On Meromorphic Harmonic Functions with Respect to -Symmetric Points

Theorem 2.1. Let $f = h + \overline{g}$ be of the form (1.3) and $f_k = h_k + \overline{g_k}$ where $h_k$ and $g_k$ are given by (1.9). If $$\sum_{n=1}^{\infty} \left[ \left[ (n-1)k + 1 + \alpha \right] |a_{(n-1)k+1}| + \left[ (n-1)k + 1 - \alpha \right] |b_{(n-1)k+1}| \right] \Omega_{m}^{j}(n,k)$$ $$+ \sum_{\substack{n=2\\n \neq lk+1}}^{\infty} n^{j+1} C(m,n) \left[ |a_{n}| + |b_{n}| \right] \le 1 - \alpha,$$ (2.1) where $j, m \in \mathbb{N}_0$ , $0 \le \alpha < 1$ , $k \ge 1$ and $\Omega_m^j(n, k) = ((n-1)k+1)^j C(m, nk+1)$ , then f is harmonic univalent, sense preserving in $\mathbb{U} \setminus \{0\}$ and $f \in \mathcal{MHS}_s^{(k)}(j, m, \alpha)$ .

Theorem 2.2. Let $f_j = h_j + \overline{g_j}$ , where $h_j$ and $g_j$ are given by (1.11), and $f_{k_j} = h_{k_j} + \overline{g_{k_j}}$ where $h_{k_j}$ and $g_{k_j}$ are given by (1.12). Then, $f_j \in \mathcal{MLS}_s^{(k)}(j, m, \alpha)$ , if and only if the inequality (2.1) holds for the coefficient of $f_j = h_j + \overline{g_j}$ and $f_{k_j} = h_{k_j} + \overline{g_{k_j}}$ .

Theorem 3.1. If $f_j = h_j + \overline{g_j} \in \overline{\mathcal{MHS}_s^{(k)}}(j, m, \alpha) \text{ and } 0 < |z| = r < 1, \text{ then}$ $\frac{1}{r} - \frac{1 - \alpha}{2^j (m+1)(2-\alpha)} r \le |f_j(z)| \le \frac{1}{r} + \frac{1 - \alpha}{2^j (m+1)(2-\alpha)} r. \tag{3.1}$

Corollary 3.2. If $f_j \in \overline{\mathcal{MHS}_s^{(k)}}(j, m, \alpha)$ , then $$f_j(\mathbb{U}\setminus\{0\}) \subset \left\{w: |w| < \frac{2^j(m+1)(2-\alpha) - (1-\alpha)}{2^j(m+1)(2-\alpha)}\right\}.$$ (3.3) Next, we determine the extreme points of closed convex hulls of $\mathcal{MHS}_s^{(k)}(j, m, \alpha)$ denoted by $\operatorname{cloo} \mathcal{MHS}_s^{(k)}(j, m, \alpha)$ .

Theorem 3.3. Let $f_j = h_j + \overline{g_j}$ where $h_j$ and $g_j$ are given by (1.11). Then, $f_j \in \overline{\mathcal{MS}_s^{(k)}}(j, m, \alpha)$ if and only if $$f_{j,n}(z) = \sum_{n=0}^{\infty} (x_n h_{j_n}(z) + y_n g_{j_n}(z)), \tag{3.4}$$ where $h_{j,0} = g_{j,0}(z) = (-1)^j/z$ , $h_{j,n}(z) = (-1)^j/z + ((1-\alpha)/n^jC(m,n)(n+\alpha\Phi_n))z^n$ (n = 1,2,3,...), $g_{j,n}(z) = (-1)^j/z + (-1)^j((1-\alpha)/n^jC(m,n)(n-\alpha\Phi_n))\overline{z}^k$ (n = 1,2,3,...), $\sum_{n=0}^{\infty}(x_n+y_n) = 1$ , $x_n \ge 0$ , $y_n \ge 0$ . In particular, the extreme points of $\mathcal{MHS}_s^{(k)}(j,m,\alpha)$ are $\{h_{j,n}\}$ and $\{g_{j,n}\}$ .

Theorem 4.1. For $0 \le \beta \le \alpha < 1$ , let $f_j \in \overline{\mathcal{MHS}_s^{(k)}}(j, m, \alpha)$ and $F_j \in \overline{\mathcal{MHS}_s^{(k)}}(j, m, \beta)$ . Then, $f_j * F_j \in \overline{\mathcal{MHS}_s^{(k)}}(j, m, \alpha) \subset \overline{\mathcal{MHS}_s^{(k)}}(j, m, \beta)$ .

Theorem 4.2. Let the functions $f_{j,t}$ defined by (4.3) be in the class $\overline{\mathcal{MS}_s^{(k)}}(j, m, \alpha)$ for every $t = 1, 2, ..., \rho$ . Then, the functions $\xi_t(z)$ defined by $$\xi_t(z) = \sum_{t=1}^{\rho} c_t f_{j_n}(z), \quad (0 \le c_t \le 1), \tag{4.4}$$ are also in the class $\overline{\mathcal{MSS}_s^{(k)}}(j, m, \alpha)$ , where $\sum_{t=1}^{\rho} c_t = 1$ .

Corollary 4.3. The class $\overline{\mathcal{MS}_s^{(k)}}(j, m, \alpha)$ is close under convex linear combination.

Definition 1.1. For $j,m \in \mathbb{N}_0$ , $0 \le \alpha < 1$ and $k \ge 1$ , let $\mathcal{MHS}_s^{(k)}(j,m,\alpha)$ denote the class of meromorphic harmonic functions f of the form (1.3) such that $$\operatorname{Re}\left\{-\frac{\mathfrak{D}_{m}^{j+1}f(z)}{\mathfrak{D}_{m}^{j}f_{k}(z)}\right\} > 0, \quad (z \in \mathbb{U} \setminus \{0\}), \tag{1.7}$$ where $$\mathfrak{D}_m^j f_k(z) = \mathfrak{D}_m^j h_k + (-1)^j \overline{\mathfrak{D}_m^j g_k} \quad (j, m \in \mathbb{N}_0, k \ge 1), \tag{1.8}$$ $$h_k(z) = \frac{(-1)^j}{z} + \sum_{n=1}^{\infty} a_n \Phi_n z^n, \qquad g_k(z) = \sum_{n=1}^{\infty} \Phi_n z^n,$$ (1.9) $$\Phi_n = \frac{1}{k} \sum_{\nu=0}^{k-1} \varepsilon^{(n-1)\nu}, \quad \left(k \ge 1; \varepsilon = \exp\left(\frac{2\pi i}{k}\right)\right). \tag{1.10}$$ For more details about harmonic functions with respect to k-symmetric points, see papers [7, 8] given by the authors. Also, note that $\mathcal{MHS}_{s}^{(2)}(j,0,\alpha) \subset MHS_{s}^{*}(n,\alpha)$ was introduced by Bostancı and Öztürk [9]. Finally, let $\overline{\mathcal{MHS}_s^{(k)}}(j, m, \alpha)$ denote the subclass of $\mathcal{MHS}_s^{(k)}(j, m, \alpha)$ consist of harmonic functions $f_j = h_j + \overline{g_j}$ such that $h_j$ and $g_j$ are of the form $$h_j(z) = \frac{(-1)^j}{z} + \sum_{n=1}^{\infty} |a_n| z^n, \qquad g_j(z) = (-1)^j \sum_{n=1}^{\infty} |b_n| z^n.$$ (1.11) Also, let $f_{k_i} = h_{k_i} + \overline{g_{k_i}}$ where $h_{k_i}$ and $g_{k_i}$ are of the form $$h_{k_j}(z) = \frac{(-1)^j}{z} + \sum_{n=1}^{\infty} \Phi_n |a_n| z^n, \qquad g_{k_j}(z) = (-1)^j \sum_{n=1}^{\infty} \Phi_n |b_n| z^n,$$ (1.12) where $\Phi_n$ is given by (1.10). In this paper, we will give a sufficient condition for functions $f = h + \overline{g}$ , where h and g given by (1.3) to be in the class $\mathcal{MAS}_s^{(k)}(j,m,\alpha)$ . Indeed, it is shown that this coefficient condition is also necessary for functions to be in the class $\overline{\mathcal{MAS}_s^{(k)}}(j,m,\alpha)$ . Also, we obtain distortion bounds and characterize the extreme points for functions in $\overline{\mathcal{MAS}_s^{(k)}}(j,m,\alpha)$ . Convolution and closure theorems are also obtained.