### Article

KYUNGPOOK Math. J. 2019; 59(3): 481-491

**Published online** September 23, 2019 https://doi.org/10.5666/KMJ.2019.59.3.481

Copyright © Kyungpook Mathematical Journal.

### Some Coefficient Inequalities Related to the Hankel Determinant for a Certain Class of Close-to-convex Functions

Yong Sun∗, Zhi-Gang Wang

School of Science, Hunan Institute of Engineering, Xiangtan, 411104, Hunan, People’s Republic of China

e-mail : yongsun2008@foxmail.com

Mathematics and Computing Science, Hunan First Normal University, Changsha, 410205, Hunan, People’s Republic of China

e-mail : wangmath@163.com

**Received**: November 14, 2017; **Revised**: February 1, 2019; **Accepted**: March 4, 2019

### Abstract

In the present paper, we investigate the upper bounds on third order Hankel determinants for certain class of close-to-convex functions in the unit disk. Furthermore, we obtain estimates of the Zalcman coefficient functional for this class.

**Keywords**: coeﬃcient inequality, Hankel determinant, Zalcman’s conjecture, close-to-convex functions.

### Introduction

Let be the class of functions analytic in the unit disk of the form

A function is said to be starlike of order

Recall that a function is close-to-convex in if it is univalent and the range is a close-to-convex domain, i.e., the complement of can be written as the union of nonintersecting half-lines. A normalized analytic function

In [11], Gao and Zhou investigated the following class of close-to-convex functions.

### Definition 1.1

Suppose that is analytic in of the form

### Theorem A. ([11])

### Theorem B. ([11])

Noonan and Thomas [24] studied the Hankel determinant _{q,n}(_{q,n}(_{2,n}(^{1/2}, where

We note that _{2}_{,}_{1}(_{2}_{,}_{2}(

By the definition, _{3}_{,}_{1}(_{1} = 1 so that _{3}_{,}_{1}(_{2}_{,}_{1}(_{2}_{,}_{2}(

### Theorem C

In 1960, Lawrence Zalcman posed a conjecture that the coefficients of satisfy the sharp inequality ^{2} and its rotations. We call _{2}(_{2}_{,}_{1}(

In the present investigation, our purpose is to develop similar results on the Hankel determinants in the context the close-to-convex functions . Further-more, the upper bounds to the Zalcman functional for this class are obtained.

### Preliminary Results

Denote by the class of Carathéodory functions

### Lemma 2.1. ([8])

### Lemma 2.2.([13])

### Lemma 2.3.([20, 21])

### The Upper Bounds of the Hankel Determinant

In this section, we first give an upper bound of the functional |_{2}_{3} −_{4}| for functions .

### Theorem 3.1

**Proof**

Let _{2}, _{3} and _{4} from _{1}| ≤ 2. By setting _{1}, we may assume without loss of generality that _{1}(_{2}(_{2}(

### Theorem 3.2

**Proof**

Using the values of _{2}, _{3} and _{4} from _{1}

Let . Then using the above results in theorem

### Theorem 3.3

Let

### Remark 3.1

In Theorem 3.1, Theorem3.2 and Theorem 3.3, we have obtained the upper bounds for the Hankel determinant. However, these results are far from sharp.

### The Upper Bounds of the Zalcman Functional

In this section, we consider the Zalcman functional for functions .

### Theorem 4.1

**Proof**

Let _{1} = _{0} = 1.

For the case of

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