Improved Upper Bounds of the Third-Order Hankel Determinant for Ozaki Close-to-Convex Functions

: L etN be the class of functions that convex in one direction and M denote the class of functions zf (cid:48) ( z ) , where f ∈ N . In the paper, the third-order Hankel determinants for these classes are estimated. The estimates of H 3,1 ( f ) obtained in the paper are improved.

In the paper, we study the upper bounds of the third-order Hankel determinant H 3,1 ( f ) for the following classes This problem was studied by Prajapat et al. [9] (see [18]).
In 1941, Ozaki [25] introduced and studied the class N . Later, Sakaguchi [26] and R. Singh and S. Singh [27] showed, respectively, that functions in N are close to convex and starlike. In 2013, Obradović [28] derived the sharp bound of |a n | ≤ 1 n(n−1) in the class N . Ponnusamy [29] obtained the bounds of initial logarithmic coefficients for f ∈ N .
In this paper, we use a method based on the estimates of the coefficients of the Schwartz function. This method is different from the commonly used method, which is the main reason for the improvement in the estimate for the class mentioned above.
To obtain the main results, we will need the following, almost forgotten, result of Carleson ( [30]).

Main Results
We begin with improvements in the upper bound of the third Hankel determinant for the class M.

Proof.
For a function f ∈ M, there exists a Schwarz function ω(z), such that By comparing the coefficients in the above expression, we receive From (2) and (3), we achieve By using triangle inequality and Lemma in (4), we come across By putting x = |c 1 | and y = |c 2 | in above expression, we obtain We continue by finding the maximum of the function F on the region Ω = {(x, y) : Differentiating F partially with respect to x and y, we obtain By putting ∂F ∂x = 0, ∂F ∂y = 0 and simplifying, we receive Applying Newton's methods to the above equations in Maple Software, we obtain x 0 ≈ 0.458445, y 0 ≈ 0.631054, On the edge y = 1 − x 2 , F(x, y) becomes Thus, we get We complete the proof of Theorem 1.

If f ∈ N , then
Proof. Assume that f ∈ N . From the definition, we know there is a Schwarz function ω such that Using some easy computation, comparing the coefficients in the above expression, we receive From (2) and (5), we achieve Applying the triangle inequality and Lemma in (6), we obtain Putting |c 1 | = x and |c 2 | = y in above expression, we come across In order to caculate the maximum of the function Υ on the region Ω = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x 2 }, we take the partial derivative with respect to x and y, respectively, and we receive By putting ∂Υ ∂x = 0, ∂Υ ∂y = 0 and simplifying, we come across By applying Newton's methods to the above equations in Maple Software, we receive x 0 ≈ 0.417110, y 0 ≈ 0.514879, Then, there is a critical point (x 0 , y 0 ) satisfying y ≤ 1 − x 2 at which Υ(x, y) obtains its maximum. Thus, we have Υ(x, y) ≤ Υ(x 0 , y 0 ) ≈ 88.353 . . . Therefore, we continue studying Υ on the edges of Ω. For x = 0, Υ(0, y) = 60y 4 + 4y 3 − 120y 2 + 72y + 60 ≤ Υ(0, 1) = 76.