Some properties of certain close-to-convex harmonic mappings

Lemma 2.1. ([23, p. 41]) For a function $p \in \mathcal{P}$ of the form (1.2), the sharp inequality $|p_n| \leq 2$ holds for each $n \geq 1$ . Equality holds for the function $$p(z) = \frac{1+z}{1-z}.$$

Lemma 2.2. ([24, Theorem 1]) Let $p \in \mathcal{P}$ be of the form (1.2) and $\mu \in \mathbb{C}$ . Then $$|p_n - \mu p_k p_{n-k}| \le 2 \max\{1, |2\mu - 1|\} \quad (1 \le k \le n - 1).$$ If $|2\mu - 1| \ge 1$ , then the inequality is sharp for the function $$p(z) = \frac{1+z}{1-z}$$ or its rotations. If $|2\mu - 1| < 1$ , then the inequality is sharp for $$p(z) = \frac{1+z^n}{1-z^n}$$ or its rotations.

Lemma 2.3. ([\[56\]](#page-16-1)) Let f = h + g ∈ M(α, ζ, n). Then the coefficients a^k (k ∈ N \ {1}) of h satisfy $$|a_k| \le \frac{1}{k!} \prod_{j=2}^k (j - 2\alpha) \ (k \in \mathbb{N} \setminus \{1\}). \tag{2.1}$$ Moreover, the coefficients b^k (k = n + 1, n + 2, · · · ; n ∈ N) of g satisfy $$|b_{n+1}| \le \frac{|\zeta|}{n+1}$$ and $|b_{k+n}| \le \frac{|\zeta|}{(k+n)(k-1)!} \prod_{j=2}^{k} (j-2\alpha) \ (k \in \mathbb{N} \setminus \{1\}; \ n \in \mathbb{N}).$ (2.2) The bounds are sharp for the extremal function given by $$f(z) = \int_0^z \frac{dt}{(1 - \delta t)^{2 - 2\alpha}} + \overline{\int_0^z \frac{\zeta t^n}{(1 - \delta t)^{2 - 2\alpha}} dt} \quad (|\delta| = 1; z \in \mathbb{D}).$$ (2.3)

Lemma 2.4. ([\[56\]](#page-16-1)) Let f ∈ M(α, ζ, n) with 0 ≤ α < 1 and 0 ≤ ζ < ¹ 2n−1 (n ∈ N). Then $$\Phi(r; \alpha, \zeta, n) \le |f(z)| \le \Psi(r; \alpha, \zeta, n) \quad (r = |z| < 1), \tag{2.4}$$ where $$\Phi(r;\alpha,\zeta,n) = \begin{cases} \log(1+r) - \frac{\zeta r^{n+1} {}_{2}F_{1}(1,\,n+1;\,n+2;\,-r)}{n+1} & (\alpha=1/2), \\ \frac{(1+r)^{2\alpha-1} - 1}{2\alpha-1} - \frac{\zeta r^{n+1} {}_{2}F_{1}(n+1,\,2-2\alpha;\,n+2;\,-r)}{n+1} & (\alpha\neq1/2), \end{cases}$$ and $$\Psi(r;\alpha,\zeta,n) = \begin{cases} -\log(1-r) + \frac{\zeta r^{n+1} {}_{2}F_{1}(1, n+1; n+2; r)}{n+1} & (\alpha = 1/2), \\ \frac{1 - (1-r)^{2\alpha-1}}{2\alpha - 1} + \frac{\zeta r^{n+1} {}_{2}F_{1}(n+1, 2-2\alpha; n+2; r)}{n+1} & (\alpha \neq 1/2). \end{cases}$$ All these bounds are sharp, the extremal function is fα,ζ,n = h^α + gα,ζ,n or its rotations, where $$f_{\alpha,\zeta,n}(z) = \begin{cases} -\log(1-z) + \frac{\overline{\zeta z^{n+1} {}_{2}F_{1}(1, n+1; n+2; z)}}{n+1} & (\alpha = 1/2), \\ \frac{1 - (1-z)^{2\alpha-1}}{2\alpha - 1} + \frac{\overline{\zeta z^{n+1} {}_{2}F_{1}(n+1, 2-2\alpha; n+2; z)}}{n+1} & (\alpha \neq 1/2). \end{cases}$$ $$(2.5)$$ The following two results are due to Ma and Minda [\[42\]](#page-15-3).

Lemma 2.5. Let f ∈ C(φ). Then zf00(z)/f⁰ (z) ≺ zK00(z)/K⁰ (z) and f 0 (z) ≺ K⁰ (z), where K is given by [\(1.4\)](#page-1-1).

Lemma 2.6. Assume that f ∈ C(φ) and |z| = r < 1. Then $$K'(-r) \le |f'(z)| \le K'(r),$$ (2.6) where K is given by [\(1.4\)](#page-1-1). Equality holds for some z 6= 0 if and only if f is a rotation of K.

Lemma 2.7. ([13]) Let $f(z) = \sum_{n=0}^{\infty} a_n z^n$ and $g(z) = \sum_{n=0}^{\infty} b_n z^n$ be two analytic functions in $\mathbb{D}$ and $g \prec f$ . Then $$\sum_{n=0}^{\infty} |b_n| r^n \le \sum_{n=0}^{\infty} |a_n| r^n \tag{2.7}$$ for $|z| = r \le 1/3$ . Remark 2.1. Lemma 2.7 continues to hold for quasi-subordination (cf. [5]). Moreover, the bound 1/3 is optimal as shown by [48, Lemma 1].

Theorem 3.1. Let $f \in \mathcal{M}(\alpha, \zeta, n)$ . Then $$|T_2(n)[h]| \le \left(\frac{1}{n!} \prod_{j=2}^n (j-2\alpha)\right)^2 + \left(\frac{1}{(n+1)!} \prod_{j=2}^{n+1} (j-2\alpha)\right)^2 \quad (n \in \mathbb{N} \setminus \{1\}), \quad (3.1)$$ and $$|T_2(n)[g]| \le \frac{1}{[(2n-1)(n+1)]^2}.$$ (3.2) The inequalities in (3.1) and (3.2) are sharp for the extremal function given by (2.3).

Corollary 3.1. Let $f \in \mathcal{M}(\alpha, \zeta, 2)$ . Then $$|T_2(2)[h]| \le \frac{2}{9}(1-\alpha)^2(2\alpha^2 - 6\alpha + 9),$$ (3.5) and $$|T_2(2)[g]| \le \frac{1}{81}. (3.6)$$ The inequalities in (3.5) and (3.6) are sharp for the extremal function given by (2.3) with n = 2.

Theorem 3.2. Let $f \in \mathcal{M}(\alpha, \zeta, 1)$ . Then $$|T_3(1)[h]| \le \begin{cases} \frac{1}{9} \left( 8\alpha^4 - 34\alpha^3 + 71\alpha^2 - 72\alpha + 36 \right) & \left( -\frac{1}{2} \le \alpha \le \frac{1}{2} \right), \\ \frac{1}{9} \left( -2\alpha^3 + 25\alpha^2 - 44\alpha + 30 \right) & \left( \frac{1}{2} \le \alpha < 1 \right), \end{cases}$$ (3.7) and $$|T_3(1)[g]| \le \frac{1}{3}(1-\alpha).$$ (3.8) The inequality in (3.7) is sharp for the function h given by (3.4), and the inequality in (3.8) is sharp for the function g defined by $$g(z) = \int_0^z \frac{\zeta t}{(1 - \delta t)^{2 - 2\alpha}} dt \quad (|\delta| = 1; |\zeta| \le 1; z \in \mathbb{D}).$$ (3.9)

Theorem 3.3. Let $f \in \mathcal{M}(\alpha, \zeta, 2)$ . Then $$|T_3(2)[h]| \le \begin{cases} \frac{1}{108} \left(1 - \alpha\right)^3 (2\alpha^2 - 7\alpha + 12)(10\alpha^2 - 27\alpha + 36\right) & \left(-\frac{1}{2} \le \alpha \le \frac{1}{7}\right), \\ \frac{5}{108} \left(1 - \alpha\right)^3 (2\alpha^2 - 7\alpha + 12)(2\alpha^2 - 4\alpha + 7) & \left(\frac{1}{7} \le \alpha < 1\right), \end{cases}$$ (3.16) and $$|T_3(2)[g]| = |2b_3^2b_4| \le \frac{1}{243}(1-\alpha).$$ (3.17) The inequality in (3.16) is sharp for the function h given by (3.4), and the inequality in (3.17) is sharp for the function g defined by $$g(z) = \int_0^z \frac{\zeta t^2}{(1 - \delta t)^{2 - 2\alpha}} dt \quad \left( |\delta| = 1; \, |\zeta| \le \frac{1}{3}; \, z \in \mathbb{D} \right). \tag{3.18}$$ •

Proposition 4.1. Let $f \in \mathcal{HC}_n(\phi)$ . Then $$L(\zeta, n, r) \le |f(z)| \le R(\zeta, n, r), \tag{4.1}$$ where $$L(\zeta, n, r) = -K(-r) - |\zeta| \int_0^r t^n K'(-t) dt, \tag{4.2}$$ and $$R(\zeta, n, r) = K(r) + |\zeta| \int_{0}^{r} t^{n} K'(t) dt.$$ (4.3) The bounds are sharp for the extremal function $f_{\zeta} = h_{\zeta} + \overline{g_{\zeta}}$ with $h_{\zeta} = K$ , where K satisfies (1.4) or its rotations and $g_{\zeta}$ satisfies $g'_{\zeta} = \zeta z^n h'_{\zeta}$ .

Proposition 4.2. Let $f \in \mathcal{HC}_n(\phi)$ and $S_r$ be the area of the image $f(\mathbb{D}_r)$ ( $\mathbb{D}_r := \{z \in \mathbb{C} : |z| < r \leq 1\}$ ). Then $$2\pi \int_0^r t \left(1 - |\zeta|^2 t^{2n}\right) (K'(-t))^2 dt \le S_r \le 2\pi \int_0^r t \left(1 - |\zeta|^2 t^{2n}\right) (K'(t))^2 dt. \tag{4.10}$$

Theorem 4.1. Let f ∈ HCn(φ). Then the majorant series of f satisfies the inequality $$|z| + \sum_{n=2}^{\infty} (|a_n| + |b_n|)|z|^n \le d(f(0), \partial f(\mathbb{D}))$$ (4.13) for |z| = r ≤ min{1/3, r^f }, where r^f is the smallest positive root in (0, 1) of $$L(\zeta, n, 1) = M_K(r) + |\zeta| \int_0^r t^n M_{K'}(t) dt,$$ and L(ζ, n, 1) is given by [\(4.2\)](#page-7-3) with r = 1. Proof. Let f = h + g ∈ HCn(φ). Since h ∈ C(φ), from Lemma [2.5,](#page-3-4) we know that $$h' \prec K'. \tag{4.14}$$ Let K(z) = z + P∞ n=2 knz n . In view of Lemma [2.7](#page-4-0) and [\(4.14\)](#page-9-0), we have $$1 + \sum_{n=2}^{\infty} n|a_n|r^{n-1} = M_{h'}(r) \le M_{K'}(r) = 1 + \sum_{n=2}^{\infty} n|k_n|r^{n-1}$$ (4.15) for |z| = r ≤ 1/3. By integrating [\(4.15\)](#page-9-1) with respect to r from 0 to r, we get $$M_h(r) = r + \sum_{n=2}^{\infty} |a_n| r^n \le r + \sum_{n=2}^{\infty} |k_n| r^n = M_K(r) \quad (r \le 1/3).$$ (4.16) From the definition of HCn(φ), we know that $$g'(z) = \zeta z^n h'(z).$$ This relationship along with [\(4.15\)](#page-9-1) yields $$\sum_{n=2}^{\infty} n|b_n|r^{n-1} = M_{g'}(r) = |\zeta|r^n M_{h'}(r) \le |\zeta|r^n M_{K'}(r) \quad (r \le 1/3). \tag{4.17}$$ By integrating [\(4.17\)](#page-9-2) with respect to r from 0 to r, it follows that $$M_g(r) = \sum_{n=2}^{\infty} |b_n| r^n \le |\zeta| \int_0^r t^n M_{K'}(t) dt \quad (r \le 1/3).$$ (4.18) Therefore, for |z| = r ≤ 1/3, from [\(4.16\)](#page-9-3) and [\(4.18\)](#page-9-4), we obtain $$M_f(r) = |z| + \sum_{n=2}^{\infty} (|a_n| + |b_n|)r^n \le M_K(r) + |\zeta| \int_0^r t^n M_{K'}(t)dt = R_{\mathcal{C}}(n, r).$$ (4.19) In view of [\(4.1\)](#page-7-4), it is evident that the Euclidean distance between f(0) and the boundary of f(D) is given by $$d(f(0), \partial f(\mathbb{D})) = \liminf_{|z| \to 1} |f(z) - f(0)| \ge L(\zeta, n, 1). \tag{4.20}$$ We note that RC(n, r) ≤ L(ζ, n, 1) whenever r ≤ r^f , where r^f is the smallest positive root of RC(n, r) = L(ζ, n, 1) in (0, 1). Let $$H_1(n,r) = R_{\mathcal{C}}(n,r) - L(\zeta,n,1).$$ Then H1(n, r) is a continuous function in [0, 1]. Since $$M_K(r) \ge K(r) > -K(-r),$$ it follows that $$H_{1}(n,1) = R_{\mathcal{C}}(n,1) - L(\zeta, n, 1)$$ $$= M_{K}(1) + K(-1) + |\zeta| \int_{0}^{r} t^{n} \left( M_{K'}(t) + K'(t) \right) dt$$ $$\geq K(1) + K(-1) + |\zeta| \int_{0}^{r} t^{n} \left( M_{K'}(t) + K'(t) \right) dt > 0.$$ (4.21) On the other hand, $$H_1(n,0) = -L(\zeta, n, 1) = K(-1)(1 - |\zeta|) + n|\zeta| \int_0^1 t^{n-1} K(-t) dt < 0.$$ (4.22) Therefore, H¹ has a root in (0, 1). Let r^f be the smallest root of H¹ in (0, 1). Then RC(n, r) ≤ L(ζ, n, 1) for r ≤ r^f . Now, in view of the inequalities [\(4.19\)](#page-9-5) and [\(4.20\)](#page-9-6) with the relationship RC(n, r) ≤ L(ζ, n, 1) for r ≤ r^f , we obtain $$|z| + \sum_{n=2}^{\infty} (|a_n| + |b_n|)r^n \le d(f(0), \partial f(\mathbb{D}))$$ for $$|z| = r \le \min\{1/3, r_f\}.$$ For a particular choice of φ in Theorem [4.1,](#page-9-7) we get the following result.

Corollary 4.1. Let f ∈ M(α, ζ, n) with 0 ≤ α < 1 and 0 ≤ ζ < 1/(2n − 1). Then the inequality [\(4.13\)](#page-9-8) holds for |z| = r ≤ r^f , where r^f is the smallest root in (0, 1) of $$F_n(r) := R(\alpha, \zeta, n, r) - L(\alpha, \zeta, n, 1) = 0.$$ The radius r^f is sharp. Proof. From Lemma [2.4,](#page-3-5) the Euclidean distance between f(0) and the boundary of f(D) shows that $$d(f(0), \partial f(\mathbb{D})) = \liminf_{|z| \to 1} |f(z) - f(0)| \ge L(\alpha, \zeta, n, 1).$$ (4.23) We note that r^f is the root of the equation R(α, ζ, n, r) = L(α, ζ, n, 1) in (0, 1). The existence of the root is ensured by the relation R(α, ζ, n, 1) > L(α, ζ, n, 1) with [\(2.4\)](#page-3-6). For 0 < r ≤ r^f , it is evident that R(α, ζ, n, r) ≤ L(α, ζ, n, 1). In view of Lemma [2.3](#page-3-1) and [\(4.23\)](#page-10-0), for |z| = r ≤ r^f , we have $$|z| + \sum_{n=2}^{\infty} (|a_n| + |b_n|)|z|^n \le r_f + (|a_2| + |b_2|)r_f^2 + \sum_{n=3}^{\infty} (|a_n| + |b_n|)r_f^n$$ $$= R(\alpha, \zeta, n, r_f) \le L(\alpha, \zeta, n, 1) \le d(f(0), \partial f(\mathbb{D})).$$ To show the sharpness of the radius r^f , we consider the function f = fα,ζ,n, which is defined in Lemma [2.4.](#page-3-5) We see that fα,ζ,n belongs to M(α, ζ, n). Since the left side of the growth inequality in Lemma [2.4](#page-3-5) holds for f = fα,ζ,n or its rotations, we have d(f(0), ∂f(D)) = L(α, ζ, n, 1). Therefore, the function f = fα,ζ,n for |z| = r^f gives $$|z| + \sum_{n=2}^{\infty} (|a_n| + |b_n|)|z|^n = r_f + (|a_2| + |b_2|)r_f^2 + \sum_{n=3}^{\infty} (|a_n| + |b_n|)r_f^n$$ $$= R(\alpha, \zeta, n, r_f) = L(\alpha, \zeta, n, 1) = d(f(0), \partial f(\mathbb{D})),$$ which reveals that the radius r^f is the best possible. The roots $r_f$ of $F_n(r) = 0$ for different values of $\alpha$ , $\zeta$ and n have been shown in Table 1, Table 2 and Figure 2, respectively.  FIGURE 2. The graphs of $F_n(r)$ , respectively, for $\alpha = 0.5$ and $\alpha = 0.9$ when n = 1, 2, 3, 4, 5. | 1 | n | 1 | 2 | 3 | 4 | 5 | 10 | 100 | 1000 | |---|--------|----------|----------|----------|----------|----------|----------|----------|----------| | ( | ٠<br>• | 1/2 | 1/4 | 1/6 | 1/8 | 1/10 | 1/20 | 1/200 | 1/2000 | | 1 | $r_f$ | 0.386555 | 0.468176 | 0.486196 | 0.492459 | 0.495252 | 0.498809 | 0.499988 | 0.500000 | Table 1. The roots $r_f$ of $F_n(r) = 0$ for different values of $\zeta$ when $\alpha = 0.5$ . | n | 1 | 2 | 3 | 4 | 5 | 10 | 100 | 1000 | |-------|----------|----------|----------|----------|----------|----------|----------|----------| | ζ | 1/2 | 1/4 | 1/6 | 1/8 | 1/10 | 1/20 | 1/200 | 1/2000 | | $r_f$ | 0.567721 | 0.731273 | 0.774894 | 0.792253 | 0.800709 | 0.812036 | 0.815292 | 0.815323 | Table 2. The roots $r_f$ of $F_n(r) = 0$ for different values of $\zeta$ when $\alpha = 0.9$ . Remark 4.1. For n=1 and $|\zeta| \leq 1$ , $r_f$ can be found in [9]. For $\alpha=0.5$ , when $n\to\infty$ , the sharp radius is 0.500000. For $\alpha=0.9$ , when $n\to\infty$ , the sharp radius is 0.815323. For n=1, when $\alpha\to 1$ , the sharp radius is 0.645751. Now, we give an improved version of Bohr inequality for the class $\mathcal{HC}_n(\phi)$ . By adding area quantity $S_r/(2\pi)$ with the majorant series of $f \in \mathcal{HC}_n(\phi)$ , the sum is still less than $d(f(0), \partial f(\mathbb{D}))$ for some radius $r \leq \min\{1/3, \widetilde{r}_f\} < 1$ . Note that the additional term such as $S_r/(2\pi)$ to the majorant sum was first mooted by Kayumov and Ponnusamy [31] to refine and improve the Bohr inequality. This variation of Bohr inequality was proved for harmonic mappings in [25]. Subsequently, several extensions were made by many authors (cf. [39]).

Theorem 4.2. Let $f \in \mathcal{HC}_n(\phi)$ and $S_r$ be the area of the image $f(\mathbb{D}_r)$ . Then the inequality $$M_f(r) + \frac{S_r}{2\pi} \le d(f(0), \partial f(\mathbb{D}))$$ holds for $|z| = r \leq \min\{1/3, \widetilde{r}_f\}$ , where $\widetilde{r}_f$ is the smallest positive root in (0,1) of $$L(\zeta, n, 1) = M_K(r) + |\zeta| \int_0^r t^n M_{K'}(t) dt + \int_0^r t \left(1 - |\zeta|^2 t^{2n}\right) (K'(t))^2 dt,$$ and $L(\zeta, n, 1)$ is given by (4.2) with r = 1.

Definition 1.1. A harmonic mapping f = h+g ∈ H is said to be in the class M(α, ζ, n) if h ∈ K(α), for some α ∈ [−1/2, 1), given by [\(1.3\)](#page-0-0) and g satisfies the condition $$g'(z) = \zeta z^n h'(z) \quad \left(\zeta \in \mathbb{C} \text{ with } |\zeta| \le \frac{1}{2n-1}; n \in \mathbb{N} := \{1, 2, 3, \dots\}\right). \tag{1.5}$$ For n = 1, α = −1/2 and |ζ| = 1, the class M(−1/2, ζ, 1) was introduced by Bharanedhar and Ponnusamy [\[12\]](#page-13-0). For n = 1, the class M(α, ζ, 1) was studied in [\[9,](#page-13-1) [52\]](#page-15-6). In 2020, Allu and Halder [\[9\]](#page-13-1) introduced and investigated the following subclass HC(φ) of close-to-convex harmonic mappings.

Definition 1.2. For ζ ∈ C with |ζ| ≤ 1, let HC(φ) denote the class of harmonic mappings f = h + g in D of the form [\(1.1\)](#page-0-1), whose analytic part h belongs to C(φ) and h 0 (0) 6= 0, along with the condition g 0 (z) = ζzh⁰ (z). Motivated essentially by the classes M(α, ζ, n) and HC(φ), we define a new subclass HCn(φ) of close-to-convex harmonic mappings as follows:

Definition 1.3. A harmonic mapping f = h + g ∈ H is said to be in the class HCn(φ) if h ∈ C(φ) and g satisfies the condition [\(1.5\)](#page-1-0). In 2019, Sun et al. [\[51\]](#page-15-7) investigated upper bounds of the third Hankel determinants for the class M(α, 1, 1) of close-to-convex harmonic mappings. In recent years, the Toeplitz determinants and Hankel determinants of functions in the class S or its subclasses have attracted many researchers' attention (see [\[11,](#page-13-2) [17,](#page-13-3) [19,](#page-14-3) [20,](#page-14-4) [28,](#page-14-5) [29,](#page-14-6) [33–](#page-14-7)[36\]](#page-14-8)). Among them, the symmetric Toeplitz determinant |Tq(n)| for subclasses of S with small values of n and q, are investigated by [\[2,](#page-13-4) [7,](#page-13-5) [10,](#page-13-6) [49,](#page-15-8) [54,](#page-15-9) [58\]](#page-16-3). The symmetric Toeplitz determinant $T_q(n)$ for analytic functions f is defined as follows: $$T_q(n)[f] := \begin{vmatrix} a_n & a_{n+1} & \cdots & a_{n+q-1} \\ a_{n+1} & a_n & \cdots & a_{n+q-2} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n+q-1} & a_{n+q-2} & \cdots & a_n \end{vmatrix},$$ where $n, q \in \mathbb{N}$ and $a_1 = 1$ . In particular, for functions in starlike and convex classes, $T_2(2)[f]$ , $T_3(1)[f]$ and $T_3(2)[f]$ were studied by Ali et al. [7]. Let $\mathcal{B}$ be the class of analytic functions f in $\mathbb{D}$ such that |f(z)| < 1 for all $z \in \mathbb{D}$ , and let $\mathcal{B}_0 = \{f \in \mathcal{B} : f(0) = 0\}$ . In 1914, Bohr [16] proved that if $f \in \mathcal{B}$ is of the form $f(z) = \sum_{n=0}^{\infty} a_n z^n$ , then the majorant series $M_f(r) = \sum_{n=0}^{\infty} |a_n| |z|^n$ of f satisfies $$M_{f_0}(r) = \sum_{n=1}^{\infty} |a_n| |z|^n \le 1 - |a_0| = d(f(0), \partial f(\mathbb{D}))$$ (1.6) for all $z \in \mathbb{D}$ with $|z| = r \le 1/3$ , where $f_0(z) = f(z) - f(0)$ . Bohr actually obtained the inequality (1.6) for $|z| \le 1/6$ . Moreover, Wiener, Riesz and Schur, independently, established the Bohr inequality (1.6) for $|z| \le 1/3$ (known as Bohr radius for the class $\mathcal{B}$ ) and proved that 1/3 is the best possible. The Bohr phenomenon was reappeared in the 1990s due to Dixon [21]. Furthermore, Boas and Khavinson [15] found bounds for Bohr's radius in any complete Reinhard domains. Other works one can see [3, 4, 14, 44, 45]. In recent years, Bohr inequality and Bohr radius have become an active research field in geometric function theory (see [6, 8, 27, 31, 38, 40, 43]). Furthermore, initiated by the work of [32], the Bohr's phenomenon for the complex-valued harmonic mappings have been widely studied (see [1, 9, 25, 26, 30, 41]). In this paper, we aim at determining the sharp estimates for Toeplitz determinants of the class $\mathcal{M}(\alpha, \zeta, n)$ . Moreover, we will derive an improved version of Bohr's inequalities for the class $\mathcal{HC}_n(\phi)$ .