Hankel, Toeplitz, and Hermitian-Toeplitz Determinants for Certain Close-to-convex Functions

Let f be analytic in D={z∈C:|z|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}$$\end{document}, and be given by f(z)=z+∑n=2∞anzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$$\end{document}. We give sharp bounds for the second Hankel determinant, some Toeplitz, and some Hermitian-Toeplitz determinants of functions in the class of Ozaki close-to-convex functions, together with a sharp bound for the Zalcman functional J2,3(f).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{2,3}(f).$$\end{document}


Introduction and definitions
Let A denote the class of functions f analytic in the unit disk D := {z ∈ C : |z| < 1} with Taylor series f (z) = ∞ n=1 a n z n , a 1 := 1. (1) Let S be the subclass of A, consisting of univalent (i.e., one-to-one) functions. A function f ∈ A is called starlike (with respect to the origin) if f (D) is starlike with respect to the origin, and convex if f (D) is convex. Let S * (α) and C(α) denote, respectively, the classes of starlike and convex functions of order α for 0 ≤ α < 1 in S. It is well known that a function f ∈ A belongs to S * (α) if, and only if, Re (zf (z)/f (z)) > α for z ∈ D, and that f ∈ C(α) if, and only if, Re(1 + zf (z)/f (z)) > α. We write S * (0) =: S * , and C(0) =: C.
Similarly, a function f ∈ A belongs to K, the class of close-to-convex functions if, and only if, there exists g ∈ S * , such that Re[e iτ (zf (z)/g(z))] > 0 for z ∈ D, and τ ∈ (−π/2, π/2), so that C ⊂ S * ⊂ K ⊂ S. When τ := 0, the resulting subclass of close-to-convex functions is denoted by K 0 . Although the class K was first formally introduced by Kaplan in 1952 [14], in 1941, Ozaki [20] showed that a function in A is univalent if it satisfies the condition It follows from the original definition of Kaplan [14], that functions satisfying (2) are close-to-convex, and therefore members of S.
We note that in contrast to the definition of K, the definition of F(λ) does not involve an independent starlike functions g, but as was shown by Umezawa [24], members of F(1) have coefficients which grow at the same rate as those in K.
In this paper, we will give the sharp bound for the second Hankel determinant, together with sharp bounds for various Toeplitz and Hermitian-Toeplitz determinants defined below, whose elements are coefficients of functions in F O (λ).
where a k := a k . When a n is a real number, T q,n (f ) is qth Hermitian-Toeplitz determinant. In particular Finding sharp bounds for the Hankel determinants of functions in A has been the subject of a great many papers in the recent years. In particular, many results are known concerning the second Hankel determinant H 2 (2) = a 2 a 4 − a 2 3 when f ∈ S and its subclasses, and a summary of some of the more important results can be found in [23]. On the other hand, investigations concerning Toeplitz determinants were introduced only recently in [2]. Similarly, problems concerning Hermitian-Toeplitz determinants were first considered in [9].
We next discuss the Zalcman functional, its relationship with the Zalcman conjecture, and a generalization due to Ma [19]. In the early 70s, Lawrence Zalcman posed the conjecture that if f ∈ S, and is given by (1), then with equality for the Koebe function k(z) = z/(1−z) 2 for z ∈ D, or a rotation. This conjecture implies the celebrated Bieberbach conjecture |a n | ≤ n for f ∈ S (see [7]). The elementary area theorem shows that the conjecture is true when n = 2 (see [10]). Kruskal established the conjecture when n = 3 (see [15]), and more recently for n = 4, 5, 6 (see [16]). However, the Zalcman conjecture for n > 6 remains an open problem. The conjecture has been proved for several subclasses of S, e.g., starlike, typically real, and close-to-convex functions [7,18], and it is known that the Zalcman conjecture is asymptotically true (see [12]). Recently, Abu Muhana et al. [1] proved the conjecture for the class F O (1).
Relevant to this paper is Ma's generalization of the Zalcman functional a 2 n − a 2n−1 , defined as follows.
Definition 1.5. Let f ∈ A, and be given by (1). For m, n ∈ N \ {1}, let J m,n (f ) := a m a n − a m+n−1 , and in particular, In [19], and established the conjecture when f ∈ S * , and if f ∈ S, provided that the coefficients in (1) are real. In a recent paper, Cho et al. [8] considered the case m = 2, n = 3, for a very wide set of functions in A, obtaining sharp bounds for |J 2,3 (f )| in all cases, where we note that Theorem 4.1 C provides a sharp bound for |J 2,3 (f )|, when f satisfies inequality (3) for λ ∈ (−1/2, 1/2].

Lemmas
We will use the following results for functions p ∈ P, the class of functions with positive real part in D, given by and since we will be concerned primarily with the coefficients a 2 , a 3 , and a 4 , we also need Lemma 2.4, which can easily be deduced from (1), (3), and (6).

The second Hankel determinant H 2 (2)(f )
The inequality is sharp.
Proof. First note that from (3), we can write Thus, from Lemma 2.4, we have Noting that both the class F O (λ) and the functional H 2 (2)(f ) are rotationally invariant, we now use Lemma 2.3 to express the coefficients p 3 and p 2 in terms of p 1 , and write t := p 1 to obtain with 0 ≤ t ≤ 2 (1 + 2λ) 2 (4 − t 2 ) 2 y 2 We now take the modulus to obtain For t = 2, we have For 0 ≤ t < 2, the function [0, 1] |y| → φ(t, |y|) is easily seen to be increasing, so Thus, the function [0, 2) t → φ(t, 1) has critical points at To see that the inequality is sharp, take a function for which Choosing λ = 1/2, and λ = 1, we deduce the following sharp inequalities.

Toeplitz determinants
In this section, we extend the results in [2] for f ∈ C to f ∈ F O (λ). We first define the function f 1 ∈ F O (λ) for z ∈ D, which serves as the extreme function for all the following results: where The inequality is sharp when f = f 1 .
To see that this is sharp, recall that Then , which, using (14), shows that the inequality for T 2 (n)(f ) is sharp.
The inequality is sharp when f = f 1 . Proof. We first note that where we have used Lemmas 2.1 and 2.4. It therefore remains to estimate |a 3 − 2a 2 2 |. Note first that Thus, taking μ = 2(1 + 2λ), we deduce from Lemma 2.1 that 4λ), and so, from (15), we obtain For equality in Theorem 4.2, we choose f = f 1 defined in (13).
The inequality is sharp when f = f 1 .
The inequality is sharp on again choosing f = f 1 defined in (13).

Hermitian-Toeplitz determinants
In this section, we compute sharp lower and upper bounds for over the class F O (λ). and Both inequalities are sharp.