The sharp refined Bohr inequalities for a subclass of close-to-convex harmonic mappings

Theorem 1.3. [31] Suppose that $f(z) = \sum_{m=0}^{\infty} a_m z^m$ is analytic in the unit disk $\mathbb{D}$ and |f(z)| < 1 in $\mathbb{D}$ . Then $$|f(z)| + \sum_{m=N}^{\infty} |a_m| r^m \le 1, \quad \text{for } r \le R_N,$$ where $R_N$ is the positive root of the equation $2(1+r)r^N - (1-r^2) = 0$ . The radius $R_N$ is the best possible. Moreover, $$|f(z)|^2 + \sum_{m=N}^{\infty} |a_m| r^m \le 1, \quad \text{for } r \le R'_N,$$ where $R'_N$ is the positive root of the equation $(1+r)r^N - (1-r^2) = 0$ . The radius $R'_N$ is the best possible. For more results on Bohr-Rogosinski inequality, we refer to [8, 9, 10, 13, 29, 44] and references therein. Let $\lfloor x \rfloor$ denotes the largest integer no more than x, where x is the real number. A refined version of Bohr-Rogosinski inequality was proved by Liu et al. [36]

Theorem 1.4. [36] Suppose that $f \in \mathfrak{A}$ and $f(z) = \sum_{n=1}^{\infty} a_n z^n$ . For $N \in \mathbb{N}$ , let $t = \lfloor (N-1)/2 \rfloor$ . Then $$|f(z)| + \sum_{m=N}^{\infty} |a_m| r^m + sgn(t) \sum_{m=1}^{t} |a_m|^2 \frac{r^N}{1-r} + \left(\frac{1}{1+a_0} + \frac{r}{1-r}\right) \sum_{m=t+1}^{\infty} (|a_m| + |b_m|)^2 r^{2m} \le 1$$ for $|z| = r \le R_N$ , where $R_N$ is an in Theorem 1.3. The radius $R_N$ is sharp. Also, $$|f(z)|^{2} + \sum_{m=N}^{\infty} |a_{m}| r^{m} + sgn(t) \sum_{m=1}^{t} |a_{m}|^{2} \frac{r^{N}}{1-r} + \left(\frac{1}{1+a_{0}} + \frac{r}{1-r}\right) \sum_{m=t+1}^{\infty} (|a_{m}| + |b_{m}|)^{2} r^{2m} \le 1$$ for $|z| = r \le R'_N$ , where $R'_N$ is an in Theorem 1.3. The radius $R'_N$ is sharp. Recently, Ahamed [17] proved the inequalities $S_{\mu,\beta,n,N}^f(r) \leq d(f(0),\partial f(\mathbb{D}))$ , where $$S_{\mu,\beta,n,N}^{f}(r) := |f(z)|^{n} + \sum_{m=N}^{\infty} (|a_{m}| + |b_{m}|) r^{m} + \mu \operatorname{sgn}(t) \sum_{m=1}^{t} (|a_{m}| + |b_{m}|)^{2} \frac{r^{N}}{1 - r} + \beta \left(1 + \frac{r}{1 - r}\right) \sum_{m=t+1}^{\infty} (|a_{m}| + |b_{m}|)^{2} r^{2m},$$ $$(1.3)$$ to establish harmonic analog of Theorem 1.3 and Theorem 1.4 for certain harmonic classes. Chichra [22] first introduced the class $W(\alpha)$ in 1977, which includes normalized analytic functions h that satisfy the condition $\operatorname{Re}(h'(z) + \alpha z h''(z)) > 0$ for $z \in \mathbb{D}$ and $\alpha > 0$ . Furthermore, Chichra [22] demonstrated that functions within the $W(\alpha)$ class form a subset of functions that are close to being convex in D. In 2014, Nagpal and Ravichandran [40] explored the class $$W_{\mathcal{H}}^0 = \{ f = h + \overline{g} \in \mathcal{H} : \operatorname{Re}(h'(z) + zh''(z)) > |g'(z) + zg''(z)| \text{ for } z \in \mathbb{D} \}$$ and derived coefficient bounds for functions in the class $W_{\mathcal{H}}^0$ . In 2019, Ghosh and Vasudevarao examined the class $W_{\mathcal{H}}^0(\alpha)$ , where $$W_{\mathcal{H}}^0(\alpha) = \{ f = h + \overline{g} \in \mathcal{H} : \operatorname{Re}(h'(z) + \alpha z h''(z)) > |g'(z) + \alpha z g''(z)| \text{ for } z \in \mathbb{D} \}.$$ Later, some authors [45, 48] generalized the class $W^0_{\mathcal{H}}(\alpha)$ . In 2021, Çakmak et al. [21] introduced a new class $R_H^0(\gamma, \delta, \lambda)$ for $0 \le \lambda < \gamma \le \delta$ of complex-valued harmonic functions $f \in \mathcal{H}_0$ satisfying the inequality $$Re\left(\gamma h'(z) + \delta z h''(z) + \left(\frac{\delta - \gamma}{2}\right) z^2 h'''(z) - \lambda\right) > \left|\gamma g'(z) + \delta z g''(z) + \left(\frac{\delta - \gamma}{2}\right) z^2 g'''(z)\right|$$ It was proved that the members of the class are close-to-convex. The authors obtained coefficient bounds, growth estimates, and sufficient condition for this class. Further, they investigated the radii of fully starlikeness and fully convexity of the class. In 2022 Wang et al. [47] investigated the class $$\mathcal{W}_{\mathscr{H}}^{0}(\alpha,\beta,\gamma) = \left\{ f = h + \overline{g} \in \mathscr{H}^{0} : \Re\left(h'(z) + \alpha z h''(z) + \gamma z^{2} h'''(z) - \beta\right) \right.$$ $$\left. > \left|g'(z) + \alpha z g''(z) + \gamma z^{2} g'''(z)\right| \left.\left(z \in \mathbb{D}\right)\right\}.$$ Authors obtained different results associated with this class related to Bohr Phenomenon. The primary objective of this paper is to explore various Bohr inequalities related to harmonic functions comprehensively. Motivated by the work done in the papers [4, 17], in the present paper we determine improved Bohr phenomenon, Bohr-Rogosinski radius and refined Bohr radius, for the class $R_H^0(\gamma, \delta, \lambda)$ in Section 3, 4 and 5 respectively.

Lemma 2.1. [21] Let $f \in R_H^0(\gamma, \delta, \lambda)$ . Then for any $m \geq 2$ : (1) $$|a_m| + |b_m| \le \frac{4(\gamma - \lambda)}{m^2[2\gamma + (\delta - \gamma)(m - 1)]};$$ (2) $||a_m| - |b_m|| \le \frac{4(\gamma - \lambda)}{m^2[2\gamma + (\delta - \gamma)(m - 1)]};$ (3) $|a_m| \le \frac{4(\gamma - \lambda)}{m^2[2\gamma + (\delta - \gamma)(m - 1)]}.$ All these inequalities are sharp and equality holds for the function f given by $$f(z) = z + \sum_{m=2}^{\infty} \frac{2(\gamma - \lambda)z^m}{m^2[2\gamma + (\delta - \gamma)(m - 1)]}.$$ (2.1)

Lemma 2.2. [21] Let $f \in R_H^0(\gamma, \delta, \lambda)$ . Then $$|z| + 4(\gamma - \lambda) \sum_{m=2}^{\infty} \frac{(-1)^{m-1}|z|^m}{m^2 [2\gamma + (\delta - \gamma)(m-1)]} \le |f(z)|$$ $$\le |z| + 4(\gamma - \lambda) \sum_{m=2}^{\infty} \frac{|z|^m}{m^2 [2\gamma + (\delta - \gamma)(m-1)]}.$$ (2.2) Both the inequalities are sharp for the function f given by $$f(z) = z + \sum_{m=2}^{\infty} \frac{2(\gamma - \lambda)\overline{z}^m}{m^2[2\gamma + (\delta - \gamma)(m-1)]}.$$

Lemma 2.3. [21] Let $f \in R_H^0(\gamma, \delta, \lambda)$ . Then for any $m \geq 2$ ; $$|b_m| \le \frac{2(\gamma - \lambda)}{m^2[2\gamma + (\delta - \gamma)(m - 1)]}.$$ (2.3) The inequality is sharp for the function f given by $$f(z) = z + \sum_{m=2}^{\infty} \frac{2(\gamma - \lambda)\overline{z}^m}{m^2[2\gamma + (\delta - \gamma)(m - 1)]}.$$ (2.4) Before presenting the primary findings of this paper, we first outline the definition of the dilogarithm. The dilogarithm $\text{Li}_2(z)$ is expressed through the power series: $$\text{Li}_2(z) = \sum_{m=1}^{\infty} \frac{z^m}{m^2}, \quad |z| < 1.$$ This definition and its nomenclature arise from the similarity to the Taylor series expansion of the ordinary logarithm around the point 1: $$-\log(1-z) = \sum_{m=1}^{\infty} \frac{z^m}{m}, \quad |z| < 1,$$ which similarly leads to the definition of the polylogarithm: $$Li_n(z) = \sum_{m=1}^{\infty} \frac{z^m}{m^n}, \quad |z| < 1, \quad n = 1, 2, 3, \dots$$ Additionally, the relationship $$\frac{d}{dz}\left(Li_n(z)\right) = \frac{1}{z}Li_{n-1}(z), \quad n \ge 2,$$ is noteworthy. This relationship is crucial for understanding the behavior of the polylogarithm. The extension of the domain for $Li_n(z)$ is evident and can be shown through induction, allowing us to expand its definition to the cut plane $\mathbb{C} \setminus [1, \infty)$ . Specifically, the analytic continuation of the dilogarithm is given by: $$\operatorname{Li}_2(z) = -\int_0^z \frac{\log(1-u)}{u} du, \quad z \in \mathbb{C} \setminus [1, \infty).$$

Theorem 3.1. Let $f \in R_H^0(\gamma, \delta, \lambda)$ $(0 \le \lambda < \gamma \le \delta)$ . Then for any $p \ge 1$ , $$|z| + \sum_{m=2}^{\infty} (|a_m| + |b_m|)|z|^m + \sum_{m=2}^{\infty} (|a_m| + |b_m|)^p |z|^{pm} \le d(f(0), \partial f(\mathbb{D}))$$ for $|z| = r \le r_p(\gamma, \delta, \lambda)$ , where $r_p(\gamma, \delta, \lambda)$ is the unique root in (0, 1) of $$r + 4(\gamma - \lambda) \sum_{m=2}^{\infty} \frac{r^m}{m^2 [2\gamma + (\delta - \gamma)(m - 1)]} + \sum_{m=2}^{\infty} \left[ 4(\gamma - \lambda) \frac{r^m}{m^2 [2\gamma + (\delta - \gamma)(m - 1)]} \right]^p$$ $$= 1 + 4(\gamma - \lambda) \sum_{m=2}^{\infty} \frac{(-1)^{m-1}}{m^2 [2\gamma + (\delta - \gamma)(m - 1)]}.$$ The radius $r_p(\gamma, \delta, \lambda)$ is the best possible.

Theorem 3.6. Let $f \in R_H^0(\gamma, \delta, \lambda) (0 \le \lambda < \gamma \le \delta)$ . If $f = h + \overline{g}$ , then $$|z| + |h(z)| + \sum_{m=2}^{\infty} |a_m||z|^m \le d(f(0), \partial f(\mathbb{D}))$$ for $r \leq R_h$ , where $R_h$ is the unique root of equation $$2r + \sum_{m=2}^{\infty} \frac{8(\gamma - \lambda)r^m}{m^2[2\gamma + (\delta - \gamma)(m - 1)]} - 1 - \sum_{m=2}^{\infty} \frac{4(\gamma - \lambda)(-1)^{m-1}}{m^2[2\gamma + (\delta - \gamma)(m - 1)]} = 0$$ in (0,1). Here $R_h$ is the best possible. (2) $$|z| + |g(z)| + \sum_{m=2}^{\infty} |b_m||z|^m \le d(f(0), \partial f(\mathbb{D}))$$ for $r \leq R_g$ , where $R_g$ is the unique root of equation $$r + \sum_{m=2}^{\infty} \frac{4(\gamma - \lambda)r^m}{m^2[2\gamma + (\delta - \gamma)(m - 1)]} - 1 - \sum_{m=2}^{\infty} \frac{4(\gamma - \lambda)(-1)^{m-1}}{m^2[2\gamma + (\delta - \gamma)(m - 1)]} = 0$$ in (0,1). Here $R_g$ is the best possible.

Theorem 4.2. Let $N \geq 2$ be integer. If $f \in R_H^0(\gamma, \delta, \lambda)$ ( $0 \leq \lambda < \gamma \leq \delta$ ), then $$|f(z)|^2 + \sum_{m=N}^{\infty} (|a_m| + |b_m|)|z|^m \le d(f(0), \partial f(\mathbb{D}))$$ for $|z| = r \le R_N(\gamma, \delta, \lambda)$ , where $R_N(\gamma, \delta, \lambda)$ is the unique root in (0, 1) of $$\left(r + \sum_{m=2}^{\infty} \frac{4(\gamma - \lambda)r^m}{m^2[2\gamma + (\delta - \gamma)(m - 1)]}\right)^2 + \sum_{m=N}^{\infty} \frac{4(\gamma - \lambda)r^m}{m^2[2\gamma + (\delta - \gamma)(m - 1)]} - 1 - 4(\gamma - \lambda)\sum_{m=2}^{\infty} \frac{(-1)^{m-1}}{m^2[2\gamma + (\delta - \gamma)(m - 1)]} = 0$$ Here $R_N(\gamma, \delta, \lambda)$ is the best possible.

Theorem 5.1. Let $N \geq 5$ be a positive integer, $t = \lfloor \frac{N-1}{2} \rfloor$ , $\mu$ , $\beta \in (0, \infty)$ and $$S_{\mu,\beta,n,N}^{f}(r) := |f(z)|^{n} + \sum_{m=N}^{\infty} (|a_{m}| + |b_{m}|)r^{m} + \mu \operatorname{sgn}(t) \sum_{m=1}^{t} (|a_{m}| + |b_{m}|)^{2} \frac{r^{N}}{1 - r} + \beta \left(1 + \frac{r}{1 - r}\right) \sum_{m=t+1}^{\infty} (|a_{m}| + |b_{m}|)^{2} r^{2m}.$$ If $f \in R_H^0(\gamma, \delta, \lambda)$ $(0 \le \lambda < \gamma \le \delta)$ be given by (1.2), then $S_{\mu,\beta,n,N}^f(r) \le d(f(0), \partial f(\mathbb{D}))$ for $|z| = r \le R_{\mu,\beta,\gamma,\delta,\lambda}^{n,N,t}$ is the unique root of equation $\psi_{\mu,\beta,\gamma,\delta,\lambda}^{n,N,t}(r)$ in (0,1), where $$\psi_{\mu,\beta,\gamma,\delta,\lambda}^{n,N,t}(r) := (H_{\gamma,\delta,\lambda}(r))^n + \sum_{m=N}^{\infty} \frac{4(\gamma - \lambda)r^m}{m^2[2\gamma + (\delta - \gamma)(m-1)]} + \mu F_{t,\gamma,\delta,\lambda}^N(r) + \beta G_{t,\gamma,\delta,\lambda}(r) - 1 - \sum_{m=2}^{\infty} \frac{(-1)^{m-1}4(\gamma - \lambda)}{m^2[2\gamma + (\delta - \gamma)(m-1)]}$$ and $$\begin{cases} H_{\gamma,\delta,\lambda}(r) = r + \sum_{m=2}^{\infty} \frac{4(\gamma-\lambda)r^m}{m^2[2\gamma + (\delta-\gamma)(m-1)]} \\ F_{t,\gamma,\delta,\lambda}^N(r) = sgn(t) \sum_{m=1}^t \frac{16(\gamma-\lambda)^2}{m^4[2\gamma + (\delta-\gamma)(m-1)]^2} \frac{r^N}{1-r} \\ G_{t,\gamma,\delta,\lambda}(r) = (1 + \frac{r}{1-r}) \sum_{m=t+1}^{\infty} \frac{16(\gamma-\lambda)^2 r^{2m}}{m^4[2\gamma + (\delta-\gamma)(m-1)]^2} \end{cases}$$ The result is sharp.

Theorem 5.2. Let $f \in R_H^0(\gamma, \delta, \lambda)$ be given by (1.2) $(0 \le \lambda < \gamma \le \delta)$ and $\mu, \beta \in (0, \infty)$ : (1) If N=1, then $S^f_{\mu,\beta,n,1}(r) \leq d(f(0),\partial f(\mathbb{D}))$ for $|z|=r \leq R^{n,1,0}_{\mu,\beta,\gamma,\delta,\lambda}$ is the unique root of equation $$(H_{\gamma,\delta,\lambda}(r))^n + H_{\gamma,\delta,\lambda}(r) - r + \beta G_{0,\gamma,\delta,\lambda}(r) + \frac{2(\gamma-\lambda)r}{\gamma} - 1 - \sum_{m=2}^{\infty} \frac{(-1)^{m-1}4(\gamma-\lambda)}{m^2[2\gamma + (\delta-\gamma)(m-1)]} = 0.$$ (2) If N=2, then $S^f_{\mu,\beta,n,2}(r) \leq d(f(0),\partial f(\mathbb{D}))$ for $|z|=r \leq R^{n,2,0}_{\mu,\beta,\gamma,\delta,\lambda}$ is the unique root of equation $$(H_{\gamma,\delta,\lambda}(r))^n + H_{\gamma,\delta,\lambda}(r) - r + \beta G_{0,\gamma,\delta,\lambda}(r) - 1 - \sum_{m=2}^{\infty} \frac{(-1)^{m-1} 4(\gamma - \lambda)}{m^2 [2\gamma + (\delta - \gamma)(m-1)]} = 0.$$ (3) If N=3, then $S^f_{\mu,\beta,n,3}(r) \leq d(f(0),\partial f(\mathbb{D}))$ for $|z|=r \leq R^{n,3,1}_{\mu,\beta,\gamma,\delta,\lambda}$ is the unique root of equation $$(H_{\gamma,\delta,\lambda}(r))^{n} + H_{\gamma,\delta,\lambda}(r) - r - \frac{(\gamma - \lambda)r^{2}}{2\gamma + (\delta - \gamma)} + \mu \frac{8(\gamma - \lambda)^{2}}{\gamma} \frac{r^{3}}{1 - r} + \beta G_{1,\gamma,\delta,\lambda}(r) - 1 - \sum_{m=2}^{\infty} \frac{(-1)^{m-1}4(\gamma - \lambda)}{m^{2}[2\gamma + (\delta - \gamma)(m - 1)]} = 0.$$ (4) If N=4, then $S^f_{\mu,\beta,n,4}(r) \leq d(f(0),\partial f(\mathbb{D}))$ for $|z|=r \leq R^{n,4,1}_{\mu,\beta,\gamma,\delta,\lambda}$ is the unique root of equation $$(H_{\gamma,\delta,\lambda}(r))^{n} + \sum_{m=4}^{\infty} \frac{4(\gamma - \lambda)r^{m}}{m^{2}[2\gamma + (\delta - \gamma)(m - 1)]} + \mu \frac{8(\gamma - \lambda)^{2}}{\gamma} \frac{r^{4}}{1 - r} + \beta G_{1,\gamma,\delta,\lambda}(r) - 1 - \sum_{m=2}^{\infty} \frac{(-1)^{m-1}4(\gamma - \lambda)}{m^{2}[2\gamma + (\delta - \gamma)(m - 1)]} = 0.$$ All radii are sharp.

Definition 1.1. ([39]) For any analytic functions $f = \sum_{n=0}^{\infty} a_n z^n$ and $g = \sum_{n=0}^{\infty} b_n z^n \in \mathcal{S}(f)$ , the class $\mathcal{S}(f)$ is said to satisfy the Bohr phenomenon if there is a $r$ , $0 < r^ \le 1$ , such that the inequality $\sum_{n=0}^{\infty} |b_n z^n| \le d(f(0), f(\mathbb{D}))$ , holds for $|z| < r^*$ . Here, $d(f(0), f(\mathbb{D}))$ denote the Euclidean distance between f(0) and the boundary of the image of the unit disk under the mapping f. In particular, for $f(\mathbb{D}) = \mathbb{D}$ , $d(f(0), \partial f(\mathbb{D})) = 1 - |f(0)|$ and the inequality $\sum_{n=0}^{\infty} |a_n| \leq d(f(0), \partial f(\mathbb{D}))$ reduces to $\sum_{n=0}^{\infty} |a_n z^n| \leq 1$ . Thus the equation (1.1) can be rewritten as $$d\left(\sum_{n=0}^{\infty} |a_n|, |a_0|\right) = \sum_{n=1}^{\infty} |a_n| \le 1 - |f(0)| = d(f(0), \partial f(\mathbb{D})),$$ where d is the Euclidean distance. Kayumov et al. [33] defined the similar definition for harmonic functions.

Definition 1.2. ([33]) Let $f \in \mathcal{H}_0$ be given be (1.2). Then the Bohr phenomenon is to find a constant $0 < \rho^ \le 1$ such that the inequality $|z| + \sum_{n=2}^{\infty} (|a_n| + |b_n|)|z|^n \le d(f(0), \partial f(\mathbb{D}))$ holds for $|z| = r \le \rho$ . The largest such radius $\rho^*$ is called the Bohr radius for the class $\mathcal{H}_0$ . In a similar vein to the Bohr radius, the Bohr–Rogosinski radius has been established [46] and is defined as follows: If $f \in \mathfrak{A}$ , then for $N \geq 1$ , we have $|S_N(z)| < 1$ within the disk $\mathbb{D}_{1/2}$ , and this radius is referred to as the Bohr–Rogosinski sum $R_N^f(z)$ of the function f, given by $$R_N^f(z) := |f(z)| + \sum_{m=N}^{\infty} |a_m| r^m, \quad |z| = r.$$ It is noteworthy that for N=1, the expression simplifies to the classical Bohr sum, where f(0) is replaced by |f(z)|. In 2017, Kayumov and Ponnusamy [31] demonstrated the following significant result regarding the Bohr–Rogosinski radius for analytic functions.