Introducing a Novel Subclass of Harmonic Functions with Close-to-Convex Properties

Lemma 3.1. Let $\mathfrak{u}$ and $\mathfrak{v}$ be analytic functions in $\mathbb{E}$ , such that $|\mathfrak{v}'(0)| < |\mathfrak{u}'(0)|$ , and for each $\varepsilon$ ( $|\varepsilon| = 1$ ), $F_{\varepsilon} = \mathfrak{u} + \varepsilon \mathfrak{v}$ is close-to-convex. Then, $\mathfrak{f} = \mathfrak{u} + \overline{\mathfrak{v}}$ is close-to-convex in $\mathbb{E}$ . The result we will present now establishes a connection between the $\mathcal{KH}^0(k,\gamma)$ harmonic function class and the $\mathcal{K}(k,\gamma)$ analytic function class.

Theorem 3.2. $\mathfrak{f} = \mathfrak{u} + \overline{\mathfrak{v}} \in \mathcal{KH}^0(k,\gamma)$ if and only if $F_{\varepsilon} = \mathfrak{u} + \varepsilon \mathfrak{v} \in \mathcal{K}(k,\gamma)$ for each $\varepsilon$ ( $|\varepsilon| = 1$ ).

Lemma 3.3. [24] Let $\phi(z) = z + \sum_{m=2}^{\infty} c_m z^m \in \mathcal{S}^* \left( \frac{k-1}{k} \right)$ for $k \ge 1$ . Then, (3.1) $$\Phi_k(z) = \frac{\phi_k(z)}{z^{k-1}} = z + \sum_{m=2}^{\infty} C_m z^m \in S^*$$ where $\phi_k(z)$ is given by (1.2).

Theorem 3.4. If F is a function in the class $K(k, \gamma)$ , then F is close-to-convex of order $\gamma$ in the region $\mathbb{E}$ .

Theorem 3.5. Every function in the class $\mathcal{KH}^0(k,\gamma)$ is close-to-convex within the region $\mathbb{E}$ .

Theorem 3.6. Let $\mathfrak{f} = \mathfrak{u} + \overline{\mathfrak{v}} \in \mathcal{KH}^0(k,\gamma)$ . For $m \geq 2$ , the following inequalities hold: $$|u_m| + |v_m| \le \gamma + m(1 - \gamma).$$ For the function $f(z) = z + [\gamma + m(1 - \gamma)]z^m$ , every outcome is sharp and every equality is holds.

Theorem 3.7. Let $\mathfrak{f} = \mathfrak{u} + \overline{\mathfrak{v}} \in \mathcal{SH}^0$ with the series expansions given by (1.1). If the following inequality holds: <span id="page-7-0"></span>(3.6) $$\sum_{m=2}^{\infty} 2m (|u_m| + |v_m|) + \sum_{m=2}^{\infty} (|1 - 2\gamma| + 1) |C_m| \le 2(1 - \gamma),$$ then $\mathfrak{f} \in \mathcal{KH}^0(k,\gamma)$ .

Theorem 3.8. Assuming $\mathfrak{f} = \mathfrak{u} + \overline{\mathfrak{v}} \in \mathcal{KH}^0(k,\gamma)$ , the following inequalities hold for all z: $$|z| + \sum_{m=2}^{\infty} (-1)^{m-1} [m(1-\gamma) + \gamma] |z|^m \le |\mathfrak{f}(z)| \le |z| + \sum_{m=2}^{\infty} [m(1-\gamma) + \gamma] |z|^m.$$ These are sharp inequality for the function $f(z) = z + \sum_{m=2}^{\infty} [m(1-\gamma) + \gamma]z^m$ .

Theorem 3.9. The class $\mathcal{KH}^0(k,\gamma)$ is closed under convex combinations.

Definition 1.1. The class $\mathcal{KH}^0(k,\gamma)$ is defined as the collection of functions $\mathfrak{f} = \mathfrak{u} + \overline{\mathfrak{v}} \in \mathcal{SH}^0$ that adhere to the following inequality: (1.3) $$\operatorname{Re}\left\{\frac{z^{k}\mathfrak{u}'(z)}{\phi_{k}(z)} - \gamma\right\} > \left|\frac{z^{k}\mathfrak{v}'(z)}{\phi_{k}(z)}\right|,$$ where $0 \le \gamma < 1$ and $\phi_k(z)$ is given by (1.2). Specifically, when $\mathfrak{v}(z) \equiv 0$ , the class $\mathcal{KH}^0(k,\gamma)$ reduces to the class $\mathcal{K}(k,\gamma)$ . Also, by setting $\mathfrak{v}(z) \equiv 0$ , k=2 and $\gamma=0$ , we obtain $\mathcal{KH}^0(2,0)=\mathcal{K}_s$ . The inclusion of the co-analytic term $\overline{\mathfrak{v}(z)}$ extends these classes, providing a more general framework that accommodates both analytic and co-analytic components. Additionally, when $\phi(z)=z$ , and by appropriately selecting the parameters, the class $\mathcal{KH}^0(k,\gamma)$ can be reduced to several well-known subclasses of harmonic functions, as outlined below. ``` \begin{split} & \text{i} \ \mathcal{KH}^{0}(k,0) = \mathcal{PH}^{0} \ [6]. \\ & \text{ii} \ \mathcal{KH}^{0}(k,\gamma) = \mathcal{PH}^{0}(\gamma) \ [7,\,8]. \\ & \text{ii} \ \mathcal{KH}^{0}(k,0) = \mathcal{WH}^{0}(0) \ ([9]). \\ & \text{iv} \ \mathcal{KH}^{0}(k,\gamma) = \mathcal{WH}^{0}(0,\gamma) \ ([10]). \\ & \text{v} \ \mathcal{KH}^{0}(k,\gamma) = \mathcal{AH}^{0}(1,0,\gamma) \ ([11]). \\ & \text{vi} \ \mathcal{KH}^{0}(k,0) = \mathcal{RH}^{0}(0,0) \ ([12]). \end{split} ``` For further details on harmonic function classes defined by differential inequality, see [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. In this work, we investigate the distortion theorems and coefficient bounds for functions in the class $\mathcal{KH}^0(k,\gamma)$ and demonstrate that functions within this class exhibit close-to-convex behavior. 2. Examples of Functions in the Class $\mathcal{KH}^0(k,\gamma)$ <span id="page-2-0"></span>Example 2.1. Let $\mathfrak{f} = \mathfrak{u} + \overline{\mathfrak{v}} = z + \frac{1-\gamma}{m}\overline{z}^m$ and $\phi(z) = z$ . For $0 \leq \gamma < 1$ and |z| < 1, we have $$\operatorname{Re}\left\{\frac{z^{k}\mathfrak{u}'(z)}{\phi_{k}(z)} - \gamma\right\} = 1 - \gamma > (1 - \gamma)\left|z\right|^{m-1} = \left|\frac{z^{k}\mathfrak{v}'(z)}{\phi_{k}(z)}\right|.$$ Hence, $f \in \mathcal{KH}^0(k, \gamma)$ . The following examples can be given for the special case of the parameters in Example 2.1. Example 2.2. Let $\mathfrak{f}=z+\frac{99}{200}\overline{z}^2$ , $\gamma=\frac{1}{100}$ and $\phi(z)=z$ . Then $\mathfrak{f}\in\mathcal{KH}^0(k,\frac{33}{100})$ . The unit disk is mapped to a starlike region by the function f. The depiction in Figure 1 showcases the image of the set $\mathbb{E}$ under the transformation defined by $\mathfrak{f}(z)=z+\frac{99}{200}\overline{z}^2$ .  <span id="page-2-1"></span>Figure 1. Under the map $\mathfrak{f}=z+\frac{99}{200}\overline{z}^2$ , the image of the unit disk. Example 2.3. Let $\mathfrak{f}=z+\frac{1}{10}\overline{z}^2$ , $\gamma=\frac{4}{5}$ , and $\phi(z)=z$ . Then $\mathfrak{f}\in\mathcal{KH}^0(k,\frac{4}{5})$ . The unit disk is mapped to a convex region by the function f. The depiction in Figure 2 showcases the image of the set $\mathbb E$ under the transformation defined by $\mathfrak{f}=z+\frac{1}{10}\overline{z}^2$ .  <span id="page-3-0"></span>Figure 2. Under the map $\mathfrak{f}=z+\frac{1}{10}\overline{z}^2,$ the image of the unit disk. Example 2.4. Let $\mathfrak{f}=z+\frac{33}{100}\overline{z}^3,\ \gamma=\frac{1}{100}$ and $\phi(z)=z.$ Then $\mathfrak{f}\in\mathcal{KH}^0(k,\frac{33}{100}).$ The unit disk is mapped to a starlike region by the function f. The depiction in Figure 3 showcases the image of the set $\mathbb E$ under the transformation defined by $\mathfrak{f}=z+\frac{33}{100}\overline{z}^3.$  <span id="page-3-1"></span>FIGURE 3. Under the map $\mathfrak{f}=z+\frac{33}{100}\overline{z}^3,$ the image of the unit disk. Example 2.5. Let $\mathfrak{f}=z+\frac{1}{15}\overline{z}^3,\ \gamma=\frac{4}{5}$ and $\phi(z)=z$ . Then $\mathfrak{f}\in\mathcal{KH}^0(k,\frac{4}{5})$ . The unit disk is mapped to a convex region by the function f. The depiction in Figure 4 showcases the image of the set $\mathbb E$ under the transformation defined by $\mathfrak{f}=z+\frac{1}{15}\overline{z}^3$ .  <span id="page-4-0"></span>Figure 4. Under the map $\mathfrak{f}=z+\frac{1}{15}\overline{z}^3$ , the image of the unit disk. Example 2.6. Let $\mathfrak{f}=z+\frac{99}{500}\overline{z}^5,\ \gamma=\frac{1}{100}$ and $\phi(z)=z$ . Then $\mathfrak{f}\in\mathcal{KH}^0(k,\frac{99}{500})$ . The unit disk is mapped to a starlike region by the function f. The depiction in Figure 5 showcases the image of the set $\mathbb E$ under the transformation defined by $\mathfrak{f}=z+\frac{99}{500}\overline{z}^5$ .  <span id="page-4-1"></span>FIGURE 5. Under the map $\mathfrak{f}=z+\frac{99}{500}\overline{z}^5,$ the image of the unit disk. Example 2.7. Let $\mathfrak{f}=z+\frac{1}{25}\overline{z}^5,\ \gamma=\frac{4}{5}$ and $\phi(z)=z$ . Then $\mathfrak{f}\in\mathcal{KH}^0(k,\frac{4}{5})$ . The unit disk is mapped to a convex region by the function f. The depiction in Figure 6 showcases the image of the set $\mathbb E$ under the transformation defined by $\mathfrak{f}=z+\frac{1}{25}\overline{z}^5$ .