Bohr inequalities via proper combinations for a certain class of close-to-convex harmonic mappings

Theorem 1.1. [1] If $g(z) = \sum_{n=0}^{\infty} b_n z^n \in S(f)$ and $f(z) = \sum_{n=0}^{\infty} a_n z^n$ is univalent, then $$\sum_{n=0}^{\infty} |b_n z^n| \le d(f(0), \partial \Omega) \text{ for } |z| = \rho_0^* \le 3 - \sqrt{8} \approx 0.17157,$$ where $\rho_0^*$ is sharp for Koebe function $f(z) = z/(1-z)^2$ . Let $\mathcal{H}$ be the class of all complex-valued harmonic functions $f = h + \bar{g}$ defined on the unit disk $\mathbb{D}$ , where h and g are analytic in $\mathbb{D}$ with the normalization h(0) =h'(0) - 1 = 0 and g(0) = 0. Let $\mathcal{H}_0$ be defined by $\mathcal{H}_0 = \{f = h + \bar{g} \in \mathcal{H} : g'(0) = 0\}$ . Therefore, each $f = h + \bar{g} \in \mathcal{H}_0$ has the following representation <span id="page-2-1"></span>(1.4) $$f(z) = h(z) + \overline{g(z)} = \sum_{n=1}^{\infty} a_n z^n + \sum_{n=1}^{\infty} b_n z^n = z + \sum_{n=2}^{\infty} a_n z^n + \sum_{n=2}^{\infty} b_n z^n,$$ where $a_1 = 1$ and $b_1 = 0$ , since $a_1$ and $b_1$ have been appeared in later results and corresponding proofs. For a class of harmonic mappings, the Bohr radius and its corresponding Bohr inequality analogue in view of distance formulation, similar to that for analytic functions, are defined as follows.

Theorem 2.1. [37] Suppose that $f(z) = \sum_{n=0}^{\infty} a_n z^n \in \mathcal{B}$ . For $N, m \in \mathbb{N}$ , $$|f(z^m)| + B_N(f,r) \le 1$$ for $r \le R_{m,N}$ , where $R_{m,N}$ is the positive root of the equation $2(1+r^m)r^N - (1-r)(1-r^m) = 0$ . The number $R_{m,N}$ cannot be improved. Moreover, $\lim_{N\to\infty} R_{m,N} = 1$ and $\lim_{m\to\infty} R_{m,N} = A_N$ , where $A_N$ is the positive root of the equation $2r^N = 1-r$ . In addition, for $N, m \in \mathbb{N}$ , $$|f(z^m)|^2 + B_N(f,r) \le 1$$ for $r \le R'_{m,N}$ , where $R'_{m,N}$ is the positive root of the equation $(1+r^m)r^N - (1-r)(1-r^m) = 0$ . The number $R'_{m,N}$ cannot be improved. Moreover, $\lim_{N \to \infty} R'_{m,N} = 1$ and $\lim_{m \to \infty} R'_{m,N} = A'_N$ , where $A'_N$ is the positive root of the equation $r^N = 1 - r$ . 2.1. Refined versions of the Bohr inequality for the class $\mathcal{B}$ . For an extension of the results discussed above, we refer to the recent article by Ponnusamy and Vijayakumar [56]. In comparison of $\sum_{n=0}^{\infty} |a_n| r^n$ with another functional often considered in function theory, namely $\sum_{n=0}^{\infty} |a_n|^2 r^{2n}$ which is abbreviated as $||f||_r^2$ . As refinement of the classical Bohr inequality, for p=1,2, we define $$\mathcal{A}_{p,f}(r) := |a_0|^p + B_1(f,r) + A(f_0,r),$$ where $f_0(z) := f(z) - a_0$ . Recently, Ponnusamy et al. [57] have obtained the following result as a refinement of the classical Bohr inequality.

Theorem 2.2. [57] Suppose that $f \in \mathcal{B}$ with $f(z) = \sum_{n=0}^{\infty} a_n z^n$ and $f_0(z) := f(z) - a_0$ . Then $\mathcal{A}_{1,f}(r) \leq 1$ and the numbers $1/(1+|a_0|)$ and $1/(2+|a_0|)$ cannot be improved. Further, $\mathcal{A}_{2,f}(r) \leq 1$ and the numbers $1/(1+|a_0|)$ and 1/2 cannot be improved. In what follows, $\lfloor x \rfloor$ denotes the largest integer no more than x, where x is a real number. Recently, Liu et al. [48] have obtained the following refined version of the Bohr-Rogosinski inequality also.

Theorem 2.3. [48] Suppose that $f \in \mathcal{B}$ and $f(z) = \sum_{n=0}^{\infty} a_n z^n$ . For $N \in \mathbb{N}$ , let $t = \lfloor (N-1)/2 \rfloor$ . Then (2.1) $$|f(z)| + B_N(f,r) + sgn(t) \sum_{n=1}^t |a_n|^2 \frac{r^N}{1-r} + \left(\frac{1}{1+|a_0|} + \frac{r}{1-r}\right) \sum_{n=t+1}^\infty |a_n|^2 r^{2n} \le 1$$ for $|z| = r \le R_N$ , where $R_N$ is the positive root of the equation $2(1+r)r^N - (1-r)^2 = 0$ . The radius $R_N$ is best possible. Moreover, (2.2) $$|f(z)|^2 + B_N(f,r) + sgn(t) \sum_{n=1}^t |a_n|^2 \frac{r^N}{1-r} r + \left(\frac{1}{1+|a_0|} + \frac{r}{1-r}\right) \sum_{n=t+1}^\infty |a_n|^2 r^{2n} \le 1$$ for $|z| = r \le R'_N$ , where $R'_N$ is positive root of the equation $(1+r)r^N - (1-r)^2 = 0$ . The radius $R'_N$ is best possible. Remark 2.1. In particular if N=1, then it is easy to see that $R_1=\sqrt{5}-2$ and $R_1'=1/3$ . For p = 1, 2, for $f \in \mathcal{B}$ , we define the functionals $$\mathcal{B}_{p,f}(z,r) := |f(z)|^p + B_1(f,r) + A(f_0,r).$$ In the context of Theorems 2.1 and 2.3, the following result is obtained in [48] showing that the two constants can be improved for any individual function in $\mathcal{B}$ .

Theorem 2.4. [48] Suppose that $f \in \mathcal{B}$ and $f(z) = \sum_{n=0}^{\infty} a_n z^n$ . Then $\mathcal{B}_{1,f}(z,r) \leq 1$ for $|z| = r \leq r_{a_0} = 2/(3 + |a_0| + \sqrt{5}(1 + |a_0|))$ . The radius $r_{a_0}$ is best possible and $r_{a_0} > \sqrt{5} - 2$ . Moreover, $\mathcal{B}_{2,f}(z,r) \leq 1$ for $|z| = r \leq r'_{a_0}$ , where $r'_{a_0}$ is the unique positive root of the equation $$(1 - |a_0|^3) r^3 - (1 + 2|a_0|)r^2 - 2r + 1 = 0.$$ The radius $r'_{a_0}$ is best possible. Further, $1/3 < r'_{a_0} < 1/(2 + |a_0|)$ . For recent developments on the Bohr-Rogosinski inequalities, we refer to the articles [2, 14, 17, 22] and references therein. However, we see that the quantities $1/(1+|a_0|)+r/(1-r)$ and $\sum_{n=1}^{\infty}|a_n|^2r^{2n}$ for analytic functions in $\mathcal{B}$ are analogous to $1/(1+|a_0|+|b_0|)+r/(1-r)=1+r/(1-r)$ (as $a_0=0=b_0$ ) and $\sum_{n=2}^{\infty}(|a_n|+|b_n|)^2r^{2n}$ , respectively, for harmonic functions given in (1.4). The observations leads us to establish several harmonic analogues of the refined Bohr inequalities for certain class of harmonic mappings. 2.2. Improved Bohr inequalities for the class $\mathcal{B}$ . Let f be holomorphic in $\mathbb{D}$ , and for 0 < r < 1, let $\mathbb{D}_r = \{z \in \mathbb{C} : |z| < r\}$ . The quantity $S_r := S_r(f)$ denoted as the planar integral has the following integral representation $$S_r := \int_{\mathbb{D}_r} |f'(z)|^2 dA(z).$$ Note that if $f(z) = \sum_{n=0}^{\infty} a_n z^n$ , then $S_r := \pi \sum_{n=1}^{\infty} n |a_n|^2 r^{2n}$ (see [27]). It is well-known that if f is a univalent function, then $S_r$ is the area of the image of the sub-disk $\mathbb{D}_r := \{z \in \mathbb{D} : |z| < r\}$ under the mapping f (see [27]). The quantity $S_r$ is non-negative and has been used extensively to study the improved versions of Bohr inequality and Bohr radius for the class of analytic functions (see e.g. [4,34,36,37]) as well as harmonic mappings (see e.g. [6,7]). In the following, we recall some of the interesting results in this regard. Recently, Kayumov and Ponnusamy [36] have obtained the following improved version of Bohr's inequality in terms of $S_r$ .

Theorem 2.5. [36] Suppose that $f \in \mathcal{B}$ with $f(z) = \sum_{n=0}^{\infty} a_n z^n$ . Then <span id="page-5-0"></span>(2.3) $$B_0(f,r) + \frac{16}{9} \left( \frac{S_r}{\pi} \right) \le 1 \quad \text{for} \quad r \le \frac{1}{3}$$ and the numbers 1/3, 16/9 cannot be improved. Moreover, <span id="page-5-1"></span>(2.4) $$|a_0|^2 + B_1(f,r) + \frac{9}{8} \left(\frac{S_r}{\pi}\right) \le 1 \text{ for } r \le \frac{1}{2}$$ and the numbers 1/2, 9/8 cannot be improved. Moreover, Ismagilov et al. [34] investigated on the inequalities (2.3) and (2.4) of Theorem 2.5 and obtained the following improved results.

Theorem 2.6. [34] Suppose that $f \in \mathcal{B}$ with $f(z) = \sum_{n=0}^{\infty} a_n z^n$ . Then (2.5) $$B_0(f,r) + \frac{16}{9} \left(\frac{S_r}{\pi}\right) + \lambda \left(\frac{S_r}{\pi}\right)^2 \le 1 \text{ for } r \le \frac{1}{3},$$ where $$\lambda = \frac{4(486 - 261a - 324a^2 + 2a^3 + 30a^4 + 3a^5)}{81(1+a)^3(3-5a)} = 18.6095...$$ and $a \approx 0.567284$ is the unique root in (0,1) of the equation $$\phi(t) = -405 + 473r + 402r^2 + 38r^3 + 3r^4 + r^5.$$ The equality is achieved for the function $f_a(z) := (a-z)/(1-az)$ .

Theorem 2.7. [34] Suppose that $f \in \mathcal{B}$ with $f(z) = \sum_{n=0}^{\infty} a_n z^n$ . Then (2.6) $$|f(z)|^2 + B_1(f,r) + \frac{16}{9} \left(\frac{S_r}{\pi}\right) + \lambda \left(\frac{S_r}{\pi}\right)^2 \le 1 \text{ for } r \le \frac{1}{3}$$ where $$\lambda = \frac{-81 + 1044a + 54a^2 - 116a^3 - 5a^4}{162(a+1)^2(2a-1)} = 16.4618...$$ and $a \approx 0.537869$ is the unique root in (0,1) of the equation $$-513 + 910t + 80t^2 + 2t^3 + t^4 = 0.$$ The equality is achieved for the function $f_a$ . We also recall here one result proved by Ismagilov et al. [34] which is an improved version of the classical Bohr inequality, where $|a_0|$ is replaced by |f(z)|.

Theorem 2.8. [34] Suppose that $f \in \mathcal{B}$ with $f(z) = \sum_{n=0}^{\infty} a_n z^n$ . Then (2.7) $$|f(z)| + B_1(f,r) + p\left(\frac{S_r}{\pi}\right) \le 1 \text{ for } |z| = r \le \sqrt{5} - 2,$$ where the constants $r_0 = \sqrt{5} - 2 \approx 0.236068$ and $p = 2(\sqrt{5} - 1)$ are sharp. Since, the quantity $S_r/\pi$ is instrumental in the study of improved Bohr inequalities for its use to achieve the sharpness, it is observed in [35] the following inequality (2.8) $$\frac{S_r}{\pi - S_r} \le \frac{r^2 (1 - |a_0|^2)^2}{(1 - r^2)(1 - r^2|a_0|^4)}.$$ In view of the bound of the quantity $S_r/(\pi - S_r)$ , Ismagilov et al. [35] have investigated on Theorem 2.5 and obtained the following sharp result.

Theorem 2.9. [35] Suppose that $f(z) = \sum_{n=0}^{\infty} a_n z^n \in \mathcal{B}$ . Then $$B_0(f,r) + \frac{16}{9} \left( \frac{S_r}{\pi - S_r} \right) \le 1 \text{ for } r \le \frac{1}{3},$$ and the number 16/9 cannot be improved. Moreover, $$|a_0|^2 + B_1(f,r) + \frac{9}{8} \left( \frac{S_r}{\pi - S_r} \right) \le 1 \quad \text{for} \quad r \le \frac{1}{2},$$ and the number 9/8 cannot be improved. In recent developments, an refined version of the inequality (2.3) mentioned in [36] has been put forth. This improvement involves the substitution of the coefficient $|a_0|$ with |f(z)|, while employing the context of $S_r/(\pi - S_r)$ instead of $S_r/\pi$ , as detailed below.

Theorem 2.10. [34] Suppose that $f \in \mathcal{B}$ with $f(z) = \sum_{n=0}^{\infty} a_n z^n$ . Then $$(2.9) |f(z)| + B_1(f,r) + 2(\sqrt{5} - 1)\left(\frac{S_r}{\pi - S_r}\right) \le 1 \text{for } |z| = r \le \sqrt{5} - 2,$$ where the constants $r_0 = \sqrt{5} - 2 \approx 0.236068$ and $2(\sqrt{5} - 1)$ are sharp. The motivation of the study Bohr phenomenon via proper combination for harmonic mappings is based on the discussions above and observation of the following remark concerning improved Bohr inequality. <span id="page-7-0"></span>Remark 2.2. Ismagilov et al. remarked in Theorem 2.6 that for any function $F:[0,\infty)\to[0,\infty)$ such that F(t)>0 for t>0, there exists analytic function $f:\mathbb{D}\to\mathbb{D}$ for which the inequality $$\sum_{n=0}^{\infty} |a_n| r^n + \frac{16}{9} \left( \frac{S_r}{\pi} \right) + \lambda \left( \frac{S_r}{\pi} \right)^2 + F(S_r) \le 1 \text{ for } r \le \frac{1}{3}$$ is false, there $\lambda$ is given in Theorem 2.6. Considering Remark 2.2, it is noteworthy to observe that augmenting a non-negative quantity with the Bohr inequality does not yield the desired inequality for the class $\mathcal{B}$ . This observation motivates us to pose the following question for further study on the topic. <span id="page-7-1"></span>Question 1. What can be deduced about the refine harmonic analogue of Theorem 2.3 by incorporating a general power of $|f(z)|^p$ for $p \in \mathbb{N}$ and adding higher power terms of $S_r/\pi$ for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ ? Consequently, can we obtain the harmonic analogue of the Theorems 2.4 to 2.8 for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ ? In the past few years, there has been a noteworthy investigation into substituting $S_r/(\pi - S_r)$ for $S_r/\pi$ in mazorent series and exploring the determination of the Bohr radius in this modified framework, making it a compelling subject in geometric function theory. Motivated from the work of Ismagilov et al. [35], it is natural to raise the following question also. <span id="page-7-2"></span>Question 2. What can be deduced about the refine harmonic counterpart of Theorem 2.3 by incorporating a general power of $|f(z)|^p$ for $p \in \mathbb{N}$ and adding higher power terms of $S_r/(\pi - S_r)$ instead of the term $S_r/\pi$ for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ ? Moreover, can we obtain the harmonic analogue of the Theorems 2.9 and 2.10 for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ ? The organization of this paper is as follows. In Section 3, we state the main results to address Question 1 and Question 2 for the close to convex class $\mathcal{P}^0_{\mathcal{H}}(M)$ of harmonic mappings. Furthermore, we shall discuss some consequences of the main results showing Bohr radii by graphs and tables for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ for different values of M. The section 4 contains the proof of our main results.

Lemma 3.1. Let $f = h + \overline{g} \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4) for some M > 0. Then for $n \geq 2$ , (i) $|a_n| + |b_n| \leq \frac{2M}{n(n-1)}$ ; (ii) $||a_n| - |b_n|| \leq \frac{2M}{n(n-1)}$ ; (iii) $|a_n| \leq \frac{2M}{n(n-1)}$ . The inequalities are sharp with extremal function $f_M$ given by $f'_M(z) = 1 - 2M \ln(1-z)$ .

Lemma 3.2. Let $f \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4). Then (3.1) $$|z| + 2M \sum_{n=2}^{\infty} \frac{(-1)^{n-1}|z|^n}{n(n-1)} \le |f(z)| \le |z| + 2M \sum_{n=2}^{\infty} \frac{|z|^n}{n(n-1)}.$$ Both inequalities are sharp for the function $f_M$ given by $f_M(z) = z + 2M \sum_{n=2}^{\infty} \frac{z^n}{n(n-1)}$ . The following is our first main result, which addresses the Question 1 and estimates the generalized refine Bohr-Rogosinski inequality for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ .

Theorem 3.1. Let $f \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4) and $0 \leq M < 1/(2(\ln 4 - 1))$ . Then for $\mu$ , $\lambda, \lambda_j \in \mathbb{R}_{\geq 0} := \{x \in \mathbb{R} : x \geq 0\}$ , where $j = 1, 2, \dots, q$ and $N \geq 5$ , we have $G^{f,\lambda_1,\lambda_2,\dots,\lambda_q}_{\beta,\mu,\lambda,m,N}(r) \leq d\left(f(0),\partial\mathbb{D}\right)$ for $|z| = r \leq R^{\lambda_1,\lambda_2,\dots,\lambda_q}_{\beta,\mu,\lambda,m,N}(M)$ , where $R^{\lambda_1,\lambda_2,\dots,\lambda_q}_{\beta,\mu,\lambda,m,N}(M)$ is the unique root in (0,1) of the equation $\Phi^{\lambda_1,\lambda_2,\dots,\lambda_q}_{\beta,\mu,\lambda,m,N}(r) = 0$ , where $$\Phi_{\beta,\mu,\lambda,m,N}^{\lambda_1,\lambda_2,\cdots\lambda_q}(r) := \beta \left( G_M(r) \right)^m + 2M \left( r + (1-r)\ln(1-r) - \sum_{n=2}^{N-1} \frac{r^n}{n(n-1)} \right) + \mu \Phi_{M,t}^N(r)$$ $$(3.2) \qquad + 4M^2 \lambda \left( 1 + \frac{r}{1-r} \right) G_{2,t}(r) + P_q(F_M(r)) - 1 - 2M \left( 1 - 2\ln 2 \right)$$ and <span id="page-9-3"></span> $$\begin{cases} G_M(r) := r + 2M \sum_{n=2}^{\infty} \frac{r^n}{n(n-1)} = r + 2M \left( r + (1-r) \ln(1-r) \right); \\ \Phi_{M,t}^N(r) := \frac{4M^2 r^N}{1-r} sgn(t) \sum_{n=1}^t \frac{1}{n^2(n-1)^2}; \\ G_t(r) := \left( r^2 + 1 \right) \text{Li}_2\left( r^2 \right) + 2 \left( r^2 - 1 \right) \ln\left( 1 - r^2 \right) - 3r^2 - \sum_{n=2}^t \frac{r^{2n}}{n^2(n-1)^2}; \\ F_M(r) := r^2 + 4M \left[ r^2 Li_2(r^2) - (r^2 + (1-r^2) \log(1-r^2)) \right]. \end{cases}$$ The constant $R_{\beta,\mu,\lambda,m,N}^{\lambda_1,\lambda_2,\cdots,\lambda_q}(M)$ is best possible. For a sake of simplification, we have used the following assumptions in order to present certain consequences of Theorem 3.1 here: $$\begin{cases} \mathcal{J}_1^M(r) := r^2 + 4M^2[(1+r^2)\operatorname{Li}_2(r^2) - 2(1-r^2)\ln(1-r^2) - 3r^2]; \\ \mathcal{J}_2^M(m,r) := (G_M(r))^m - 1 - 2M(1-2\ln 2); \\ \mathcal{J}_3(r) := r + (1-r)\ln(1-r). \end{cases}$$ In fact, the following corollary of Theorem 3.1, we obtain the following result in which the cases for N = 1, 2, 3, 4 are discussed completely. Corollary 3.1. Let $f \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4) and $0 \leq M < 1/(2(\ln 4 - 1))$ , $\mu, \lambda \in \mathbb{R}_{>0}.$ (i) If N = 1, then $G_{\beta,\mu,\lambda,m,1}^{f,\lambda_1,\lambda_2,\cdots\lambda_q}(r) \leq d(f(0),\partial\mathbb{D})$ for $|z| = r \leq R_{\beta,\mu,\lambda,m,1}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ , where $R_{2,\mu,\lambda,m,1}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ is the unique root in (0,1) of the equation $$r + 2M\mathcal{J}_3(r) + \mathcal{J}_2^M(m,r) + \lambda \left(1 + \frac{r}{1-r}\right)\mathcal{J}_1^M(r) + P_q(F_M(r)) = 0.$$ (ii) If N=2, then $G_{\beta,\mu,\lambda,m,2}^{f,\lambda_1,\lambda_2,\cdots\lambda_q}(r) \leq d(f(0),\partial\mathbb{D})$ for $|z|=r \leq R_{\beta,\mu,\lambda,m,2}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ , where $R_{\beta,\mu,\lambda,m,2}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ is the unique root in (0,1) of the equation $$2M\mathcal{J}_3(r) + \mathcal{J}_2^M(m,r) + \lambda \left(1 + \frac{r}{1-r}\right) \mathcal{J}_1^M(r) + P_q(F_M(r)) = 0.$$ (iii) If N=3, then $G_{\beta,\mu,\lambda,m,3}^{f,\lambda_1,\lambda_2,\cdots\lambda_q}(r) \leq d\left(f(0),\partial\mathbb{D}\right)$ for $|z|=r \leq R_{\beta,\mu,\lambda,m,3}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ , where $R_{\beta,\mu,\lambda,m,3}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ is the unique root in (0,1) of the equation $$2M\left(\mathcal{J}_{3}(r) - \frac{r^{2}}{2}\right) + \mathcal{J}_{2}^{M}(m,r) + \mu \frac{r^{3}}{1-r} + \lambda \left(1 + \frac{r}{1-r}\right) \left(\mathcal{J}_{1}^{M}(r) - r^{2}\right) + P_{q}(F_{M}(r)) = 0.$$ (iv) If N=4, then $G_{\beta,\mu,\lambda,m,4}^{f,\lambda_1,\lambda_2,\cdots\lambda_q}(r) \leq d\left(f(0),\partial\mathbb{D}\right)$ for $|z|=r \leq R_{\beta,\mu,\lambda,m,4}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ , where $R_{\beta,\mu,\lambda,m,4}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ is the unique root in (0,1) of the equation $$2M\left(\mathcal{J}_{3}(r) - \frac{r^{2}}{2} - \frac{r^{3}}{6}\right) + \mathcal{J}_{2}^{M}(m,r) + \mu \frac{r^{4}}{1-r} + \lambda \left(1 + \frac{r}{1-r}\right) \left(\mathcal{J}_{1}^{M}(r) - r^{2}\right) + P_{q}(F_{M}(r)) = 0.$$ The constants $R_{\beta,\mu,\lambda,m,1}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ , $R_{\beta,\mu,\lambda,m,2}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ , $R_{\beta,\mu,\lambda,m,3}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ and $R_{\beta,\mu,\lambda,m,4}^{\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ all are best possible. For m=1,2, we define a functional $G_m^f$ $$G_m^f := |f(z)|^m + r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n + \left(1 + \frac{r}{1-r}\right) \sum_{n=1}^{\infty} (|a_n| + |b_n|)^2 r^{2n}.$$ The following corollary is a harmonic analogue of the Theorem 2.4 for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ . A part of Question 1 is thus answered.  FIGURE 1. The figure exhibits the roots $R_{1,\mu,1,1,1}^{0,0,\cdots 0}(M)$ of equation (3.3) and the roots $R_{1,\mu,1,2,1}^{0,0,\cdots 0}(M)$ of (3.4) respectively for different values of M as defined in Table 1 Corollary 3.2. Let $f \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4) and $0 \leq M < 1/(2(\ln 4 - 1))$ . Then (i) $G_1^f \leq d(f(0), \partial \mathbb{D})$ for $|z| = r \leq R_{1,\mu,1,1,1}^{0,0,\cdots 0}(M)$ , where $R_{1,\mu,1,1,1}^{0,0,\cdots 0}(M)$ is the unique root in (0,1) of the equation <span id="page-11-0"></span> $$(3.3) \quad 2r - 1 + 2M(2r - 1 + \ln 4 + (1 - r)\ln(1 - r)^{2}) + \left(1 + \frac{r}{1 - r}\right)\mathcal{J}_{1}^{M}(r) = 0$$ where the constant $R_{1,\mu,1,1,1}^{0,0,\cdots 0}(M)$ is best possible. (ii) $G_2^f \leq d(f(0), \partial \mathbb{D})$ for $|z| = r \leq R_{1,\mu,1,2,1}^{0,0,\cdots 0}(M)$ , where $R_{1,\mu,1,2,1}^{0,0,\cdots 0}(M)$ is the unique root in (0,1) of the equation <span id="page-11-1"></span>(3.4) $$(G_M(r))^2 + r - 1 + 2M((1-r)(\ln(1-r) - 1) + \ln 4 + ) + \left(1 + \frac{r}{1-r}\right)\mathcal{J}_1^M(r) = 0$$ where the constant $R_{1,\mu,1,2,1}^{0,0,\cdots 0}(M)$ is best possible. | M | | 0.14 | 0.28 | 0.42 | 0.56 | 0.70 | 0.84 | 0.98 | 1.12 | 1.26 | |----------------------------------|----|--------|--------|--------|--------|--------|--------|--------|--------|--------| | $R_{1,\mu,1,1,1}^{0,0,\cdots 0}$ | M) | 0.3398 | 0.2993 | 0.2601 | 0.2217 | 0.1834 | 0.1446 | 0.1042 | 0.0610 | 0.0129 | | $R_{1,\mu,1,2,1}^{0,0,\cdots 0}$ | M) | 0.4041 | 0.3642 | 0.3250 | 0.2858 | 0.2455 | 0.2029 | 0.1556 | 0.0996 | 0.0245 | <span id="page-11-2"></span>TABLE 1. The table exhibits the roots of the equations (3.3) and (3.4) for different values of M. We obtain the next corollary as a harmonic analogue of the Theorem 2.5 for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ . Thus a part of Question 1 is answered. Corollary 3.3. Let $f \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4) and $0 \leq M < 1/(2(\ln 4 - 1))$ . Then we have  FIGURE 2. The figure exhibits the roots $R_{0,\mu,0,m,1}^{\frac{16}{9},0,\cdots 0}(M)$ of equation (3.5) and the roots $R_{0,\mu,0,m,1}^{\frac{9}{8},0,\cdots 0}(M)$ of (3.6) respectively for different values of M as defined in Table 2 (a) $$r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n + \frac{16}{9} \left( \frac{S_r}{\pi} \right) \le d(f(0), \partial \mathbb{D})$$ for $|z| = r \le R_{0,\mu,0,m,1}^{\frac{16}{9},0,\cdots 0}(M)$ , where $R_{0,\mu,0,m,1}^{\frac{16}{9},0,\cdots 0}(M)$ is the unique root in (0,1) of the equation <span id="page-12-0"></span>(3.5) $$r - 1 + 2M((1-r)(\log(1-r) - 1) + \ln 4) + \frac{16}{9}(F_M(r)) = 0.$$ The constant $R_{0,\mu,0,1,1}^{\frac{16}{9},0,\cdots 0}(M)$ is best possible. Moreover (b) $$r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n + \frac{9}{8} \left( \frac{S_r}{\pi} \right) \le d \left( f(0), \partial \mathbb{D} \right)$$ for $|z| = r \le R_{0,\mu,0,m,1}^{\frac{9}{8},0,\cdots 0}(M)$ , where $R_{0,\mu,0,m,1}^{\frac{9}{8},0,\cdots 0}(M)$ is the unique root in (0,1) of the equation <span id="page-12-1"></span>(3.6) $$r - 1 + 2M((1-r)(\log(1-r) - 1) + \ln 4) + \frac{9}{8}(F_M(r)) = 0.$$ The constant $R_{0,\mu,0,2,1}^{\frac{9}{8},0,\cdots 0}(M)$ is best possible. | M | 0.14 | 0.28 | 0.42 | 0.56 | 0.70 | 0.84 | 0.98 | 1.12 | 1.26 | |-------------------|--------|--------|--------|--------|--------|--------|--------|--------|--------| | Ι Ο, μο, Ο, πο, Σ | 0.4658 | 0.4140 | 0.3642 | 0.3158 | 0.2675 | 0.2178 | 0.1643 | 0.1030 | 0.0246 | | 9 0 0 | 0.5271 | 0.4625 | 0.4026 | 0.3459 | 0.2904 | 0.2342 | 0.1746 | 0.1077 | 0.0250 | <span id="page-12-2"></span>TABLE 2. The table exhibits the roots of the equations (3.5) and (3.6) for different values of M.  FIGURE 3. The figure exhibits the roots of equation (3.7) and roots of the equation (3.8) as defined in Table 3 and Table 4 respectively. The following corollary is a harmonic analogue of the Theorem 2.8 for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ . A part of Question 1 is thus answered. Corollary 3.4. Let $f \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4) and $0 \leq M < 1/(2(\ln 4 - 1))$ . Then we have $$|f(z)| + r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n + 2(\sqrt{5} - 1) \left(\frac{S_r}{\pi}\right) \le d(f(0), \partial \mathbb{D})$$ for $|z| = r \le R_{1,\mu,0,1,1}^{2(\sqrt{5}-1),0,\cdots 0}(M)$ , where $R_{1,\mu,0,1,1}^{2(\sqrt{5}-1),0,\cdots 0}(M)$ is the unique root in (0,1) of the equation <span id="page-13-0"></span> $$(3.7) r - 1 + 2M(r - 1 + \ln 4 + (1 - r)\ln(1 - r)) + 2(\sqrt{5} - 1)(F_M(r)) = 0.$$ The constant $R_{1,\mu,0,1,1}^{2(\sqrt{5}-1),0,\cdots 0}(M)$ is best possible. | M | 0.14 | 0.28 | 0.42 | 0.56 | 0.70 | 0.84 | 0.98 | 1.12 | 1.26 | |-------------------------------------------------|--------|--------|--------|--------|--------|--------|--------|--------|--------| | $R_{1,\mu,0,1,1}^{2(\sqrt{5}-1),0,\cdots 0}(M)$ | 0.3108 | 0.2743 | 0.2392 | 0.2050 | 0.1708 | 0.1358 | 0.0991 | 0.0590 | 0.0128 | <span id="page-13-1"></span>TABLE 3. The table exhibits the roots of the equation (3.7) for different values of M. The following corollary is a harmonic analogue of the Theorem 2.7 for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ . Consequently, a part of Question 1 is answered. Corollary 3.5. Let $f \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4) and $0 \leq M < 1/(2(\ln 4 - 1))$ . Then we have $$|f(z)|^2 + r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n + \frac{16}{9} \left(\frac{S_r}{\pi}\right) + \lambda_2 \left(\frac{S_r}{\pi}\right)^2 \le d(f(0), \partial \mathbb{D})$$ Bohr inequalities via proper combinations for certain class of close-to-convex harmonic mappings 15 for $|z| = r \le R_{1,\mu,0,2,1}^{16/9,\lambda_2,0,\cdots 0}(M)$ , where $R_{1,\mu,0,2,1}^{16/9,\lambda_2,0,\cdots 0}(M)$ is the unique root in (0,1) of the equation <span id="page-14-0"></span> $$(G_M(r))^2 + r - 1 + 2M(r - 1 + \ln 4 + (1 - r)\ln(1 - r)) + \frac{16}{9}(F_M(r)) + \lambda_2(F_M(r))^2 = 0,$$ where $\lambda_2 = \lambda$ is as in Theorem 2.7. The constant $R_{1,\mu,0,2,1}^{16/9,\lambda_2,0,\cdots 0}(M)$ is best possible. | M | 0.14 | 0.28 | 0.42 | 0.56 | 0.70 | 0.84 | 0.98 | 1.12 | 1.26 | |--------------------------------------------------|--------|--------|--------|--------|--------|--------|--------|--------|--------| | $R_{1,\mu,0,2,1}^{16/9,\lambda_2,0,\cdots 0}(M)$ | 0.3358 | 0.3095 | 0.2818 | 0.2525 | 0.2211 | 0.1862 | 0.1456 | 0.0951 | 0.0241 | <span id="page-14-1"></span>TABLE 4. The table exhibits the roots of the equation (3.8) for different values of M, where $\lambda_2 = \lambda$ is as in Theorem 2.7. Inspired by the idea of considering the quantity in Theorem 2.9, our next aim is to explore sharp Bohr inequalities for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ . Henceforth, we define the following functional (3.9) $$\mathcal{S}_{\beta,\mu,\lambda,m,N}^{f,\lambda_1,\lambda_2,\cdots\lambda_q}(r) := H_{\beta,\mu,\lambda,m,N}^f(r) + P_q\left(\frac{S_r}{\pi - S_r}\right).$$ To address the Question 2 and serve our purpose, we obtain the following result for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ .

Theorem 3.2. Let $f \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4) and $0 \leq M < 1/(2(\ln 4 - 1))$ . Then for $\mu$ , $\lambda, \lambda_j \in \mathbb{R}_{\geq 0} := \{x \in \mathbb{R} : x \geq 0\}$ , where $j = 1, 2, \dots, q$ and $N \geq 5$ , we have $$\mathcal{S}_{\beta,\mu,\lambda,m,N}^{f,\lambda_1,\lambda_2,\cdots\lambda_q}(r) \leq d\left(f(0),\partial\mathbb{D}\right) \text{ for } |z| = r \leq R_{\beta,\mu,\lambda,m,N}^{*,\lambda_1,\lambda_2,\cdots\lambda_q}(M)$$ where $R^{,\lambda_1,\lambda_2,\cdots\lambda_q}_{\beta,\mu,\lambda,m,N}(M)$ is the unique root of the equation $\Phi^{,\lambda_1,\lambda_2,\cdots\lambda_q}_{\beta,\mu,\lambda,m,N}(r)=0$ , where $$\Phi_{\beta,\mu,\lambda,m,N}^{*,\lambda_1,\lambda_2,\cdots\lambda_q}(r) := \beta \left( G_M(r) \right)^m + 2M \left( r + (1-r)\ln(1-r) - \sum_{n=2}^{N-1} \frac{r^n}{n(n-1)} \right)$$ $$+ \mu \Phi_{M,t}^N(r) + 4M^2 \lambda \left( 1 + \frac{r}{1-r} \right) G_{2,t}(r) + P_q \left( \frac{F_M(r)}{1-F_M(r)} \right)$$ $$- 1 - 2M \left( 1 - 2\ln 2 \right).$$ The constant $R_{\beta,\mu,\lambda,m,N}^{*,\lambda_1,\lambda_2,\cdots\lambda_q}(M)$ is best possible. As a corollary of Theorem 3.2, we obtain the following result in which the cases for N = 1, 2, 3, 4 are discussed completely. Corollary 3.6. Let $f \in \mathcal{P}^0_{\mathcal{H}}(M)$ be given by (1.4) and $0 \leq M < 1/(2(\ln 4 - 1))$ , $\mu, \lambda \in \mathbb{R}_{\geq 0}$ . (i) If N = 1, then S f,λ1,λ2,···λ^q β,µ,λ,m,1 (r) ≤ d (f(0), ∂D) for |z| = r ≤ R ∗,λ1,λ2,···λ^q β,µ,λ,m,1 (M), where R ∗,λ1,λ2,···λ^q β,µ,λ,m,1 (M) is the unique root in (0, 1) of the equation $$r + 2M\mathcal{J}_3(r) + \mathcal{J}_2^M(m,r) + \lambda \left(1 + \frac{r}{1-r}\right)\mathcal{J}_1^M(r) + P_q\left(\frac{F_M(r)}{1-F_M(r)}\right) = 0.$$ (ii) If N = 2, then S f,λ1,λ2,···λ^q β,µ,λ,m,2 (r) ≤ d (f(0), ∂D) for |z| = r ≤ R ∗,λ1,λ2,···λ^q β,µ,λ,m,2 (M), where R ∗,λ1,λ2,···λ^q β,µ,λ,m,2 (M) is the unique root in (0, 1) of the equation $$2M\mathcal{J}_{3}(r) + \mathcal{J}_{2}^{M}(m,r) + \lambda \left(1 + \frac{r}{1-r}\right)\mathcal{J}_{1}^{M}(r) + P_{q}\left(\frac{F_{M}(r)}{1 - F_{M}(r)}\right) = 0.$$ (iii) If N = 3, then S f,λ1,λ2,···λ^q β,µ,λ,m,3 (r) ≤ d (f(0), ∂D) for |z| = r ≤ R ∗,λ1,λ2,···λ^q β,µ,λ,m,3 (M), where R ∗,λ1,λ2,···λ^q β,µ,λ,m,3 (M) is the unique root in (0, 1) of the equation $$2M\left(\mathcal{J}_{3}(r) - \frac{r^{2}}{2}\right) + \mathcal{J}_{2}^{M}(m,r) + \mu \frac{r^{3}}{1-r} + \lambda \left(1 + \frac{r}{1-r}\right) \left(\mathcal{J}_{1}^{M}(r) - r^{2}\right) + P_{q}\left(\frac{F_{M}(r)}{1 - F_{M}(r)}\right) = 0.$$ (iv) If N = 4, then S f,λ1,λ2,···λ^q β,µ,λ,m,N (r) ≤ d (f(0), ∂D) for |z| = r ≤ R ∗,λ1,λ2,···λ^q β,µ,λ,m,4 (M), where R ∗,λ1,λ2,···λ^q β,µ,λ,m,4 (M) is the unique root in (0, 1) of the equation $$2M\left(\mathcal{J}_{3}(r) - \frac{r^{2}}{2} - \frac{r^{3}}{6}\right) + \mathcal{J}_{2}^{M}(m,r) + \mu \frac{r^{4}}{1-r} + \lambda \left(1 + \frac{r}{1-r}\right) \left(\mathcal{J}_{1}^{M}(r) - r^{2}\right) + P_{q}\left(\frac{F_{M}(r)}{1 - F_{M}(r)}\right) = 0.$$ The constants R ∗,λ1,λ2,···λ^q β,µ,λ,m,1 (M), R ∗,λ1,λ2,···λ^q β,µ,λ,m,2 (M), R ∗,λ1,λ2,···λ^q β,µ,λ,m,3 (M) and R ∗,λ1,λ2,···λ^q β,µ,λ,m,4 (M) all are best possible. Remark 3.2. For particular choice of β = 1 and λ^j = 0 ∀ j = 1, 2, · · · q and µ, λ ∈ R^≥⁰ in Theorem [3.1](#page-9-0) and [3.2,](#page-14-2) then Theorem [3.1](#page-9-0) and [3.2](#page-14-2) coincides with [\[3,](#page-23-13) Theorem 2.2]. Hence [\[3,](#page-23-13) Theorem 2.2] becomes a particular case of the Theorem 3.1 and 3.2. The following corollary partially addresses the Question [2,](#page-7-2) while also serving as a harmonic counterpart to Theorem [2.9](#page-6-1) for the class P 0 ^H(M). Corollary 3.7. Let f ∈ P⁰ ^H(M) be given by [\(1.4\)](#page-2-1) and 0 ≤ M < 1/(2(ln 4 − 1)). Then we have (a) $$r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n + \frac{16}{9} \left( \frac{S_r}{\pi - S_r} \right) \le d(f(0), \partial \mathbb{D})$$  FIGURE 4. The figure exhibits the roots of equation (3.10) and the roots of equation (3.11) as defined in Table 5. for $|z| = r \le R_{0,\mu,0,m,1}^{,16/9,0,\cdots 0}(M)$ , where $R_{0,\mu,0,m,1}^{,16/9,0,\cdots 0}(M)$ is the unique root in (0.1) of the equation <span id="page-16-0"></span>(3.10) $$r - 1 + 2M((1-r)(\log(1-r) - 1) + \ln 4) + \frac{16}{9} \left(\frac{F_M(r)}{1 - F_M(r)}\right) = 0.$$ The constant $R_{0,\mu,0,m,1}^{*,16/9,0,\cdots 0}(M)$ is best possible. Moreover, we have (b) $$r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n + \frac{9}{8} \left( \frac{S_r}{\pi - S_r} \right) \le d(f(0), \partial \mathbb{D})$$ for $|z| = r \le R_{0,\mu,0,m,1}^{,9/8,0,\cdots 0}(M)$ , where $R_{0,\mu,0,m,1}^{,9/8,0,\cdots 0}(M)$ is the unique root in (0,1) <span id="page-16-1"></span> $$(3.11) r - 1 + 2M((1-r)(\log(1-r) - 1) + \ln 4) + \frac{9}{8} \left(\frac{F_M(r)}{1 - F_M(r)}\right) = 0.$$ The constant $R_{0,\mu,0,m,1}^{*,9/8,0,\cdots 0}(M)$ is best possible. | M | 0.14 | 0.28 | 0.42 | 0.56 | 0.70 | 0.84 | 0.98 | 1.12 | 1.26 | |------------------------------------------|--------|--------|--------|--------|--------|--------|--------|--------|--------| | $R_{0,\mu,0,m,1}^{*,16/9,0,\cdots 0}(M)$ | 0.4368 | 0.3947 | 0.3518 | 0.3083 | 0.2633 | 0.2158 | 0.1635 | 0.1028 | 0.0246 | | $R_{0,\mu,0,m,1}^{*,9/8,0,\cdots 0}(M)$ | 0.4902 | 0.4394 | 0.3886 | 0.3377 | 0.2861 | 0.2323 | 0.1740 | 0.1076 | 0.0250 | <span id="page-16-2"></span>TABLE 5. The table exhibits the roots of the equations (3.5) and (3.6) for different values of M. The following corollary of Theorem 3.2 is a harmonic analogue of the Theorem 2.10 for the class $\mathcal{P}^0_{\mathcal{H}}(M)$ . Thus a part of Question 2 is answered.  Figure 5. The figure exhibits the roots of equation [\(3.12\)](#page-17-0) as defined in Table [6](#page-17-1) . Corollary 3.8. Let f ∈ P⁰ ^H(M) be given by [\(1.4\)](#page-2-1) and 0 ≤ M < 1/(2(ln 4 − 1)). Then we have $$|f(z)| + r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n + 2(\sqrt{5} - 1) \left(\frac{S_r}{\pi - S_r}\right) \le d(f(0), \partial \mathbb{D})$$ for |z| = r ≤ R ∗,2(^√ 5−1),0,···0 1,µ,0,1,1 (M), where R ∗,2(^√ 5−1),0,···0 1,µ,0,1,1 (M) is the unique root in (0, 1) of the equation <span id="page-17-0"></span>(3.12) $$r - 1 + 2M(r - 1 + \ln 4 + (1 - r)\ln(1 - r)) + 2(\sqrt{5} - 1)\left(\frac{F_M(r)}{1 - F_M(r)}\right) = 0.$$ The constant R ∗,2(^√ 5−1),0,···0 1,µ,0,1,1 (M) is best possible. | M | 0.14 | 0.28 | 0.42 | 0.56 | 0.70 | 0.84 | 0.98 | 1.12 | 1.26 | |-----------------------------------------------|--------|--------|--------|--------|--------|--------|--------|--------|--------| | ∗,2(√<br>5−1),0,···0<br>R<br>(M)<br>1,µ,0,1,1 | 0.3045 | 0.2704 | 0.2369 | 0.2037 | 0.1701 | 0.1355 | 0.0990 | 0.0590 | 0.0128 | <span id="page-17-1"></span>Table 6. The table exhibits the roots of the equations [\(3.12\)](#page-17-0) for different values of M.

Definition 1.1. [32] Let $A \subset \hat{\mathbb{C}}$ be a set, a function $f: A \to \mathbb{C}$ is said to be harmonic in an open set $\Omega$ provided $\Omega \subset A$ , f is continuous in $\Omega$ and f is twice continuously differentiable in $\Omega \setminus \{\infty\}$ and satisfies the Laplace equation $$\frac{\partial^2 f(z)}{\partial x^2} + \frac{\partial^2 f(z)}{\partial y^2} = 0, \ z = x + iy \in \Omega \setminus \{\infty\}.$$ In particular, f is said to be harmonic provided A is an open set and f is harmonic in A. Harmonic mappings are instrumental because of its applications in many Science and Engineering branches. To be specific, methods of harmonic mappings have been applied to study and solve the fluid flow problems (see [8,25]). For example, in 2012, Aleman and Constantin [8] established a connection between harmonic mappings and ideal fluid flows. In fact, Aleman and Constantin have developed an ingenious technique to solve the incompressible two dimensional Euler equations in terms of univalent harmonic mappings (see [25] for details). 1.2. Bohr inequality for harmonic mappings. Let $\mathcal{H}(\Omega)$ be the class of complexvalued functions harmonic in $\Omega$ . It is well-known that functions f in the class $\mathcal{H}(\Omega)$ has the following representation $f = h + \overline{g}$ , where h and g both are analytic functions in $\Omega$ . The famous Lewy's theorem (see [45]) states that a harmonic mapping $f = h + \overline{g}$ is locally univalent on $\Omega$ if, and only if, the determinant $|J_f(z)|$ of its Jacobian matrix $J_f(z)$ does not vanish on $\Omega$ , where $$|J_f(z)| := |f_z(z)|^2 - |f_{\bar{z}}(z)|^2 = |h'(z)|^2 - |g'(z)|^2 \neq 0.$$ In view of this result, a locally univalent harmonic mapping is sense-preserving if $|J_f(z)| > 0$ and sense-reversing if $|J_f(z)| < 0$ in $\Omega$ . For detailed information about the harmonic mappings, we refer the reader to [24, 28]. In [41], Kayumov et al. first established the harmonic extension of the classical Bohr theorem, since then investigating on the Bohr-type inequalities for certain class of harmonic mappings becomes an interesting topic of research in geometric function theory. <span id="page-2-0"></span>We see that (1.1) can be written as (1.3) $$d\left(\sum_{n=0}^{\infty} |a_n z^n|, |a_0|\right) = \sum_{n=1}^{\infty} |a_n z^n| \le 1 - |f(0)| = d(f(0), \partial(\mathbb{D})).$$ More generally, a class $\mathcal{F}$ of analytic functions $f(z) = \sum_{n=0}^{\infty} a_n z^n$ mapping $\mathbb{D}$ into a domain $\Omega$ is said to satisfy a Bohr phenomenon if an inequality of type (1.3) holds uniformly in $|z| \leq \rho_0$ , where $0 < \rho_0 \leq 1$ for functions in the class $\mathcal{F}$ . Similar definition makes sense for classes of harmonic mappings (see [41]) also. In view of the distance formulation of the Bohr inequality as mentioned in (1.3), Abu-Muhanna (see [1]) have established the following result for subordination class S(f) in case of when f is univalent function showing that the corresponding inequality is sharp.

Definition 1.2. Let $f \in \mathcal{H}_0$ be given by (1.4). In the study of it is the Bohr phenomenon is to find a constant $r_f \in (0,1]$ such that the inequality $$r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n \le d(f(0), \partial\Omega) \text{ for } |z| = r \le r_f.$$ The largest such radius $r_f$ is called the Bohr radius for the class $\mathcal{H}_0$ . Before stating the main result of the paper, we need some background of Bohr phenomenon for the class $\mathcal{B}$ of all analytic self-maps on $\mathbb{D}$ . In the section, we discuss in details several refined and improved Bohr inequalities established in recent years.