finer than that was obtained by Thomas and Halim [5] for the class of functions of form (1) that are close-to-convex in .

Next, we determine a sharp bound for the coefficient body .

Theorem 3. Let be given by (1). Then we have the sharp bound

Proof. Note that, by (2), (7), and (8), we have Making use of Lemma 1 and triangle inequality, we obtain where, without loss of generality, we let and .
Differentiating and using a simple calculus shows that for and fixed . It follows that is an increasing function of . So . Upon letting , a simple algebraic manipulation yields This bound is sharp and the extremal function is given by .

No bounds for was obtained by Thomas and Halim [5] for the class of functions of form (1) that are close-to-convex in . In our next theorem we determine a sharp bound for the coefficient body .

Theorem 4. Let be given by (1). Then we have the sharp bound

Proof. Expanding the determinant and letting we obtain As before, without loss of generality, we assume that , where . Then, by using the triangle inequality and the fact that , we obtain Considering the modulus as positive, we get One can apply an elementary calculus to show that attains its maximum value of on when Similarly, considering the modulus as negative, we obtain Again, using an elementary calculus argument shows that this expression has a maximum value of on when . The sharp bound is achieved for and .

We remark that the sharp bound given by Theorem 4 is much finer than obtained by Thomas and Halim [5] for the class of functions of form (1) that are close-to-convex in . Finally, an upper bound for the coefficient body is presented in the following.

Theorem 5. Let be given by (1). Then we have the upper bound

Proof. WriteUsing the same techniques as above, one can obtain with simple computations that . Thus we need to show that . In view of (6), (7), and (8), a simple computation leads to Expressing and in terms of as earlier and using Lemma 1 with and , we obtain Applying the triangle inequality and assuming that , we obtain We need to find the maximum value of on . First, assume that there is a maximum at an interior point of . Then differentiating with respect to and equaling it to 0 would imply that , which is a contradiction. Thus to find the maximum of , we need only to consider the end points of .
When ,
When ,
When , , which has maximum value on
When , , which has maximum value on

No bounds for were obtained by Thomas and Halim [5] for the class of functions of form (1) that are close-to-convex in .

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

The work of the second author is supported by a grant from Department of Science and Technology, Government of India vide ref: SR/FTP/MS-022/2012 under fast track scheme.