# Hankel Determinants for Univalent Functions Related to the Exponential Function

## Abstract

**:**

## 1. Introduction

- if ${\varphi}_{1}\left(z\right)=\frac{1+(1-2\alpha )z}{1-z}$, then ${\mathcal{S}}^{*}\left({\varphi}_{1}\right)={\mathcal{S}}^{*}\left(\alpha \right)$ is the class of starlike functions of order $\alpha $,
- if ${\varphi}_{2}\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\alpha}$, then ${\mathcal{S}}^{*}\left({\varphi}_{2}\right)={\mathcal{S}}_{\alpha}^{*}$ is the class of strongly starlike functions of order $\alpha $,
- if ${\varphi}_{3}\left(z\right)=\frac{1+Az}{1+Bz}$, then ${\mathcal{S}}^{*}\left({\varphi}_{3}\right)={\mathcal{S}}^{*}[A,B]$ is the class of Janowski starlike functions.

## 2. Auxiliary Lemmas

**Lemma**

**1**

- 1.
- $2{p}_{2}={p}_{1}{}^{2}+x(4-{p}_{1}{}^{2})$,
- 2.
- $4{p}_{3}={p}_{1}{}^{3}+2{p}_{1}(4-{p}_{1}{}^{2})x-{p}_{1}(4-{p}_{1}{}^{2}){x}^{2}+2(4-{p}_{1}{}^{2}){(1-|x|}^{2})y$,

**Lemma**

**2**

**.**If $P\in \mathcal{P}$ is of the form (9) and $\mu \in [0,1]$, then the following sharp estimates hold

- 1.
- $|{p}_{n+m}-\mu {p}_{m}{p}_{n}|\le 2\phantom{\rule{1.em}{0ex}}$ for $\phantom{\rule{1.em}{0ex}}n,m=1,2,\dots $,
- 2.
- $|{p}_{2}-\mu {p}_{1}{}^{2}|\le \left\{\begin{array}{cc}2-\mu |{p}_{1}{|}^{2}\hfill & ,\phantom{\rule{4pt}{0ex}}\mu \in [0,1/2]\phantom{\rule{4pt}{0ex}},\hfill \\ 2-(1-\mu )|{p}_{1}{|}^{2}\hfill & ,\phantom{\rule{4pt}{0ex}}\mu \in [1/2,1]\phantom{\rule{4pt}{0ex}}.\hfill \end{array}\right.$

## 3. Coefficient Problems for ${\mathcal{S}}_{\mathit{e}}^{\mathbf{*}}$

**Theorem**

**1.**

- 1.
- $|{a}_{3}|\le {\textstyle \frac{1}{4}}(2+{\textstyle \frac{1}{4}}{p}^{2})$
- 2.
- $|{a}_{4}|\le {\textstyle \frac{1}{6}}(2+{\textstyle \frac{1}{2}}p-{\textstyle \frac{1}{48}}{p}^{3})$
- 3.
- $|{a}_{5}|\le {\textstyle \frac{1}{8}}(2+{\textstyle \frac{1}{3}}p+{\textstyle \frac{1}{144}}{p}^{4})$.

**Proof.**

**Corollary**

**1.**

- 1.
- $|{a}_{2}|\le 1$
- 2.
- $|{a}_{3}|\le {\textstyle \frac{3}{4}}$
- 3.
- $|{a}_{4}|\le {\textstyle \frac{17}{36}}$
- 4.
- $|{a}_{5}|\le {\textstyle \frac{25}{72}}$.

**Theorem**

**2.**

**Theorem**

**3.**

**Corollary**

**2.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Corollary**

**2.**

**Theorem**

**4.**

**Corollary**

**3.**

**Proof**

**of**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

## 4. Coefficient Problems for ${\mathcal{K}}_{\mathit{e}}$

**Theorem**

**6.**

- 1.
- $|{b}_{3}|\le {\textstyle \frac{1}{12}}(2+{\textstyle \frac{1}{4}}{p}^{2})$
- 2.
- $|{b}_{4}|\le {\textstyle \frac{1}{24}}(2+{\textstyle \frac{1}{2}}p-{\textstyle \frac{1}{48}}{p}^{3})$
- 3.
- $|{b}_{5}|\le {\textstyle \frac{1}{40}}(2+{\textstyle \frac{1}{3}}p+{\textstyle \frac{1}{144}}{p}^{4})$.

**Corollary**

**4.**

- 1.
- $|{b}_{2}|\le {\textstyle \frac{1}{2}}$
- 2.
- $|{b}_{3}|\le {\textstyle \frac{1}{4}}$
- 3.
- $|{b}_{4}|\le {\textstyle \frac{17}{144}}$
- 4.
- $|{b}_{5}|\le {\textstyle \frac{5}{72}}$.

**Theorem**

**7.**

**Theorem**

**8.**

**Corollary**

**5.**

**Proof**

**of**

**Theorem**

**8.**

**Proof**

**of**

**Corollary**

**5.**

**Theorem**

**9.**

**Corollary**

**6.**

**Proof**

**of**

**Theorem**

**9.**

**Theorem**

**10.**

**Proof.**

## 5. Concluding Remark

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Zaprawa, P.
Hankel Determinants for Univalent Functions Related to the Exponential Function. *Symmetry* **2019**, *11*, 1211.
https://doi.org/10.3390/sym11101211

**AMA Style**

Zaprawa P.
Hankel Determinants for Univalent Functions Related to the Exponential Function. *Symmetry*. 2019; 11(10):1211.
https://doi.org/10.3390/sym11101211

**Chicago/Turabian Style**

Zaprawa, Paweł.
2019. "Hankel Determinants for Univalent Functions Related to the Exponential Function" *Symmetry* 11, no. 10: 1211.
https://doi.org/10.3390/sym11101211