# Third Hankel Determinant for the Logarithmic Coefficients of Starlike Functions Associated with Sine Function

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- 1.
- If we let$$\Phi \left(z\right)=1+sinz,$$$${\mathcal{S}}_{sin}^{*}={\mathcal{S}}^{*}\left(1+sinz\right),$$
- 2.
- For the choice$$\Phi \left(z\right)=1+z-\frac{1}{3}{z}^{3},$$$${\mathcal{S}}_{nep}^{*}={\mathcal{S}}^{*}\left(1+z-\frac{1}{3}{z}^{3}\right),$$
- 3.
- If we put$$\Phi \left(z\right)=\sqrt{1+z},$$$${\mathcal{S}}_{\mathcal{L}}^{*}={\mathcal{S}}^{*}\left(\sqrt{1+z}\right),$$
- 4.
- Moreover, if we take$$\Phi \left(z\right)=1+\frac{4}{3}z+\frac{2}{3}{z}^{2},$$$${\mathcal{S}}_{car}^{*}={\mathcal{S}}^{*}\left(1+\frac{4}{3}z+\frac{2}{3}{z}^{2}\right),$$
- 5.
- Furthermore, if we pick $\Phi \left(z\right)={e}^{z}$ we obtain the class ${\mathcal{S}}_{exp}^{*}={\mathcal{S}}^{*}\left({e}^{z}\right),$ which was introduced and studied by Mendiratta et al. [6].
- 6.
- If we put $\Phi \left(z\right)=\sqrt{1+z}+z$, then we have the class of starlike functions associated with the crescent-shaped region as discussed in [7].

## 2. A Set of Lemmas

**Lemma 1.**

**Lemma 2.**

**Lemma 3.**

## 3. **Main Results**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 4. Concluding Remarks and Observations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Khan, B.; Aldawish, I.; Araci, S.; Khan, M.G.
Third Hankel Determinant for the Logarithmic Coefficients of Starlike Functions Associated with Sine Function. *Fractal Fract.* **2022**, *6*, 261.
https://doi.org/10.3390/fractalfract6050261

**AMA Style**

Khan B, Aldawish I, Araci S, Khan MG.
Third Hankel Determinant for the Logarithmic Coefficients of Starlike Functions Associated with Sine Function. *Fractal and Fractional*. 2022; 6(5):261.
https://doi.org/10.3390/fractalfract6050261

**Chicago/Turabian Style**

Khan, Bilal, Ibtisam Aldawish, Serkan Araci, and Muhammad Ghaffar Khan.
2022. "Third Hankel Determinant for the Logarithmic Coefficients of Starlike Functions Associated with Sine Function" *Fractal and Fractional* 6, no. 5: 261.
https://doi.org/10.3390/fractalfract6050261