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Abstract
In this paper, we introduce a new subclass of close-to-convex harmonic functions. We present a sufficient coefficient condition for a function to be a member of this class. Furthermore, we establish a distortion theorem. These results lay the groundwork for extending the findings to function classes involving higher-order derivatives.
Results & Lemmas (9)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 3.1
Lemma 3.1. Let and be analytic functions in, such that, and for each ( ), is close-to-convex. Then, is close-to-convex in. The result we…
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
Theorem 3.2. if and only if for each ( ).
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
Lemma 3.3. [24] Let for. Then, (3.1) where is given by (1.2).
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
Theorem 3.4. If F is a function in the class, then F is close-to-convex of order in the region.
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
Theorem 3.5. Every function in the class is close-to-convex within the region.
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
Theorem 3.6. Let. For, the following inequalities hold: For the function, every outcome is sharp and every equality is holds.
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
Theorem 3.7. Let with the series expansions given by (1.1). If the following inequality holds: (3.6) then.
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:
(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
Theorem 3.8. Assuming, the following inequalities hold for all z: These are sharp inequality for the function.
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
Theorem 3.9. The class is closed under convex combinations.
Theorem 3.9. The class $\mathcal{KH}^0(k,\gamma)$ is closed under convex combinations.
Definitions (1)
Def 1.1
Definition 1.1. The class is defined as the collection of functions that adhere to the following inequality: (1.3) where and is given by…
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 $\mathcal{KH}^0(k,\gamma)$
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 $\mathfrak{f}(z)=z+\frac{99}{200}\overline{z}^2$ .
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 $\mathfrak{f}=z+\frac{1}{10}\overline{z}^2$ .
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 $\mathfrak{f}=z+\frac{33}{100}\overline{z}^3.$
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 $\mathfrak{f}=z+\frac{1}{15}\overline{z}^3$ .
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 $\mathfrak{f}=z+\frac{99}{500}\overline{z}^5$ .
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 $\mathfrak{f}=z+\frac{1}{25}\overline{z}^5$ .
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