# Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function

## Abstract

**:**

## 1. Introduction

- $H\left(U\right)$ denotes the class of analytic functions in the unit disc of the complex plane;
- For n a positive integer and $a\in \mathbb{C}$, let$$H[a,n]=\{f\in H\left(U\right):\phantom{\rule{4pt}{0ex}}f\left(z\right)=a+{a}_{n}{z}^{n}+{a}_{n+1}{z}^{n+1}+\dots ,\phantom{\rule{4pt}{0ex}}z\in U\},\mathrm{with}{H}_{0}=H[0,1];$$
- ${A}_{n}=\{f\in H\left(U\right):\phantom{\rule{4pt}{0ex}}f\left(z\right)=z+{a}_{n+1}{z}^{n+1}+\dots ,\phantom{\rule{4pt}{0ex}}z\in U\},\mathrm{with}{A}_{1}=A;$
- $K=\left\{{\displaystyle f\in H\left(U\right):\phantom{\rule{4pt}{0ex}}\mathrm{Re}\frac{z{f}^{\prime \prime}\left(z\right)}{{f}^{\prime}\left(z\right)}+1>0,\phantom{\rule{4pt}{0ex}}z\in U}\right\}$ denotes the class of convex functions in the unit disc;
- $S=\{f\in H\left(U\right):\phantom{\rule{4pt}{0ex}}f\left(z\right)=z+{a}_{2}{z}^{2}+{a}_{3}{z}^{3}+\dots ,\phantom{\rule{4pt}{0ex}}f\mathrm{univalent}\mathrm{in}U\}$.

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Lemma**

**1.**

## 2. Results

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof.**

**Remark**

**2.**

**Corollary**

**1.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

**Corollary**

**2.**

**Proof.**

**Remark**

**5.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Example**

**1.**

## 3. Discussion

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Oros, G.I.
Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function. *Symmetry* **2021**, *13*, 259.
https://doi.org/10.3390/sym13020259

**AMA Style**

Oros GI.
Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function. *Symmetry*. 2021; 13(2):259.
https://doi.org/10.3390/sym13020259

**Chicago/Turabian Style**

Oros, Georgia Irina.
2021. "Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function" *Symmetry* 13, no. 2: 259.
https://doi.org/10.3390/sym13020259