Improved Bohr radius for the class of starlike log-harmonic mappings

Theorem 1.1. [3] If $g(z) = \sum_{n=0}^{\infty} b_n z^n \in \mathcal{S}(f)$ and $f(z) = \sum_{n=0}^{\infty} a_n z^n$ is univalent, then (1.3) $$\sum_{n=1}^{\infty} |b_n z^n| \le d(f(0), \partial \Omega) \text{ for } |z| \le r_0 = 3 - \sqrt{8} = 0.17157.$$ Here $r_0$ is sharp for the Koebe function $f_K(z) = z/(1-z)^2$ . In [3], Abu-Muhanna has proved the following lemma to find the lower bound of the distance $d(f(0), \partial\Omega)$ .

Lemma 1.4. [3] Let $f(z) = \sum_{n=0}^{\infty} a_n z^n$ be an analytic univalent function from $\mathbb{D}$ onto a simply connected domain $\Omega$ . Then <span id="page-2-1"></span>(1.5) $$\frac{1}{4}|f'(0)| \le d(f(0), \partial\Omega) \le |f'(0)|.$$ Next we discuss improved Bohr radius for starlike log-harmonic mappings. A function $f: \mathbb{D} \to \mathbb{C}$ is said to be log-harmonic if there is a $w \in \mathcal{B}$ such that f is a non-constant solution of the non-linear elliptic partial differential equation (1.6) $$\bar{f}_{\bar{z}}(z)/\bar{f}(z) = w(z)f_z(z)/f(z),$$ where the second dilation function w is such that |w(z)| < 1 for all z ∈ D. The Jacobian $$J_f = |f_z|^2 - |f_{\bar{z}}|^2 = |f_z|^2 (1 - |w(z)|^2)$$ is positive, and therefore all the non-constant log-harmonic mappings are sensepreserving and open in D. In 2013, Li et al. [\[34\]](#page-23-12) proved a necessary and sufficient condition for a function to be log-p-harmonic and also studied local log-p-harmonic mappings. Mao et al. [\[38\]](#page-23-13) have established Schwarz' lemma for log-harmonic mappings, through which they proved two versions of the Landau's theorem for these functions. In 2018, Liu and Ponnusamy [\[35\]](#page-23-14) obtained the coefficient estimates and hence studied Bohr radius for log-harmonic mappings. Inner mapping radius by constructing a family of 1-slit log-harmonic mappings have been established in [\[35\]](#page-23-14). Several interesting properties have been established in [\[35\]](#page-23-14) of log-harmonic mappings. In 2019, Liu and Ponnusamy [\[36\]](#page-23-15) obtained the precise ranges of log-harmonic Koebe mapping, log-harmonic right half-plane mapping and log-harmonic two-slits mappings. Further, the coefficient estimates for univalent log-harmonic starlike mappings has been established in [\[36\]](#page-23-15). Let h⁰ and g⁰ be two functions defined by <span id="page-3-2"></span>(1.7) $$h_0(z) = \frac{1}{1-z} \exp\left(\frac{2z}{1-z}\right) = \exp\left(\sum_{n=1}^{\infty} \left(2 + \frac{1}{n}\right) z^n\right)$$ <span id="page-3-3"></span> $$(1.8) g_0(z) = (1-z) \exp\left(\frac{2z}{1-z}\right) = \exp\left(\sum_{n=1}^{\infty} \left(2-\frac{1}{n}\right) z^n\right).$$ Then the function f⁰ defined by <span id="page-3-1"></span>(1.9) $$f_0(z) = zh_0(z)\overline{g_0(z)} = \frac{z(1-\bar{z})}{1-z} \exp\left(\operatorname{Re}\left(\frac{4z}{1-z}\right)\right) \text{ for } z \in \mathbb{D}$$ is the log-harmonic Koebe function. In 2011, Duman [\[25\]](#page-23-16) obtained the upper bound for |h(z)| and |g(z)|. In 2016, Ali et al. [\[10,](#page-22-15) Theorem 2] established the sharp lower bounds and exhibited the corresponding extremal functions h0, g⁰ and f0. Ali et al. [\[10\]](#page-22-15) extended the Bohr phenomenon to the context of starlike univalent log-harmonic mappings of the form <span id="page-3-0"></span>(1.10) $$f(z) = zh(z)\overline{g(z)} \text{ in } \mathcal{ST}_{LH}^{0},$$ and proved the following interesting result. <span id="page-4-1"></span>Theorem 1.2. [\[10\]](#page-22-15) Let f be a function given by [\(1.10\)](#page-3-0). Also, let H(z) = zh(z) and G(z) = zg(z). Then $$\begin{cases} \frac{1}{2e} \le d(0, \partial H(\mathbb{D})) \le 1 \\ \frac{2}{e} \le d(0, \partial G(\mathbb{D})) \le 1 \\ \frac{1}{e^2} \le d(0, \partial f(\mathbb{D})) \le 1. \end{cases}$$ Equalities occur if, and only if, h, g and f are suitable rotation of h0, g⁰ and f0. In 1989, Abdulhadi and Hengartner [\[2\]](#page-22-3) established the sharp coefficient bounds for the function in the class ST ⁰ LH. <span id="page-4-0"></span>Theorem 1.3. [\[2\]](#page-22-3) Let f be a function given by [\(1.10\)](#page-3-0). Then $$|a_n| \le 2 + \frac{1}{n}$$ and $|b_n| \le 2 - \frac{1}{n}$ for all $n \ge 1$ . Equalities hold for rotation of the function f0. In 2016, Ali et al. [\[10\]](#page-22-15) obtained Bohr radius for log-harmonic mappings of the class ST ⁰ LH. Theorem 1.4. [\[10\]](#page-22-15) Let f(z) = zh(z)g(z) ∈ ST ⁰ LH and H(z) = zh(z) and G(z) = zg(z). Then (i) the inequality $$M_h(r) := |z| \exp\left(\sum_{n=1}^{\infty} |a_n||z|^n\right) \le d(0, \partial H(\mathbb{D}))$$ holds for |z| ≤ r^H ≈ 0.1222, where r^H is the unique root in (0, 1) of $$\frac{r}{1-r}\exp\left(\frac{2r}{1-r}\right) = \frac{1}{2e}.$$ (ii) the inequality $$M_g(r) := |z| \exp\left(\sum_{n=1}^{\infty} |b_n||z|^n\right) \le d(0, \partial G(\mathbb{D}))$$ holds for |z| ≤ r^G ≈ 0.3659, where r^G is the unique root in (0, 1) of $$r(1-r)\exp\left(\frac{2r}{1-r}\right) = \frac{2}{e}.$$ Both the radii are sharp and are attained by appropriate rotation of the functions H0(z) = zh0(z) and G0(z) = zg0(z). Theorem 1.5. [\[10\]](#page-22-15) Let f be a function given by [\(1.10\)](#page-3-0). Then for any real t, the inequality $$|z| \exp\left(\sum_{n=1}^{\infty} |a_n + e^{it}b_n||z|^n\right) \le d(0, \partial f(\mathbb{D}))$$ holds for |z| ≤ r^f ≈ 0.09078, where r^f is the unique root in (0, 1) of $$r \exp\left(\frac{4r}{1-r}\right) = \frac{1}{e^2}.$$ The bound is sharp and is attained by suitable rotation of the log-harmonic Koebe function f0. Our another interest in this paper is to study Bohr radius for the class of analytic functions f which map unit disk D into a concave-wedge domain. The concave-wedge domain is defined (see [\[4\]](#page-22-16)) by $$W_{\alpha} = \left\{ w \in \mathbb{C} : |\arg w| < \frac{\alpha \pi}{2}, \ 1 \le \alpha \le 2 \right\}.$$ It is known that the conformal mapping from D onto W^α is given by <span id="page-5-0"></span>(1.11) $$F_{\alpha,t}(z) = t \left(\frac{1+z}{1-z}\right)^{\alpha} = t \left(1 + \sum_{n=1}^{\infty} A_n z^n\right) \text{ for } 1 \le \alpha \le 2 \text{ and } t > 0.$$ It is easy to see that when α = 1, the domain turns out to be a convex half-plane and when α = 2 it gives a slit domain. Let S^W^α be the class of analytic functions f which maps the unit disk D into the wedge domain Wα. In 2014, Abu-Muhana et al. [\[4\]](#page-22-16) proved the following interesting result for functions in the class S^W^α . <span id="page-5-1"></span>Theorem 1.6. [\[4\]](#page-22-16) Let α ∈ [1, 2]. If f(z) = a⁰ + P^∞ ⁿ=1 anz ⁿ ∈ S^W^α with a⁰ > 0, then the inequlaity $$\sum_{n=1}^{\infty} |a_n||z|^n \le d(a_0, \partial W_{\alpha})$$ holds for |z| ≤ r^α = (2¹/α − 1)/(2¹/α + 1). The function f = Fα,a⁰ in [\(1.11\)](#page-5-0) shows that r^α is sharp. The following lemma is useful to prove one of our main results for functions in the Class S^W^α . Lemma 1.12. [\[4\]](#page-22-16) Let Fα,t be given by [\(1.11\)](#page-5-0), where α ∈ [1, 2]. Then Aⁿ > 0 for all n ≥ 1.

Theorem 2.1. Let $\beta \in [0, 1/4)$ . If $g(z) = \sum_{n=0}^{\infty} b_n z^n \in \mathcal{S}(f)$ and $f(z) = \sum_{n=0}^{\infty} a_n z^n$ is univalent, then <span id="page-6-2"></span> $$(2.1) \quad \beta|f'(0)| + \sum_{n=0}^{\infty} |b_n z^n| \le d(f(0), \partial\Omega) \text{ for } |z| \le r_{\beta} = \frac{3 - 4\beta - \sqrt{8}\sqrt{1 - 2\beta}}{1 - 4\beta}.$$ The radius $r_{\beta}$ is sharp for the Koebe function $f_K(z) = z/(1-z)^2$ . - Remark 2.1. In particular, when $\beta=0$ , the radius $r_{\beta}$ which has been proved in Theorem 2.1 coincides exactly with $r_0=3-\sqrt{8}=0.17159$ in Theorem 1.1. Further, in particular, we obtain $r_{\beta}=5-2\sqrt{6}\approx 0.10102$ for $\beta=1/8$ , $r_{\beta}=9-4\sqrt{5}\approx 0.05572$ for $\beta=3/16$ and $r_{\beta}=17-12\sqrt{2}\approx 0.02943$ for $\beta=7/32$ . In fact, we see that $\lim_{\beta\to(1/4)^-}r_{\beta}=0$ . - 2.2. Improved Bohr radius for starlike log-harmonic mappings. It is known that if f is a non-vanishing log-harmonic mapping then f can be written as $f(z) = h(z)\overline{g(z)}$ where h and g are analytic functions in $\mathbb{D}$ . On the other hand, if f vanishes at z = 0 but is not identically zero, then f admits the following representation (2.2) $$f(z) = z^m |z|^{2\beta m} h(z) \overline{g(z)}$$ where m is a non-negative integer and Re $\beta > -1/2$ , and h, g are analytic functions in $\mathbb{D}$ with g(0) = 1 and $h(0) \neq 1$ (see [1]). The exponent $\beta$ in (2.2) depends only on w(0) and it can be expresses as <span id="page-6-1"></span> $$\beta = \overline{w(0)} \frac{1 + w(0)}{1 - |w(0)|^2}.$$ Note that $f(0) \neq 0$ if, and only if, m = 0, and that a univalent log-harmonic mapping on $\mathbb{D}$ vanishes at the origin if, and only if, m = 1. Univalent log-harmonic mappings have been studied extensively by many researchers (see [10, 25, 29]). The class of log-harmonic mappings is denoted by $\mathcal{S}_{LH}$ . Let $z|z|^{2\beta}h(z)\overline{g(z)}$ be a log-harmonic univalent function. We say that f is a starlike log-harmonic mapping if (2.3) $$\frac{\partial}{\partial \theta} \operatorname{Arg}(f(e^{i\theta})) = \operatorname{Re}\left(\frac{zf_z - \bar{z}f_{\bar{z}}}{f}\right) > 0 \text{ in } \mathbb{D}$$ and we denote the set of all strlike log-harmonic functions by $\mathcal{ST}_{LH}$ . Let $\mathcal{ST}_{LH}^0$ be a subclass of $\mathcal{ST}_{LH}$ , consisiting of functions $f \in \mathcal{ST}_{LH}$ which map the unit disk $\mathbb{D}$ onto a starlike domain (with repsect to the origin). Our main aim is to study Bohr radius for the class of sense-preserving satrlike log-harmonic mappings in $\mathbb{D}$ of the form $f(z) = zh(z)\overline{g(z)}$ with $$h(z) = \exp\left(\sum_{n=1}^{\infty} a_k z^k\right)$$ and $g(z) = \exp\left(\sum_{n=1}^{\infty} b_n z^n\right)$ , where h(z) and g(z) may be called as analytic and co-analytic factors of the function f(z). We prove the following improved Bohr radius for functions in the class $\mathcal{ST}_{LH}^0$ .

Theorem 2.2. Let f be a function given by (1.10). Then for any real t, the inequality $$|z| \exp\left(\sum_{n=1}^{\infty} \left| a_n + e^{it}b_n + \frac{n}{4n^2 - 1}a_nb_n \right| |z|^n \right) \le d(0, \partial f(\mathbb{D}))$$ holds for $|z| \le r_f \approx 0.08528$ , where $r_f$ is the unique root in (0,1) of (2.4) $$\frac{r}{1-r} \exp\left(\frac{4r}{1-r}\right) = \frac{1}{e^2} in (0,1).$$ The radius $r_f$ is sharp and is attained by a suitable rotation of the log-harmonic Koebe function $f_0$ given by (1.9). <span id="page-7-0"></span> FIGURE 1. The radius $r_f \approx 0.08528$ is a root of (2.4) in (0, 1).  FIGURE 2. Image of unit disk $\mathbb{D}$ under the Koebe function $f(z) = \frac{z}{(1-z)^2}$ and log-harmonic Koebe function $f_0(z) = \frac{z(1-\bar{z})}{1-z} \exp\left(\operatorname{Re}\left(\frac{4z}{1-z}\right)\right)$ . Next we prove the sharp Bohr radius for the class $\mathcal{ST}_{LH}^0$ in view of additional terms $|a_n|^2$ and $|b_n|^2$ in the series expansion of h and g respectively.  Figure 3. Image of unit disk D under the map h0(z) = ¹ 1−z exp 2z 1−z and g0(z) = (1 − z) exp 2z 1−z . <span id="page-8-1"></span>Theorem 2.3. Let f be a function given by [\(1.10\)](#page-3-0) and H(z) = zh(z) and G(z) = zg(z). Then (i) the inequality $$|z| \exp\left(\sum_{n=1}^{\infty} \left(|a_n| + \frac{n}{(2n+1)^2} |a_n|^2\right) |z|^n\right) \le d(0, \partial H(\mathbb{D}))$$ holds for |z| ≤ r^H ≈ 0.09735, where r^H is the unique root of (2.5) $$\frac{r}{(1-r)^2} \exp\left(\frac{2r}{1-r}\right) = \frac{1}{2e} \ in \ (0,1).$$ <span id="page-8-0"></span> Figure 4. The radii r^H ≈ 0.09735 and r^G ≈ 0.30539 are roots of [\(2.5\)](#page-8-0) and [\(2.6\)](#page-9-0) respectively in (0, 1). (ii) the inequality <span id="page-9-0"></span> $$|z| \exp\left(\sum_{n=1}^{\infty} \left(|b_n| + \frac{n}{(2n-1)^2} |b_n|^2\right) |z|^n\right) \le d(0, \partial G(\mathbb{D}))$$ holds for |z| ≤ r^G ≈ 0.30539, where r^G is the unique root of (2.6) $$r\exp\left(\frac{2r}{1-r}\right) = \frac{2}{e} \ in \ (0,1).$$ Both the radii are sharp and are attained by appropriate rotation of H0(z) = zh0(z) and G0(z) = zg0(z). We prove the next improved sharp Bohr radius for the class ST ⁰ LH adding |H(z)| and |G(z)| with Mh(r) and Mg(r) respecively. <span id="page-9-4"></span>Theorem 2.4. Let f be a function given by [\(1.10\)](#page-3-0) and H(z) = zh(z) and G(z) = zg(z). Then (i) the inequality $$|H(z)| + |z| \exp\left(\sum_{n=1}^{\infty} |a_n||z|^n\right) \le d(0, \partial H(\mathbb{D}))$$ <span id="page-9-1"></span>holds for |z| ≤ r^H ≈ 0.1073, where r^H is the unqiue root of (2.7) $$r\left(\frac{2r}{1-r} - \log(1-r) + \frac{1}{1-r}\exp\left(\frac{2r}{1-r}\right)\right) = \frac{1}{2e} in (0,1).$$ (ii) the inequality $$|G(z)| + |z| \exp\left(\sum_{n=1}^{\infty} |b_n||z|^n\right) \le d(0, \partial G(\mathbb{D}))$$ <span id="page-9-2"></span>holds for |z| ≤ r^G ≈ 0.3063, where r^G is the unique root of (2.8) $$r\left(\frac{2r}{1-r} + \log(1-r) + (1-r)\exp\left(\frac{2r}{1-r}\right)\right) = \frac{2}{e} in (0,1).$$ Both the radii are sharp and are attained by appropriate rotation of H0(z) = zh0(z) and G0(z) = zg0(z). For any positive integer m, considering |h(z)| ^m and |g(z)| ^m, next we prove the improved sharp Bohr radius for the class ST ⁰ LH. <span id="page-9-3"></span>Theorem 2.5. Let f be a function given by [\(1.10\)](#page-3-0) and H(z) = zh(z) and G(z) = zg(z). (i) If |h(z)| ≤ 1, then for any m ∈ N, the inequality $$|z| \exp\left(|h(z)|^m + \sum_{n=1}^{\infty} |a_n||z|^n\right) \le d(0, \partial H(\mathbb{D}))$$  Figure 5. The radii r^H ≈ 0.1073 and r^G ≈ 0.3063 are roots of [\(2.7\)](#page-9-1) and [\(2.8\)](#page-9-2) respectively in (0, 1). holds for |z| ≤ r^H ≈ 0.0566, where r^H is the unique root of (2.9) $$\frac{re}{1-r} \exp\left(\frac{2r}{1-r}\right) = \frac{1}{2e} \ in \ (0,1).$$ (ii) If |g(z)| ≤ 1, then for any m ∈ N, the inequality <span id="page-10-1"></span><span id="page-10-0"></span> $$|z| \exp\left(|g(z)|^m + \sum_{n=1}^{\infty} |b_n||z|^n\right) \le d(0, \partial G(\mathbb{D}))$$ holds for |z| ≤ r^G ≈ 0.1764, where r^G is the unique root of (2.10) $$re(1-r)\exp\left(\frac{2r}{1-r}\right) = \frac{2}{e} \ in \ (0,1).$$ Both the radii are sharp and are attained by a suitable rotation of H0(z) = zh0(z) and G0(z) = zg0(z).  Figure 6. The radii r^H ≈ 0.0566 and r^G ≈ 0.1764 are roots of [\(2.9\)](#page-10-0) and [\(2.10\)](#page-10-1) respectively in (0, 1). Remark 2.2. It is worth to notice in Theorem [2.5](#page-9-3) that the Bohr radii r^H and r^G are independent of the choice of the positive integer m. We prove the improved sharp Bohr radius adding |h(z) + g(z)| with the series $\sum_{n=1}^{\infty} |a_n + e^{it}b_n||z|^n$ for the class $\mathcal{ST}_{LH}^0$ .

Theorem 2.6. Let f be a function given by (1.10) with $|h(z)| + |g(z)| \le 1$ . Then for any real t, the inequality $$|z| \exp\left(|h(z) + g(z)| + \sum_{n=1}^{\infty} |a_n + e^{it}b_n||z|^n\right) \le d(0, \partial f(\mathbb{D}))$$ holds for $|z| \le r_f \approx 0.04181$ , where $r_f$ is the unique root of (2.11) $$er \exp\left(\frac{4r}{1-r}\right) = \frac{1}{e^2} in \ (0,1).$$ The Bohr radius $r_f$ is sharp and is attained by suitable rotation of the log-harmonic Koebe function $f_0$ . Next we prove the improved sharp Bohr radius for the class $\mathcal{ST}_{LH}^0$ adding |f(z)|.

Theorem 2.7. Let f be a function given by (1.10) with $|h(z)| \le 1$ and $|g(z)| \le 1$ . Then for any real t, the inequality <span id="page-11-0"></span> $$|f(z)| + |z| \exp\left(\sum_{n=1}^{\infty} |a_n + e^{it}b_n||z|^n\right) \le d(0, \partial f(\mathbb{D}))$$ holds for $|z| \leq r_f \approx 0.0592$ , where $r_f$ is the unique root of (2.12) $$r\left(1 + \exp\left(\frac{4r}{1-r}\right)\right) = \frac{1}{e^2} \ in \ (0,1).$$ The Bohr radius $r_f$ is sharp for a suitable rotation of the log-harmonic Koebe function $f_0$ .  FIGURE 7. The radius $r_f \approx 0.0592$ is the root of (2.12) in (0, 1). 2.3. Bohr radius for concave-wedge domain. We prove the following result which is an improved version of Theorem 1.6.

Theorem 2.8. Let $\alpha \in [1,2]$ and $\beta \in [0,2)$ . If $f(z) = a_0 + \sum_{n=1}^{\infty} a_n z^n \in S_{W_{\alpha}}$ with $a_0 > 0$ , then the inequality $$\frac{2\beta a_0}{\alpha \pi} |\arg f(z)| + \sum_{n=1}^{\infty} |a_n| |z|^n \le d(a_0, \partial W_\alpha)$$ holds for $|z| \le r_{\alpha,\beta} = ((2-\beta)^{1/\alpha} - 1)/((2-\beta)^{1/\alpha} + 1)$ . The function $f = F_{\alpha,a_0}$ in (1.11) shows that $r_{\alpha,\beta}$ is sharp. Remark 2.3. Since $W_{\alpha}$ turns out to be a convex half-plane when $\alpha = 1$ , it is evident that, for $\alpha = 1$ and $\beta = 0$ , the radius $r_{\alpha,\beta}$ coincides exactly with the Bohr radius 1/3.  FIGURE 8. Image of unit disk $\mathbb{D}$ under the maps $F_{1,1}(z)$ , $F_{1.5,20}(z)$ and $F_{2,3}(z)$ repsectively.