Bohr inequality and Bohr-Rogosinski inequality for K-Quasiconformal harmonic mappings

Lemma 2.1. [23] Suppose that $h(z) = \sum_{n=0}^{\infty} a_n z^n$ and $g(z) = \sum_{n=0}^{\infty} b_n z^n$ are two analytic functions in $\mathbb{D}$ and $h \prec g$ , then for $N \in \mathbb{N}$ , we have $$\sum_{n=N}^{\infty} |a_n| r^n \le \sum_{n=N}^{\infty} |b_n| r^n \text{ holds for } |z| = r \le 1/3.$$

Lemma 2.2. [7,39] Suppose that $h(z) = \sum_{n=0}^{\infty} a_n z^n$ and $g(z) = \sum_{n=0}^{\infty} b_n z^n$ are two analytic functions in $\mathbb{D}$ . If $|g'(z)| \leq k|h'(z)|$ in $\mathbb{D}$ for some $k \in (0,1]$ , then $$\sum_{n=1}^{\infty} |b_n| r^n \le k \sum_{n=1}^{\infty} |a_n| r^n \text{ for } |z| = r \le 1/3.$$ We establish the sharp Bohr inequality for harmonic mappings whose analytic part is subordinate to a function in the class $\widehat{C_p}$ , where $p \in (0, 1)$ .

Theorem 2.1. Suppose that $f(z) = h(z) + \overline{g(z)} = \sum_{n=0}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb{D}$ and $h \prec \phi$ , where $\phi \in \widehat{C_p}$ and $p \in (0,1)$ . Then $$\sum_{n=1}^{\infty} (|a_n| + |b_n|) r^n \le d(\phi(0), \partial \phi(\mathbb{D})) \text{ for } r \le r_{k,p},$$ where $r_{k,p} \in (0,p)$ is the unique root of the equation $$\left(\frac{2K}{K+1}\right)k_p(r) - \frac{p}{(1+p)^2} = 0.$$ Each number $r_{k,p}$ is sharp. Proof of Theorem 2.1. For a given $\phi \in \widehat{C}_p$ with $\phi(z) = \sum_{n=0}^{\infty} c_n z^n$ , let the Koebe transform of $\phi$ , $$F(z) = \frac{\phi\left(\frac{z+a}{1+\bar{a}z}\right) - \phi(a)}{(1-|a|^2)\phi'(a)}$$ for any $a \in \mathbb{D} \setminus \{p\}$ . Since $e^{-it}F(ze^{it}) \in C_{\lfloor \frac{p-a}{1-\bar{a}p} \rfloor}$ for some $t \in \mathbb{R}$ , it follows from [14, Chapter 15, p. 137] that $$d(\phi(0), \partial \phi(\mathbb{D})) \ge (p/(1+p)^2)|\phi'(0)|.$$ (2.1) As $F(z) = (\phi(z) - \phi(0))/\phi'(0)$ and using the inequality (2.1), we have (see [30, p. 26]) $$|c_n| \le |\phi'(0)| \frac{1 - p^{2n}}{(1 - p^2)p^{n-1}} \le, \quad n \ge 1.$$ (2.2) As $h \prec \phi$ and $\phi \in \widehat{C}_p$ , $p \in (0,1)$ , by applying Lemma 2.1 and inequality (2.2), we have $$\sum_{n=1}^{\infty} |a_n| r^n \le \sum_{n=1}^{\infty} |c_n| r^n \le |\phi'(0)| \sum_{n=1}^{\infty} \frac{1 - p^{2n}}{(1 - p^2) p^{n-1}} r^n$$ $$= |\phi'(0)| k_p(r) \text{ for } |z| = r \le 1/3.$$ (2.3) Since f is K-quasiconformal sense-preserving harmonic mapping on $\mathbb{D}$ , by applying Schwarz's Lemma, we get that the dilatation $\omega = g'/h'$ is analytic in $\mathbb{D}$ and $|\omega(z)| = |g'(z)/h'(z)| \le k$ in $\mathbb{D}$ , where $K = (1+k)/(1-k) \ge 1$ , $k \in [0,1)$ . Therefore, by Lemma 2.2, we have $$\sum_{n=1}^{\infty} |b_n| r^n \le k \sum_{n=1}^{\infty} |a_n| r^n \le k |\phi'(0)| k_p(r) \text{ for } |z| = r \le 1/3.$$ (2.4) By using (2.3) and (2.4), we have $$B_{1}(r) := \sum_{n=1}^{\infty} (|a_{n}| + |b_{n}|) r^{n} \le (k+1) |\phi'(0)| k_{p}(r)$$ $$\le (k+1) \frac{(1+p)^{2}}{p} d(\phi(0), \partial \phi(\mathbb{D})) k_{p}(r).$$ (2.5) Let $$G_1(r) := (k+1)k_p(r) - \frac{p}{(1+p)^2}.$$ Then, we see that $$G'_1(r) = (k+1)k'_p(r) > 0 \text{ for } r \in (0,p)$$ This implies that G is strictly increasing function of $r \in (0, p)$ . Also, we see that $$G_1(0) = -\frac{p}{(1+p)^2}$$ and $\lim_{r \to p} G_1(r) = \infty$ . It follows that the equation $G_1(r) = 0$ has a unique positive root $r_{k,p} \in (0,p)$ . Hence, it follows from (2.5) that $B_1(r) \leq d(\phi(0), \partial \phi(\mathbb{D}))$ for $r \leq r_{k,p}$ , where $r_{k,p}$ is the positive root of the equation $G_1(r) = 0$ . For the sharpness of the result, we consider the function $f(z) = h(z) + \overline{g(z)}$ in $\mathbb D$ such that $$h(z) = \phi(z) = k_p(z)$$ and $g(z) = k\lambda k_p(z)$ , where $|\lambda| = 1$ and k = (K-1)/(K+1). For the function $k_p$ , it is well-known that $\widehat{C} \setminus k_p(\mathbb{D}) = [-p/(1-p)^2, -p/(1+p)^2]$ (see [14, p. 137]) and hence we obtain $d(\phi(0), \partial \phi(\mathbb{D})) = p/(1+p)^2$ . Thus we have $$\sum_{n=1}^{\infty} \left( \left| \frac{1 - p^{2n}}{(1 - p^2)p^{n-1}} \right| + \left| \frac{k\lambda(1 - p^{2n})}{(1 - p^2)p^{n-1}} \right| \right) r^n = (1 + k) \sum_{n=1}^{\infty} \frac{1 - p^{2n}}{(1 - p^2)p^{n-1}} r^n$$ $$= (1 + k)k_p(r)$$ $$> \frac{p}{(1 + p)^2} = d(\phi(0), \partial\phi(\mathbb{D}))$$ for $r > r_{k,p}$ , where $r_{k,p}$ is the positive root of the equation $G_1(r) = 0$ . This shows that $r_{k,p}$ is the best possible. Remark 2.1. When we take K = 1 (that is, k = 0), we find that $$r_{0,p} = (1 + 1/p + p) - (\sqrt{p} + 1/\sqrt{p})\sqrt{p + 1/p}$$ is the root in the interval (0, p) of the equation $pr^2 - 2(1 + p + p^2)r + p = 0$ . This indicates that the result in [15, Corollary 1] is a special case of Theorem 2.1. In the following, we establish the Bohr-Rogosinski inequality for harmonic mappings whose analytic part is subordinate to a function in the class $\widehat{C}_p$ , where $p \in (0,1)$ .

Theorem 2.2. Suppose that $f(z) = h(z) + \overline{g(z)} = \sum_{n=0}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb{D}$ and $h \prec \phi$ , where $\phi \in \widehat{C_p}$ and $p \in (0,1)$ . Then $$|h(z)| + \sum_{n=1}^{\infty} (|a_n| + |b_n|) r^n \le d(\phi(0), \partial \phi(\mathbb{D})) \text{ for } r \le r_{k,p}^*,$$ where $r_{k,p}^* \in (0,p)$ is the unique root of the equation $$\left(\frac{3K+1}{K+1}\right)k_p(r) - \frac{p}{(1+p)^2} = 0.$$ Each number $r_{k,p}^*$ is sharp. Proof of Theorem 2.2. By employing arguments similar to those in the proof of Theorem 2.1 and considering Lemmas 2.1 and 2.2 and equation (2.2), we obtain the inequalities (2.3) and (2.4). Since $h \prec \phi$ , where $\phi \in \widehat{C}_p$ and $p \in (0,1)$ , we have $$|h(z)| \leq |h(0)| + |h(z) - h(0)|$$ $$\leq |\phi(0)| + |\phi(z) - \phi(0)|$$ $$= |\phi(0)| + \sum_{n=1}^{\infty} |c_n| r^n$$ $$\leq |\phi(0)| + \frac{(1+p)^2}{p} d(\phi(0), \partial \phi(\mathbb{D})) k_p(r)$$ (2.6) for $|z| = r \le 1/3$ . Therefore, by using equation (2.6) $$B_{2}(r) := |h(z)| + \sum_{n=1}^{\infty} (|a_{n}| + |b_{n}|) r^{n}$$ $$\leq |\phi(0)| + (k+2) \frac{(1+p)^{2}}{p} d(\phi(0), \partial \phi(\mathbb{D})) k_{p}(r).$$ (2.7) Let $$G_2(r) := (k+2)k_p(r) - \frac{p}{(1+p)^2}.$$ Then $$G_2'(r) = (k+2)k_p'(r) > 0 \text{ for } r \in (0,p)$$ This implies that $G_2$ is strictly increasing function of $r \in (0, p)$ . Also, $$G_2(0) = -\frac{p}{(1+p)^2}$$ and $\lim_{r \to p} G_2(r) = \infty$ . It follows that the equation $G_2(r) = 0$ has a unique positive root $r_{k,p}^ \in (0,p)$ . Hence, it follows from (2.7) that $B_2(r) \leq |\phi(0)| + d(\phi(0), \partial \phi(\mathbb{D}))$ for $r \leq r_{k,p}$ , where $r_{k,p}^*$ is the positive root of the equation $G_2(r) = 0$ . For the sharpness of the result, we consider the function $f(z) = h(z) + \overline{g(z)}$ in $\mathbb D$ such that $$h(z) = \phi(z) = k_p(z)$$ and $g(z) = k\lambda k_p(z)$ , where $|\lambda| = 1$ and k = (K-1)/(K+1). It is well-known that $d(\phi(0), \partial \phi(\mathbb{D})) = p/(1+p)^2$ . Hence, for z = r, we have $$|k_p(r)| + \sum_{n=1}^{\infty} \left( \left| \frac{1 - p^{2n}}{(1 - p^2)p^{n-1}} \right| + \left| \frac{k\lambda(1 - p^{2n})}{(1 - p^2)p^{n-1}} \right| \right) r^n$$ $$= |k_p(r)| + (1 + k) \sum_{n=1}^{\infty} \frac{1 - p^{2n}}{(1 - p^2)p^{n-1}} r^n$$ $$= (2 + k)k_p(r)$$ $$> \frac{p}{(1 + p)^2}$$ $$= d(\phi(0), \partial \phi(\mathbb{D}))$$ for $r > r_{k,p}$ , where $r_{k,p}$ is the positive root of the equation $G_2(r) = 0$ . This shows that $r_{k,p}^*$ is the best possible.

Lemma 3.1. [40] Let ϕ be a Ma-Minda function. Then there exist unique functions k^ϕ and h^ϕ in S such that $$1 + \frac{zk_{\phi}''(z)}{k'(z)} = \frac{zh_{\phi}'(z)}{h_{\phi}(z)} = \phi(z) \text{ for } z \in \mathbb{D}.$$ Since ϕ is symmetric about the real axis and maps the real line to itself, all its derivatives at 0 will be real numbers, implying that all the Taylor coefficients of ϕ are also real. Therefore, by applying Lemma 3.1, the Taylor expansion of k^ϕ and h^ϕ is of the form $$k_{\phi}(z) = z + \sum_{n=2}^{\infty} c_n z^n \text{ and } h_{\phi}(z) = z + \sum_{n=2}^{\infty} d_n z^n,$$ (3.1) where cⁿ and dⁿ are real numbers for all n. Now, look at the absolute sum of k^ϕ and hϕ, which we shall denote as follows: $$\hat{k}_{\phi}(r) = r + \sum_{n=2}^{\infty} |c_n| r^n \text{ and } \hat{h}_{\phi}(r) = r + \sum_{n=2}^{\infty} |d_n| r^n.$$ (3.2) The growth and distortion theorems for the classes C(ϕ) and S ∗ (ϕ) are provided in the following Lemmas.

Lemma 3.2. [40] Let f ∈ C(ϕ). Then f ′ (z) ≺ k ′ ϕ (z) and (a) Growth theorem: $$-k_{\phi}(-r) \leq |f(z)| \leq k_{\phi}(r)$$ , (b) Distortion theorem: k ′ ϕ (−r) ≤ |f ′ (z)| ≤ k ′ ϕ (r), for |z| = r ∈ (0, 1). Equality holds for some z ̸= 0 if and only if f is a rotation of kϕ.

Lemma 3.3. [40] Let f ∈ S^∗ (ϕ). Then $$\frac{f(z)}{z} \prec \frac{h_{\phi}(z)}{z}.$$ Moreover, $$-h_{\phi}(-r) \le |f(z)| \le h_{\phi}(r),$$ for |z| = r ∈ (0, 1). Equality holds for some z ̸= 0 if and only if f is a rotation of hϕ. It should be noted that the distortion theorem does not generally apply to all Ma-Minda functions ϕ. In [40], Ma and Minda provided a counterexample and established the distortion theorem for functions in S ∗ (ϕ) by adding two additional conditions on ϕ, which are presented in the following lemma.

Lemma 3.4. [40] Distortion theorem for the class S ∗ (ϕ): assume that $$\min_{|z|=r} |\phi(z)| = \phi(-r) \text{ and } \max_{|z|=r} |\phi(z)| = \phi(r).$$ If f ∈ S^∗ (ϕ), then $$h'_{\phi}(-r) \le |f'(z)| \le h'_{\phi}(r),$$ for |z| = r ∈ (0, 1). Equality holds for some z ̸= 0 if, and only if, f is a rotation of hϕ. According to [40], the functions −kϕ(−r) and −hϕ(−r) are increasing on (0, 1) and bounded above by 1, which ensures the existence of the limits limr→¹ −kϕ(−r) and limr→¹ −hϕ(−r), denoted by −kϕ(−1) and −hϕ(−1), respectively. The following lemma plays a key role in deriving the results for the class C(ϕ).

Lemma 3.5. [10] Let f ∈ C(ϕ) having the Taylor expansion $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n.$$ Also assume equation (3.1) gives the Taylor expansion of the function kϕ. Then $$\sum_{n=1}^{\infty} |a_n| r^n \le \sum_{n=1}^{\infty} |c_n| r^n \text{ for } r \le 1/3.$$ For further improvement of Theorems C or D, it is natural to raise the following questions about the study of classes C(ϕ) and S ∗ (ϕ). Problem 3.1. Can we establish the sharp Bohr radius for K-quasiconformal harmonic mappings in D whose analytic part is subordinate to a function in the class C(ϕ)? Problem 3.2. Can we establish the sharp Bohr radius for K-quasiconformal harmonic mappings in $\mathbb{D}$ whose analytic part is subordinate to a function in the class $S^*(\phi)$ ? We will affirmatively address the Problem 3.1 and Problem 3.2. Here, we obtain the sharp Bohr radius for harmonic mappings in which the analytic part is subordinate to a function from the class $C(\phi)$ .

Theorem 3.1. Suppose that $F(z) = f(z) + \overline{g(z)} = \sum_{n=1}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb{D}$ and $f \prec h$ , where $h \in \mathcal{C}(\phi)$ . Consider $$P_{k,\phi}(r) := \left(\frac{2K}{K+1}\right)\hat{k}_{\phi}(r) + k_{\phi}(-1)$$ where $\hat{k}_{\phi}$ has the Taylor expansion $r + \sum_{n=2}^{\infty} |c_n| r^n$ . If $\lim_{r \to 1} P_{k,\phi}(r) = 0$ , then $$\sum_{n=1}^{\infty} (|a_n| + |b_n|)r^n \le d(h(0), \partial h(\mathbb{D}))$$ (3.3) holds for $|z| = r \leq \frac{1}{3}$ . Otherwise, the inequality (3.3) satisfies for $|z| = r \leq \min\{\frac{1}{3}, r_{k,\phi}\}$ where $r_{k,\phi}$ is the root in (0,1) of the equation $P_{k,\phi}(r) = 0$ . In this case, the result is sharp if $r_{k,\phi} \leq 1/3$ and the Taylor coefficients of $k_{\phi}$ are non-negative. Remark 3.1. If we choose $\phi(z) = \frac{1+Az}{1+Bz}$ $(-1 \le B < A \le 1)$ , the Ma-Minda class of convex functions $\mathcal{C}(\phi)$ reduces to the familiar class $\mathcal{C}(A,B)$ consisting of the Janowski convex functions. For this particular $\phi$ , by using Lemma 3.1, it can be easily seen that the function $k_{\phi}(z)$ takes the following form $$k_{\phi}(z) = \begin{cases} \frac{1}{A} \left[ (1 + Bz)^{\frac{A}{B}} - 1 \right], & B \neq 0 \text{ and } A \neq 0 \\ \frac{1}{B} \log(1 + Bz), & B \neq 0 \text{ and } A = 0 \\ \frac{1}{A} (e^{Az} - 1), & B = 0. \end{cases}$$ and the absolute sum of the terms in the Taylor expansion of $k_{\phi}$ is as follows $$\hat{k}_{\phi}(r) = \begin{cases} r + \sum_{n=2}^{\infty} \prod_{m=2}^{n} |A - (m-1)B| r^{n}, & B \neq 0 \text{ and } A \neq 0 \\ r + \sum_{n=2}^{\infty} \frac{|B|^{n-1}}{n} r^{n}, & B \neq 0 \text{ and } A = 0 \\ \frac{1}{4} (e^{Ar} - 1), & B = 0. \end{cases}$$ If we let $\phi(z) = \frac{1+Az}{1+Bz}$ (-1 $\leq B < A \leq$ 1), then Theorem (3.1) leads directly to the following Corollary.

Corollary 3.1. Suppose that $F(z) = f(z) + \overline{g(z)} = \sum_{n=1}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb{D}$ and $f \prec h$ , where $h \in \mathcal{C}(A, B)$ . Consider $P_{A,B,k}(r)$ $$:= \begin{cases} \left(\frac{2K}{K+1}\right) \left[r + \sum_{n=2}^{\infty} \prod_{m=0}^{n-2} |A - (m-1)B| r^n\right] + \frac{1}{A} \left[ (1-B)^{\frac{A}{B}} - 1 \right], & B \neq 0 \text{ and } A \neq 0 \\ \left(\frac{2K}{K+1}\right) \left(r + \sum_{n=2}^{\infty} \frac{|B|^{n-1}}{n} r^n\right) + \frac{1}{B} \log(1-B), & B \neq 0 \text{ and } A = 0 \\ \frac{1}{A} \left[ \left(\frac{2K}{K+1}\right) (e^{Ar} - 1) + (e^{-A} - 1) \right], & B = 0. \end{cases}$$ If $\lim_{r\to 1} P_{A,B,k}(r) = 0$ , then $$\sum_{n=1}^{\infty} (|a_n| + |b_n|)r^n \le d(h(0), \partial h(\mathbb{D}))$$ (3.4) holds for $|z| = r \le 1/3$ . Otherwise, the inequality (3.4) satisfies for $|z| = r \le \min\{1/3, R_{A,B,k}\}$ where $R_{A,B,k}$ is the root in (0,1) of the equation $P_{A,B,k}(r) = 0$ . In this case, the result is sharp if $R_{A,B,k} \le 1/3$ and the Taylor coefficients of $k_{\phi}$ are non-negative. Remark 3.2. By setting $\phi(z) = \frac{1+(1-2\alpha)z}{1-z}$ for $\alpha \in [0,1)$ , the Ma–Minda class of convex functions $\mathcal{C}(\phi)$ coincides with the class $\mathcal{C}(\alpha)$ of convex functions of order $\alpha$ . For this $\phi$ , the function $k_{\phi}$ is given by $$k_{\phi}(z) = \begin{cases} \frac{1 - (1 - z)^{2\alpha - 1}}{2\alpha - 1}, & \alpha \neq \frac{1}{2} \\ -\log(1 - z), & \alpha = \frac{1}{2}. \end{cases}$$ and hence $$\hat{k}_{\phi}(r) = \begin{cases} \frac{1 - (1 - r)^{2\alpha - 1}}{2\alpha - 1}, & \alpha \neq \frac{1}{2} \\ -\log(1 - r), & \alpha = \frac{1}{2}. \end{cases}$$ Thus, setting $\phi(z) = \frac{1+(1-2\alpha)z}{1-z}$ in Theorem 3.1 gives rise to the following corollary.

Corollary 3.2. Suppose that $F(z) = f(z) + \overline{g(z)} = \sum_{n=1}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb{D}$ and $f \prec h$ , where $h \in \mathcal{C}(\alpha)$ . Then $$\sum_{n=1}^{\infty} (|a_n| + |b_n|) r^n \le d(h(0), \partial h(\mathbb{D}))$$ holds for $|z| = r \leq \min\{1/3, R_{k,\alpha}\}$ , where $R_{k,\alpha}$ is the root in (0,1) of the equations $$\left(\frac{2K}{K+1}\right) \left(\frac{1 - (1-r)^{2\alpha - 1}}{2\alpha - 1}\right) + \frac{1 - 2^{2\alpha - 1}}{2\alpha - 1} = 0, \text{ if } \alpha \neq \frac{1}{2}$$ $$\left(\frac{2K}{K+1}\right) \log(1-r) + \log 2 = 0, \text{ if } \alpha = \frac{1}{2}$$ The number $R_{k,\alpha}$ is sharp if $R_{k,\alpha} \leq 1/3$ . Remark 3.3. If $0 \le \alpha < 1$ , K = 1, and f = h, then the Corollary 3.2 coincides with the result in [6, Theorem 2.2]. Remark 3.4. Taking $\phi(z) = \frac{1+z}{1-z}$ , the class $\mathcal{C}(\phi)$ becomes identical to the classical convex class $\mathcal{C}$ . With this $\phi$ , it is straightforward to obtain $k_{\phi}$ as follows $$k_{\phi}(z) = \frac{z}{(1-z)^2}$$ and $\hat{k}_{\phi}(r) = \frac{r}{(1-r)^2}$ . For the choice $\phi(z) = \frac{1+z}{1-z}$ , Theorem 3.1 immediately gives the result stated in the following corollary.

Corollary 3.3. Suppose that $F(z) = f(z) + \overline{g(z)} = \sum_{n=1}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb{D}$ and $f \prec h$ , where $h \in \mathcal{C}$ . Then $$\sum_{n=1}^{\infty} (|a_n| + |b_n|) r^n \le d(\phi(0), \partial \phi(\mathbb{D})) \text{ for } r \le \frac{K+1}{5K+1},$$ The number (K+1)/(5K+1) is sharp. Remark 3.5. Theorem C provides the Bohr radius for sense-preserving K-quasiconformal harmonic mappings in $\mathbb{D}$ with analytic parts subordinate to a convex function, while Corollary 3.3 determines the same Bohr radius for such mappings when the analytic part is subordinate to a normalized convex function. Proof of Theorem 3.1. Let $h \in \mathcal{C}(\phi)$ with the Taylor expansion $h(z) = z + \sum_{n=2}^{\infty} e_n z^n$ . Since $f \prec h$ , by applying Lemma 2.1 and Lemma 3.5, we have $$\sum_{n=1}^{\infty} |a_n| r^n \le \sum_{n=1}^{\infty} |e_n| r^n \le \sum_{n=1}^{\infty} |c_n| r^n \text{ for } |z| = r \le 1/3.$$ (3.5) By applying the similar arguments as in the proof of Theorem 2.1 and Lemma 2.2, we obtain $$\sum_{n=1}^{\infty} |b_n| r^n \le k \sum_{n=1}^{\infty} |a_n| r^n \le k \sum_{n=1}^{\infty} |c_n| r^n \text{ for } |z| = r \le 1/3.$$ (3.6) By using (3.5) and (3.6), we have $$C_{1}(r) := \sum_{n=1}^{\infty} (|a_{n}| + |b_{n}|) r^{n} \le (k+1) \sum_{n=1}^{\infty} |c_{n}| r^{n}$$ $$= (k+1) \hat{k}_{p}(r)$$ $$= P_{k,\phi}(r) - k_{\phi}(-1),$$ (3.7) where $P_{k,\phi}(r) = (k+1)\hat{k}_p(r) + k_\phi(-1)$ . Clearly, $P_{k,\phi}$ is a continuous and strictly increasing function on the interval (0,1). Also, we can deduce the following inequality by letting $r \to 1$ in Lemma 3.2(a) that $$-k_{\phi}(-1) \le k_{\phi}(1) \le \hat{k}_{\phi}(1) \le (k+1)\hat{k}_{\phi}(1).$$ It follows that $(k+1)\hat{k}_{\phi}(1) + k_{\phi}(-1) \ge 0$ and hence $\lim_{r\to 1} P_{k,\phi}(r) = (k+1)\hat{k}_{\phi}(1) + k_{\phi}(-1) \ge 0$ . Case I: $\lim_{r\to 1} P_{k,\phi}(r) = 0$ . Since $P_{k,\phi}(r)$ is strictly increasing function on (0,1), we obtain $P_{k,\phi}(r) < 0$ for all $r \in (0,1)$ . Also, by letting $r \to 1$ in the inequality of Lemma 3.2(a), we obtain $-k_{\phi}(-1) \leq d(h(0), \partial h(\mathbb{D}))$ . Then from the inequality (3.7), we have $C_1(r) \leq -k_{\phi}(-1) \leq d(h(0), \partial h(\mathbb{D}))$ for $|z| = r \leq 1/3$ . Case II: $\lim_{r\to 1} P_{k,\phi}(r) > 0$ . It is noted that $P_{k,\phi}(0) = k_{\phi}(-1) < 0$ ensures the existence of a unique root of $P_{k,\phi}(r) = 0$ in (0,1), denoted by $r_{k,\phi}$ . Then we have $P_{k,\phi}(r) \leq 0$ for $|z| = r \leq r_{k,\phi}$ . This gives $$C_1(r) \le -k_{\phi}(-1) \le d(h(0), \partial h(\mathbb{D})) \text{ for } |z| = r \le \min\{1/3, r_{k,\phi}\}.$$ To establish the sharpness of the radius obtained in Case II when $r_{k,\phi} \leq 1/3$ , we consider the function $F(z) = f(z) + \overline{g(z)}$ in $\mathbb{D}$ , where $f = h = k_{\phi}$ and $g = k\lambda k_{\phi}$ with non-negative Taylor coefficient of $k_{\phi}$ , and $|\lambda| = 1$ . Then the facts $\hat{k}_{\phi} = k_{\phi}$ and $d(k_{\phi}(0), \partial k_{\phi}(\mathbb{D})) = -k_{\phi}(-1)$ give that $$\sum_{n=1}^{\infty} (|c_n| + |k\lambda c_n|) r^n = (k+1)k_{\phi}(r) > -k_{\phi}(-1) = d(k_{\phi}(0), \partial k_{\phi}(\mathbb{D}))$$ holds for $r > r_{k,\phi}$ . This shows that the radius $r_{k,\phi}$ cannot be improved. Our focus here is on deriving the sharp Bohr radius for harmonic mappings with an analytic component subordinate to a function in $\mathcal{S}^*(\phi)$ .

Theorem 3.2. Suppose that $F(z) = f(z) + \overline{g(z)} = \sum_{n=1}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb{D}$ and $f \prec h$ , where $h \in \mathcal{S}^*(\phi)$ . Consider $$Q_{k,\phi}(r) := \left(\frac{2K}{K+1}\right)\hat{h}_{\phi}(r) + h_{\phi}(-1)$$ where $\hat{h}_{\phi}$ has the Taylor expansion $r + \sum_{n=2}^{\infty} |d_n| r^n$ . If $\lim_{r \to 1} Q_{k,\phi}(r) = 0$ , then $$\sum_{n=1}^{\infty} (|a_n| + |b_n|)r^n \le d(h(0), \partial h(\mathbb{D}))$$ (3.8) holds for $|z| = r \le 1/3$ . Otherwise, the inequality (3.8) satisfies for $|z| = r \le \min\{1/3, r_{k,\phi}^\}$ where $r_{k,\phi}$ is the root in (0,1) of the equation $Q_{k,\phi}(r) = 0$ . In this case, the result is sharp if $r_{k,\phi}^* \le 1/3$ and the Taylor coefficients of $h_{\phi}$ are nonnegative. Remark 3.6. If we choose $\phi(z) = \frac{1+Az}{1+Bz}$ ( $-1 \le B < A \le 1$ ), the class $\mathcal{S}^(\phi)$ reduces to the familiar class $\mathcal{S}^(A, B)$ consisting of the Janowski starlike functions. For this particular $\phi$ , by using Lemma 3.1, we obtain that the function $h_{\phi}(z)$ takes the following form $$h_{\phi}(z) = \begin{cases} z(1+Bz)^{\frac{A-B}{B}}, & B \neq 0\\ ze^{Az}, & B = 0. \end{cases}$$ and the absolute corresponding sum of Taylor expansion of $h_{\phi}$ is given by $$\hat{h}_{\phi}(r) = \begin{cases} r + \sum_{n=2}^{\infty} \prod_{m=0}^{n-2} \frac{|(B-A) + Bm|}{m+1} r^n, & B \neq 0 \\ re^{Ar}, & B = 0. \end{cases}$$ In the special case where the function $\phi(z)$ is defined by $\phi(z) = \frac{1+Az}{1+Bz}$ , Theorem 3.2 does not hold in its most general form, but instead leads to the more specific result outlined in the following corollary.

Corollary 3.4. Suppose that $F(z) = f(z) + \overline{g(z)} = \sum_{n=1}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb{D}$ and $f \prec h$ , where $h \in \mathcal{S}^*(A, B)$ . Consider $$Q_{A,B,k}(r) := \begin{cases} \left(\frac{2K}{K+1}\right) \left(r + \sum_{n=2}^{\infty} \prod_{m=0}^{n-2} \frac{|(B-A) + Bm|}{m+1} r^n\right) - (1-B)^{(A-B)/B}, & B \neq 0\\ \left(\frac{2K}{K+1}\right) r e^{Ar} - e^{-A}, & B = 0. \end{cases}$$ If $\lim_{r\to 1} Q_{A,B,k}(r) = 0$ , then $$\sum_{n=1}^{\infty} (|a_n| + |b_n|)r^n \le d(h(0), \partial h(\mathbb{D}))$$ (3.9) holds for $|z|=r\leq 1/3$ . Otherwise, the inequality (3.9) satisfies for $|z|=r\leq \min\{1/3,R_{A,B,k}^\}$ where $R_{A,B,k}$ is the root in (0,1) of the equation $Q_{A,B,k}(r)=0$ . In this case, the result is sharp if $R_{A,B,k}^*\leq 1/3$ and the Taylor coefficients of $h_\phi$ are non-negative. Remark 3.7. If we set K = 1 (so that k = 0) and f = h, Corollary 3.4 reduces to the result established in [9, Theorem 1]. Remark 3.8. On taking $\phi(z) = \frac{1+(1-2\alpha)z}{1-z} (0 \le \alpha < 1)$ , the class $\mathcal{S}^(\phi)$ becomes the family $\mathcal{S}^(\alpha)$ of starlike functions of order $\alpha$ . With this $\phi$ , the function $h_{\phi}$ is given by $$h_{\phi}(z) = \frac{z}{(1-z)^{2(1-\alpha)}}$$ and hence $$\hat{h}_{\phi}(r) = \frac{r}{(1-r)^{2(1-\alpha)}}.$$ For the particular choice $\phi(z) = \frac{1+(1-2\alpha)z}{1-z}$ , Theorem 3.2 leads to the result outlined in the following corollary.

Corollary 3.5. Suppose that $F(z) = f(z) + \overline{g(z)} = \sum_{n=1}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb{D}$ and $f \prec h$ , where $h \in \mathcal{S}^*(\alpha)$ . Then $$\sum_{n=1}^{\infty} (|a_n| + |b_n|) r^n \le d(h(0), \partial h(\mathbb{D}))$$ holds for $|z| = r \leq \min\{1/3, R_{k,\alpha}^\}$ , where $R_{k,\alpha}$ is the root in (0,1) of the equation $$\left(\frac{2K}{K+1}\right)\frac{r}{(1-r)^{2(1-\alpha)}} - \frac{1}{2^{2(1-\alpha)}} = 0.$$ The number $R_{k,\alpha}$ is sharp if $R_{k,\alpha}^ \leq 1/3$ . For $\alpha$ in the interval $[0, \frac{1}{2}\log_3(3+3k)]$ , the Bohr radius $R_{k,\alpha}$ remains less than or equal to 1/3, whereas for $\alpha$ in $(\frac{1}{2}\log_3(3+3k), 1)$ , it becomes strictly greater than 1/3 Remark 3.9. If we set K = 1 and f = h, then Corollary 3.5 reduces to the results in [25, Corollary 3.4] and [9, Corollary 1] Remark 3.10. The result of Corollary 3.5 coincides with that of [15, Theorem 3] for K = 1 and $\alpha \in [0, 1/2]$ . Remark 3.11. If we take $\phi(z) = \frac{1+z}{1-z}$ , the class $\mathcal{S}^(\phi)$ reduces to the class $\mathcal{S}$ of starlike functions. For this choice of $\phi$ , it can be easily seen that $$h_{\phi}(z) = \frac{z}{(1-z)^2}$$ and $\hat{h}_{\phi}(r) = \frac{r}{(1-r)^2}$ . With this choice $\phi(z) = \frac{1+z}{1-z}$ , Theorem 3.2 directly leads to the result shown in the following corollary.

Corollary 3.6. Suppose that $F(z) = f(z) + \overline{g(z)} = \sum_{n=1}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}$ is a sense-preserving K-quasiconformal harmonic mapping in $\mathbb D$ and $f \prec h$ , where $h \in \mathcal S^*$ . Then $$\sum_{n=1}^{\infty} (|a_n| + |b_n|) r^n \le d(h(0), \partial h(\mathbb{D}))$$ holds for $|z| = r \le R_k = (5K + 1 - 2\sqrt{6K^2 + 2K})/(K+1)$ . The number $R_k$ is sharp. Remark 3.12. If we consider K = 1 (which corresponds to k = 0) and f = h in Corollary 3.6, then $R_0 = 3 - 2\sqrt{2}$ becomes the sharp Bohr radius for the class of starlike functions. Remark 3.13. If we take K = 1 (which corresponds to k = 0), we see that $r_{0,\phi}^*$ is the root in (0,1) of the equation $Q_{k,\phi}(r) := \hat{h}_{\phi}(r) + h_{\phi}(-1) = 0$ . This indicates that the Theorem 3.2 includes the result of [25, Theorem 3.1] as a special case. Proof of Theorem 3.2. Let $h \in \mathcal{S}^*(\phi)$ with the Taylor expansion $h(z) = z + \sum_{n=2}^{\infty} e_n z^n$ . Since $f \prec h$ , by applying Lemma 3.1, we have $$\sum_{n=1}^{\infty} |a_n| r^n \le \sum_{n=1}^{\infty} |e_n| r^n \text{ for } |z| = r \le 1/3.$$ (3.10) Since $h(z)/z \prec h_{\phi}(z)/z$ , it follows along with the inequality (3.10) that $$\sum_{n=1}^{\infty} |a_n| r^n \le \sum_{n=1}^{\infty} |e_n| r^n \le \sum_{n=1}^{\infty} |d_n| r^n \text{ for } |z| = r \le 1/3.$$ (3.11) By applying the similar arguments as in the prof of Theorem 2.1 and Lemma 2.2, we obtain $$\sum_{n=1}^{\infty} |b_n| r^n \le k \sum_{n=1}^{\infty} |a_n| r^n \le k \sum_{n=1}^{\infty} |d_n| r^n \text{ for } |z| = r \le 1/3.$$ (3.12) By using (3.11) and (3.12), we have $$C_{2}(r) := \sum_{n=1}^{\infty} (|a_{n}| + |b_{n}|) r^{n} \le (k+1) \sum_{n=1}^{\infty} |d_{n}| r^{n}$$ $$= (k+1) \hat{h}_{\phi}(r)$$ $$= Q_{k,\phi}(r) - h_{\phi}(-1),$$ (3.13) where $Q_{k,\phi}(r) = (k+1)\hat{h}_p(r) + h_\phi(-1)$ . It is obvious that $Q_{k,\phi}$ is continuous and strictly increasing function on (0,1). Also, we can deduce the following inequality by letting $r \to 1$ in the inequality of Lemma 3.3 that $$-h_{\phi}(-1) \le h_{\phi}(1) \le \hat{h}_{\phi}(1) \le (k+1)\hat{h}_{\phi}(1).$$ It follows that $(k+1)\hat{h}_{\phi}(1) + h_{\phi}(-1) \geq 0$ and hence $\lim_{r\to 1} Q_{k,\phi}(r) = (k+1)\hat{h}_{\phi}(1) + h_{\phi}(-1) \geq 0$ . Case I: $\lim_{r\to 1} Q_{k,\phi}(r) = 0$ . Since $Q_{k,\phi}(r)$ is strictly increasing function on (0,1), we obtain $Q_{k,\phi}(r) < 0$ for all $r \in (0,1)$ . Since for all $z \in \mathbb{D}$ , $|h(z)| \ge -h_{\phi}(-r)$ , we obtain $d(h(0), \partial h(\mathbb{D})) \ge -h_{\phi}(-1)$ . Then from the inequality (3.13), we have $C_2(r) \le -h_{\phi}(-1) \le d(h(0), \partial h(\mathbb{D}))$ for $|z| = r \le 1/3$ . Case II: $\lim_{r\to 1} Q_{k,\phi}(r) > 0$ . It is noted that $Q_{k,\phi}(0) = h_{\phi}(-1) < 0$ ensures the existence of a unique root of $Q_{k,\phi}(r) = 0$ in (0,1), denoted by $r_{k,\phi}$ . Then we have $Q_{k,\phi}(r) \leq 0$ for $|z| = r \leq r_{k,\phi}$ . This gives $$C_2(r) \le -h_{\phi}(-1) \le d(h(0), \partial h(\mathbb{D})) \text{ for } |z| = r \le \min\{1/3, r_{k,\phi}^*\}.$$ To establish the sharpness of the radius obtained in Case II when $r_{k,\phi}^* \leq 1/3$ , we consider the function $F(z) = f(z) + \overline{g(z)}$ in $\mathbb{D}$ , where $f = h = h_{\phi}$ and $g = k\lambda h_{\phi}$ with non-negative Taylor coefficient of $h_{\phi}$ , and $|\lambda| = 1$ . Then the facts $\hat{h}_{\phi} = h_{\phi}$ and $d(h_{\phi}(0), \partial h_{\phi}(\mathbb{D})) = -h_{\phi}(-1)$ give that $$\sum_{n=1}^{\infty} (|d_n| + |k\lambda d_n|)r^n = (k+1)\hat{h}_{\phi}(r) > -h_{\phi}(-1) = d(h_{\phi}(0), \partial h_{\phi}(\mathbb{D}))$$ holds for $r > r_{k,\phi}$ . This shows that the radius $r_{k,\phi}$ cannot be improved. Acknowledgment: The first author is supported by Science and Engineering Research Board (SERB) (File No. SUR/2022/002244), Govt. of India, and the second author is supported by UGC-JRF (NTA Ref. No.: 221610103011), New Delhi, India.

Definition 3.1. [40] Let ϕ : D → C be an analytic univalent function with ϕ(0) = 1, ϕ′ (0) > 0, Re(ϕ(z)) > 0, symmetric with respect to the real axis, and starlike with respect to 1. Define the classes $$\mathcal{C}(\phi) := \left\{ f \in \mathcal{S} : 1 + \frac{zf''(z)}{f'(z)} \prec \phi(z) \right\}$$ and $$\mathcal{S}^*(\phi) := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec \phi(z) \right\}.$$ A function ϕ is termed as a Ma-Minda function if it satisfies the assumption outlined in Definition 3.1. It is significant to mention that for certain choices of ϕ, the classes S ∗ (ϕ)and C(ϕ) generate several important subclasses of starlike and convex functions, respectively. For example, if we choose ϕ(z) = (1 + Az)/(1 + Bz), the classes C(ϕ) and S ∗ (ϕ) become the Janowski convex class C(A, B) and the Janowski starlike class S ∗ (A, B), where −1 ≤ B < A ≤ 1, respectively. The concept of these classes was introduced by Janowski in [28, 29]. For ϕ(z) = (1 + (1 − 2α)z)/(1 − z), where 0 ≤ α < 1, the class S ∗ (ϕ) becomes the family S ∗ (α) of starlike functions of order α, and C(ϕ) becomes the family C(α) of convex functions of order α. The Bohr phenomenon for these classes was explored in [8, 25].