**1. Introduction**

Let A be a class of analytic functions in the open unit disk D = {*z* ∈ C : |*z*| < <sup>1</sup>}, of the form

$$f(z) = z + \sum\_{n=2}^{\infty} a\_n z^n \qquad (z \in \mathbb{D})\,. \tag{1}$$

Let S be the class of functions *f* ∈ A which are univalent in D. A function *f* ∈ A is said to be *starlike*, if it satisfies the inequality

$$\operatorname{Re}\left(\frac{zf'(z)}{f(z)}\right) > 0 \qquad \left(z \in \mathbb{D}\right). \tag{2}$$

We denote by S∗ the class which consists of all functions *f* ∈ A that are starlike. A function *f* ∈ A is said to be *close-to-convex* if there exits a function *g* ∈ S∗ such that it satisfies the inequality

$$\operatorname{Re}\left(\frac{zf'(z)}{g(z)}\right) > 0 \qquad (z \in \mathbb{D})\,. \tag{3}$$

We denote by C the class which consists of all functions *f* ∈ A that are close-to-convex. We note that S∗ ⊂C⊂S and that |*an*| ≤ *n* for *f* ∈ S∗.

For two functions *f* and *g* which are analytic in D, we say that the function *f* is *subordinate* to *g*, and write *f*(*z*) ≺ *g*(*z*), if there exists a *Schwarz function w*, that is a function *w* analytic in D with *w*(0) = 0 and |*w*(*z*)| < 1 in D, such that *f*(*z*) = *g*(*w*(*z*)) for all *z* ∈ D. In particular, if the function *g* is univalent in D, then *f* ≺ *g* if and only if *f*(0) = *g*(0) and *f*(D) ⊆ *g*(D) [1].

In 1976, Noonan and Thomas [2] defined the *q-th Hankel determinant* for integers *n* ≥ 1 and *q* ≥ 1 by

$$H\_{\emptyset}(n) = \begin{vmatrix} a\_{\mathfrak{n}} & a\_{\mathfrak{n}+1} & \dots & a\_{\mathfrak{n}+\mathfrak{q}-1} \\ a\_{\mathfrak{n}+1} & a\_{\mathfrak{n}+2} & \dots & a\_{\mathfrak{n}+\mathfrak{q}} \\ \vdots & \vdots & \vdots & \vdots \\ a\_{\mathfrak{n}+\mathfrak{q}-1} & a\_{\mathfrak{n}+\mathfrak{q}} & \dots & a\_{\mathfrak{n}+2\mathfrak{q}-2} \end{vmatrix} \quad (a\_1 = 1) \; . $$

In general, one of the important tools in the theory of univalent functions is the *Hankel determinant*. It is used, for example, in showing that a function of bounded characteristic in D, that is, a function which is a ratio of two bounded analytic functions with its Laurent series around the origin having integral coefficients, is rational [3]. For the use of Hankel determinant in the study of meromorphic functions, see [4]. For detailed information, the readers are encouraged [5,6]. Various properties of these determinants can be found in [7] (Chapter 4). The investigations of Hankel determinants for different classes of analytic functions started in the 1960s. Pommerenke [8] proved that the Hankel determinants of univalent functions satisfy *Hq*(*n*) <sup>≤</sup> *Kn*−( 12 <sup>+</sup>*β*)*q*<sup>+</sup> 32 where *n*, *q* ∈ N, *q* ≥ 2, *β* > 1/4000 and *K* depends only on *q*. Later, Hayman [9] proved that *Hq*(*n*) <sup>≤</sup> *An*12 where *n* ∈ N and *A* is an absolute constant for areally mean univalent functions. Pommerenke [10] investigated the Hankel determinant of areally mean *p*-valent functions, univalent functions as well as of starlike functions. For results related to these determinants, see also [11,12].

Note that

$$H\_2(1) = \begin{vmatrix} a\_1 & a\_2 \\ a\_2 & a\_3 \end{vmatrix} \qquad \text{and} \qquad H\_2(2) = \begin{vmatrix} a\_2 & a\_3 \\ a\_3 & a\_4 \end{vmatrix} \land$$

where the Hankel determinants *<sup>H</sup>*2(1) = *a*3 − *a*22 and *<sup>H</sup>*2(2) = *a*2*a*4 − *a*23 are well-known as *Fekete-Szegö* and *second Hankel determinant* functionals, respectively. Further, Fekete and Szegö [13] introduced the generalized functional *a*3 − *<sup>λ</sup>a*22, where *λ* is some real number. In recent years, the research on Hankel determinants has focused on the estimation of |*<sup>H</sup>*2(2)|. Problems in this field has also been argued by several authors for various classes of univalent functions [14–24].

The Koebe one-quarter theorem [1] ensures that the image of D under every univalent function *f* ∈ S contains a disk of radius 1/4. Thus every function *f* ∈ S has an inverse *f* −1, such that

$$f^{-1}\left(f(z)\right) = z \quad \left(z \in \mathbb{D}\right), \qquad \text{and} \qquad f\left(f^{-1}(w)\right) = w \quad \left(|w| < r\_0(f); \ r\_0(f) \ge \frac{1}{4}\right),$$

where the inverse *f* −1 has the power series expansion (see [25])

$$f^{-1}(w) = w - a\_2 w^2 + \left(2a\_2^2 - a\_3\right) w^3 - \left(5a\_2^3 - 5a\_2 a\_3 + a\_4\right) w^4 + \cdots \ . \tag{4}$$

A function *f* ∈ A is said to be *bi-univalent* in D if both *f* and *f* −1 are univalent in D, in the sense that *f* −1 has a univalent analytic continuation to D. Let Σ denote the class of bi-univalent functions in D. For a brief history of functions in the class Σ and also other different characteristics of these functions and the coefficient problems, see [25–32] and the references therein.

In 2014, Hamidi and Jahangiri [33] defined the class of bi-close-to-convex functions of order *α* (0 ≤ *α* < 1) that this class is denoted by CΣ(*α*) and in particular, CΣ(0) = C<sup>Σ</sup>.

**Definition 1.** *A function f* ∈ Σ *is in the class of bi-close-to-convex functions of order α if the following conditions are satisfied:*

$$\operatorname{Re}\left(\frac{zf'(z)}{g(z)}\right) > a \qquad (z \in \mathbb{D})\tag{5}$$

*and*

$$\operatorname{Re}\left(\frac{wF'(w)}{G(w)}\right) > \mathfrak{a} \qquad \left(w \in \mathbb{D}\right),\tag{6}$$

*where the function <sup>F</sup>*(*w*) = *f* <sup>−</sup><sup>1</sup>(*w*) *is defined by* (4)*, g*(*z*) = *z* + ∞ ∑ *n*=2 *bnzn* ∈ S∗ *and*

$$G(w) = w + \sum\_{n=2}^{\infty} B\_n z^n \in \mathcal{S}^\*. \tag{7}$$

Recently, Güney et al. [34] obtained the bound for the second Hankel determinant *<sup>H</sup>*2(2) for the class CΣ of bi-close-to-convex functions as follows:

**Theorem 1.** *Let the function f given by* (1) *be in the class* CΣ *and <sup>G</sup>*(*w*) = *<sup>g</sup>*<sup>−</sup><sup>1</sup>(*w*)*. Then*

$$|H\_2(\mathbf{2})| := \left| a\_2 a\_4 - a\_3^2 \right| \le \frac{353}{36}.$$

**Remark 1.** *By means of the subordination, the conditions* (5) *and* (6) *are, respectively, equivalent to*

$$\frac{z f'(z)}{g(z)} \prec \frac{1+z}{1-z} \qquad \text{and} \qquad \frac{w F'(z)}{G(w)} \prec \frac{1+w}{1-w}.$$

The main purpose of this paper is to determine bounds for the functional *<sup>H</sup>*2(2) = *a*2*a*4 − *a*23 for functions belonging to the subclass CΣ of bi-close-to-convex functions, which is a much improved estimation than the previous result given by Güney et al. [34]. We note that our proof method is by means of the subordination and more direct than those used by others and so we ge<sup>t</sup> a smaller upper bound and more accurate estimation for the functional |*<sup>H</sup>*2(2)| for functions in the class C<sup>Σ</sup>.

#### **2. Main Results**

**Theorem 2.** *Let the function f given by* (1) *be in the class* CΣ *and <sup>G</sup>*(*w*) = *<sup>g</sup>*<sup>−</sup><sup>1</sup>(*w*)*. Then*

$$|H\_2(2)| := \left| a\_2 a\_4 - a\_3^2 \right| \le \frac{227}{36}.$$

In order to prove our main result, we need the following lemmas.

**Lemma 1.** [1] (p. 190) *Let u be analytic function in the unit disk* D*, with u*(0) = 0*, and* |*u*(*z*)| < 1 *for all z* ∈ D*, with the power series expansion*

$$u(z) = \sum\_{n=1}^{\infty} c\_n z^n.$$

*Then,* |*cn*| ≤ 1 *for all n* ∈ N*. Furthermore,* |*cn*| = 1 *for some n* ∈ N *if and only if u*(*z*) = *ei<sup>θ</sup> zn, θ* ∈ R*.*

> *s* − *ψ*1

 −

\*\*Lemma 2.\*\* [20]  $If \,\psi(z) = \sum\_{n=1}^{\infty} \psi\_n z^n, z \in \mathbb{D},$  is a Schurz function with  $\psi\_1 \in \mathbb{R}$ , then 
$$\psi\_2 = x \left(1 - \psi\_1^2\right),$$
 
$$\psi\_3 = \left(1 - \psi\_1^2\right) \left(1 - |x|^2\right) s - \psi\_1 \left(1 - \psi\_1^2\right) x^2,$$

 −

 −

*ψ*3 =

*for some x*, *s, with* |*x*| ≤ 1 *and* |*s*| ≤ 1*.* **Lemma 3.** [35] *Let the function f* ∈ S∗ *be given by* (1)*. Then, for any real number μ,*

$$\left| a\_3 - \mu a\_2^2 \right| \begin{cases} 3 - 4\mu & \text{if} \quad \mu \le \frac{1}{2} \\\\ 1 & \text{if} \quad \frac{1}{2} \le \mu \le 1 \\\\ 4\mu - 3 & \text{if} \quad \mu \ge 1 \end{cases}$$

**Lemma 4.** [19] *Let the function f* ∈ S∗ *be given by* (1)*. Then*

$$|H\_2(2)| := \left| a\_2 a\_4 - a\_3^2 \right| \le 1.$$

*Equality holds true for the Koebe function k*(*z*) = *z* (1 − *z*)<sup>2</sup> *.*

**Lemma 5.** [36] *Let the function f* ∈ S∗ *be given by* (1)*. Then*

$$|a\_2 a\_3 - a\_4| \le 2.$$

*Equality holds true for the Koebe function k*(*z*) = *z* (1 − *z*)<sup>2</sup> *.*

**Proof of Theorem 2.** As noted in Remark 1, if *f* ∈ C<sup>Σ</sup>, then by definition of subordination, there exist two Schwarz functions *u* and *v*, of the form *u*(*z*) = ∞ ∑ *<sup>n</sup>*=1 *cnz<sup>n</sup>* and *v*(*z*) = ∞ ∑ *<sup>n</sup>*=1 *dnzn*, *z* ∈ D that we can write

$$\frac{z f'(z)}{g(z)} = \frac{1 + u(z)}{1 - u(z)} = 1 + 2c\_1 z + (2c\_2 + 2c\_1^2)z^2 + (2c\_3 + 4c\_1 c\_2 + 2c\_1^3)z^3 + \cdots + 1$$

and

$$\frac{zF'(z)}{G(z)} = \frac{1+v(w)}{1-v(w)} = 1 + 2d\_1w + (2d\_2 + 2d\_1^2)w^2 + (2d\_3 + 4d\_1d\_2 + 2d\_1^3)w^3 + \dotsb \dotsb$$

Equating coefficients in two above relations then gives

$$2a\mathfrak{z} - b\mathfrak{z} = 2c\mathfrak{z}\_1.\tag{8}$$

$$2a\_3 - b\_3 - 2a\_2b\_2 + b\_2^2 = 2c\_2 + 2c\_{1'}^2\tag{9}$$

$$2a\_4 - b\_4 - 2a\_2b\_3 + 2b\_2b\_3 - 3a\_3b\_2 + 2a\_2b\_2^2 - b\_2^3 = 2c\_3 + 4c\_1c\_2 + 2c\_1^3\tag{10}$$

and

$$-2a\_2 + b\_2 = 2d\_1,\tag{11}$$

$$a - 3a\_3 + b\_3 - 2a\_2b\_2 - b\_2^2 + 6a\_2^2 = 2d\_2 + 2d\_{1'}^2 \tag{12}$$

$$-4a\_4 + b\_4 - 2a\_2b\_3 - 3b\_2b\_3 - 3a\_3b\_2 + 2a\_2b\_2^2 + 2b\_2^3 - 20a\_2^3$$

$$2 + 20a\_2a\_3 + 6a\_2^2b\_2 = 2d\_3 + 4d\_1d\_2 + 2d\_1^3\tag{13}$$

respectively. From (8) and (11), we ge<sup>t</sup> that

$$c\_1 = -d\_{1'} \tag{14}$$

Also, according to the proof of [34] (Theorem), it is enough that we set 2*c*1, 2*c*2 + <sup>2</sup>*c*21, 2*c*3 + 4*c*1*c*2 + <sup>2</sup>*c*31 instead of *c*1, *c*2, *c*3, and 2*d*1, 2*d*2 + 2*d*21, 2*d*3 + 4*d*1*d*2 + 2*d*31 instead of *d*1, *d*2, *d*3 in relations (2.5)–(2.10) in [34], respectively. Thus we can write (2.20) in [34], as given below:

$$\begin{aligned} \left| a\_2 a\_4 - a\_3^2 \right| &= \left| \frac{1}{8} (b\_2 b\_4 - b\_3^2) + \frac{7}{72} b\_3^2 + \frac{2}{8} (b\_4 - b\_2 b\_3) c\_1 - \frac{10}{48} b\_2 b\_3 c\_1 \right| \\ &+ \frac{4}{24} \left( b\_3 - \frac{13}{3} b\_2^2 \right) c\_1^2 \\ &- \frac{7}{144} \left( b\_3 - \frac{19}{14} b\_2^2 \right) b\_2^2 - \frac{2}{9} \left( b\_3 - \frac{19}{16} b\_2^2 \right) (c\_2 - d\_2) \\ &- \frac{10}{32} b\_2 c\_1 \left[ 4c\_1^2 - \frac{8}{15} (c\_2 - d\_2) - \frac{13}{15} b\_2^2 \right] \\ &- \frac{2}{16} c\_1 \left[ 8c\_1^3 - \frac{4}{3} c\_1 (c\_2 - d\_2) - 2 (c\_3 - d\_3) - 4c\_1^3 - 4c\_1 (c\_2 + d\_2) \right] \\ &+ \frac{1}{16} b\_2 \left[ 2(c\_3 - d\_3) + 4c\_1^3 + 4c\_1 (c\_2 + d\_2) \right] - \frac{4}{36} (c\_2 - d\_2)^2 \Big| . \end{aligned} \tag{15}$$

According to Lemma 2 and (14), we find that

$$c\_2 - d\_2 = \left(1 - c\_1^2\right) \left(\mathbf{x} - y\right) \qquad \text{and} \qquad c\_2 + d\_2 = \left(1 - c\_1^2\right) \left(\mathbf{x} + y\right) \tag{16}$$

and

$$c\_3 = \left(1 - c\_1^2\right)\left(1 - \left|x\right|^2\right)s - c\_1\left(1 - c\_1^2\right)x^2 \quad \text{and} \quad d\_3 = \left(1 - d\_1^2\right)\left(1 - \left|y\right|^2\right)t - d\_1\left(1 - d\_1^2\right)y^2,$$
  $\text{and}$ 

where

$$c\_3 - d\_3 = (1 - c\_1^2) \left[ (1 - |x|^2)s - (1 - |y|^2)t \right] - c\_1 (1 - c\_1^2) (x^2 + y^2) \tag{17}$$

for some *x*, *y*, *s*, *t* with |*x*| ≤ 1, |*y*| ≤ 1, |*s*| ≤ 1 and |*t*| ≤ 1. Applying (16) and (17) in (15), it follows that

*a*2*a*4 − *<sup>a</sup>*<sup>23</sup> = 18 (*b*2*b*4 − *b*23) + 172 *b*23 + 28 (*b*4 − *b*2*b*3)*<sup>c</sup>*1 − 1048 *b*2*b*3*c*1 + 4 24 *b*3 − 134 *b*22 *c*21 − 7144 *b*3 − 1914 *b*22 *b*22 − 40 32 *<sup>b</sup>*2*<sup>c</sup>*31 + 2696 *<sup>b</sup>*32*<sup>c</sup>*1 − 1616 *c*41 + 816 *c*41 + 416 *<sup>b</sup>*2*<sup>c</sup>*31 + 2 1 − *c*21 (*x* − *y*), − 19 *b*3 − 1916 *b*22 + 896 *b*2*c*1 + 448 *c*21- + 2*c*1 1 − *c*21 (*x* + *y*), 416 *c*1 + 216 *b*2- + 2(1 − *c*21) &(1 − |*x*|<sup>2</sup>)*s* − (1 − <sup>|</sup>*y*|<sup>2</sup>)*t*' , 216 *c*1 + 116 *b*2- − <sup>2</sup>*c*1(<sup>1</sup> − *c*21)(*x*<sup>2</sup> + *y*2), 216 *c*1 + 116 *b*2- − 436 1 − *c*21 2 (*x* − *<sup>y</sup>*)<sup>2</sup> .

Since by Lemma 1, |*<sup>c</sup>*1| ≤ 1, we assume that *c*1 = *c* ∈ [0, 1]. So by utilizing the triangle inequality we have

 *a*2*a*4 − *a*23 ≤1 8 *<sup>b</sup>*2*b*4 − *<sup>b</sup>*23 + 172 *<sup>b</sup>*<sup>23</sup> + 28 |*b*4 − *b*2*b*3| *c* + 1048 *<sup>b</sup>*3 − 1310 *<sup>b</sup>*<sup>22</sup> *c* |*b*2| + 4 24 *b*3 − 134 *<sup>b</sup>*<sup>22</sup> *c*<sup>2</sup> + 7144 *<sup>b</sup>*3 − 1914 *b*<sup>22</sup> *b*<sup>22</sup> + −4032 + 416 <sup>|</sup>*b*2<sup>|</sup> *c*3 <sup>+</sup> −1616 + 816 *c*<sup>4</sup> + 2 1 − *c*2 , 19 *b*3 − 1916 *<sup>b</sup>*<sup>22</sup> + 896 |*b*2|*<sup>c</sup>* + 448 *c*2- (|*x*| + |*y*|) + 2*c* 1 − *c*2 , 416 *c* + 216 |*b*2|- (|*x*| + |*y*|) + 2(1 − *c*2), 216 *c* + 116 |*b*2|- &(1 − |*x*|<sup>2</sup>)+(<sup>1</sup> − <sup>|</sup>*y*|<sup>2</sup>)' + <sup>2</sup>*c*(<sup>1</sup> − *c*2), 216 *c* + 116 |*b*2|- <sup>|</sup>*x*|<sup>2</sup> + <sup>|</sup>*y*|<sup>2</sup> + 436 1 − *c*2 2 ((|*x*| + |*y*|)<sup>2</sup> =1 8 *<sup>b</sup>*2*b*4 − *<sup>b</sup>*23 + 172 *<sup>b</sup>*<sup>23</sup> + 28 |*b*4 − *b*2*b*3| *c* + 1048 *<sup>b</sup>*3 − 1310 *<sup>b</sup>*<sup>22</sup> *c* |*b*2| + 4 24 *b*3 − 134 *<sup>b</sup>*<sup>22</sup> *c*<sup>2</sup> + 7144 *<sup>b</sup>*3 − 1914 *<sup>b</sup>*<sup>22</sup> *<sup>b</sup>*<sup>22</sup> + |*b*2| *c*3 + 12 *c*4 + 4(1 − *c*2), 216 *c* + 116 |*b*2|- + 2, 19 *b*3 − 1916 *<sup>b</sup>*<sup>22</sup> + 896 |*b*2|*<sup>c</sup>* + 448 *c*2- + 2*c*, 416 *c* + 216 <sup>|</sup>*b*2<sup>|</sup>- 1 − *c*2 (|*x*| + |*y*|) + 2 216 *c* + 116 |*b*2| (*c* − 1)(1 − *c*2) <sup>|</sup>*x*|<sup>2</sup> + <sup>|</sup>*y*|<sup>2</sup> + 436 1 − *c*2 2 (|*x*| + |*y*|)<sup>2</sup> .

We now apply Lemmas 3–5 in order to deduce that 

$$\begin{aligned} & \left| a\_2 a\_4 - a\_3^2 \right| \\ \leq & \frac{1}{8} + \frac{1}{8} + \frac{2}{4}c + \frac{22}{24}c + \frac{40}{24}c^2 + \frac{17}{36} + 2c^3 + \frac{1}{2}c^4 + \frac{8}{16}(1 - c^2)(c + 1) \\ & + \left( 2\left[\frac{7}{36} + \frac{16}{96}c + \frac{4}{48}c^2\right] + 2c\left[\frac{4}{16}c + \frac{4}{16}\right] \right) \left(1 - c^2\right) (|x| + |y|) \\ & + \frac{4}{16}\left(c + 1\right)(c - 1)\left(1 - c^2\right)\left(|x|^2 + |y|^2\right) \\ & + \frac{4}{36}\left(1 - c^2\right)^2 \left(|x| + |y|\right)^2. \end{aligned}$$

Now, for *λ* = |*x*| ≤ 1 and *μ* = |*y*| ≤ 1, we obtain

$$\left| a\_2 a\_4 - a\_3^2 \right| \le I\_1 + (\lambda + \mu) I\_2 + (\lambda^2 + \mu^2) I\_3 + (\lambda + \mu)^2 I\_4 = L(\lambda, \mu)\_{\ast}$$

where

$$\begin{aligned} f\_1 &= f\_1(c) = \frac{26}{36} + \frac{34}{24}c + \frac{40}{24}c^2 + 2c^3 + \frac{1}{2}c^4 + \frac{8}{16}(1 - c^2)(c + 1) \ge 0 \\\ f\_2 &= f\_2(c) = \left(\frac{7}{18} + \frac{5}{6}c + \frac{4}{6}c^2\right)\left(1 - c^2\right) \ge 0 \\\ f\_3 &= f\_3(c) = -\frac{1}{4}(1 - c^2)^2 \le 0 \\\ f\_4 &= f\_4(c) = \frac{1}{9}\left(1 - c^2\right)^2 \ge 0. \end{aligned}$$

We now need to maximize the function *<sup>L</sup>*(*<sup>λ</sup>*, *μ*) on the closed square [0, 1] × [0, 1] for *c* ∈ [0, 1]. With regards to *<sup>L</sup>*(*<sup>λ</sup>*, *μ*) = *<sup>L</sup>*(*μ*, *<sup>λ</sup>*), it is sufficient to show that there exists the maximum of

$$H(\lambda) = L(\lambda, \lambda) = f\_1 + 2\lambda f\_2 + 2\lambda^2 (f\_3 + 2f\_4), \tag{18}$$

on *λ* ∈ [0, 1] according to *c* ∈ [0, 1]. We let *c* ∈ [0, 1]. Considering Equation (18) for 0 < *λ* < 1 and *J*3 + 2*J*4 < 0, we consider for critical point

$$\lambda\_0 = \frac{-l\_2}{2(l\_3 + 2l\_4)} = \frac{l\_2}{2k} = \frac{18\left(\frac{7}{18} + \frac{5}{6}c + \frac{4}{6}c^2\right)\left(1 - c^2\right)}{\left(1 - c^2\right)^2} = \frac{18\left(\frac{7}{18} + \frac{5}{6}c + \frac{4}{6}c^2\right)}{\left(1 - c^2\right)} > 1$$

for any fixed *c* ∈ [0, 1], where *k* = −(*J*<sup>3</sup> + <sup>2</sup>*J*4) > 0. Therefore, for *λ*0 = *J*22*k* > 1, it follows that *k* < *J*22 ≤ *J*2, and so *J*2 + *J*3 + 2*J*4 ≥ 0. So,

$$H(0) = l\_1 \le l\_1 + 2(l\_2 + l\_3 + 2l\_4) = H(1).$$

Therefore, it follows that

$$\max \left\{ H(\lambda) : \lambda \in [0, 1] \right\} = H(1) = l\_1 + 2l\_2 + 2l\_3 + 4l\_4.$$

Therefore, max *<sup>L</sup>*(*<sup>λ</sup>*, *μ*) = *L*(1, 1) on the boundary of the square. We define the real function *W* on (0, 1) by

$$\mathcal{W}(\mathfrak{c}) = L(1,1) = f\_1 + 2f\_2 + 2f\_3 + 4f\_4.$$

Now putting *J*1, *J*2, *J*3 and *J*4 in the function *W*, we have

$$W(c) = -\frac{8}{9}c^4 - \frac{1}{6}c^3 + \frac{11}{6}c^2 + \frac{43}{12}c + \frac{70}{36}.$$

By elementary calculations, we ge<sup>t</sup> that *<sup>W</sup>*(*c*) is an increasing function of *c*. Therefore, we obtain the maximum of *<sup>W</sup>*(*c*) on *c* = 1 and

$$\max \mathcal{W}(c) = \mathcal{W}(1) = \frac{227}{36}.$$

This completes the proof.

**Example 1.** *If we choose the functions*

$$f(z) = z + \frac{z^3}{2}, \quad g(z) = z - \frac{z^3}{3}.$$

*then will have*

$$F(w) = f^{-1}(w) = w - \frac{w^3}{2}, \quad G(w) = g^{-1}(w) = w + \frac{w^3}{3}$$

*and so these functions satisfy in Definition 1. Thus function f* ∈ Σ *is bi-close-to-convex, that is, f* ∈ CΣ *(see for more details, [33]). Therefore, Theorem 2 holds for f*(*z*) = *z* + *z*32 *.*

**Remark 2.** *The obtained bound for <sup>a</sup>*2*a*4 − *a*23 *in Theorem 2 is smaller than and more accurate the estimation given in Theorem 1.*
