# New Conditions for Univalence of Confluent Hypergeometric Function

## Abstract

**:**

## 1. Introduction

**Definition**

**1**

**.**Let a and c be complex numbers with $c\ne 0,-1,-2,\dots $ and consider

**Definition**

**2**

**.**Let f and F be members of $H\left(U\right)$. The function f is said to be subordinate to F, written $f\prec F$ or $f\left(z\right)\prec F\left(z\right)$, if there exists a function w, analytic in U, with $w\left(0\right)=0$ and $\left|w\right(z\left)\right|<1$ and such that $f\left(z\right)=F\left(w\right(z\left)\right)$. If F is univalent, then $f\prec F$ if and only if $f\left(0\right)=F\left(0\right)$ and $f\left(U\right)\subset F\left(U\right)$.

**Definition**

**3**

**.**Let $\psi (r,s,t;z):{\mathbb{C}}^{2}\times U\to \mathbb{C}$ and let h be univalent in U. If p is analytic and satisfies the (second-order) differential subordination

**Lemma**

**1**

**.**Let q be univalent in U and let θ and ϕ be analytic in a domain D containing $q\left(U\right)$, with $\varphi \left(w\right)\ne 0$, when $w\in q\left(U\right)$. Set $Q\left(z\right)=z{q}^{\prime}\left(z\right)\varphi \left[q\left(z\right)\right]$, $h\left(z\right)=\theta \left[q\right(z\left)\right]+Q\left(z\right)$ and suppose that either

- (i)
- h in convex , or
- (ii)
- Q is starlike.

- (iii)
- $\mathrm{Re}\frac{z{h}^{\prime}\left(z\right)}{Q\left(z\right)}=\mathrm{Re}\left[{\displaystyle \frac{\theta \left[q\right(z\left)\right]}{\varphi \left(q\right(z\left)\right]}+\frac{zQ\left(z\right)}{Q\left(z\right)}}\right]>0$.

## 2. Results

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Corollary**

**1.**

**Proof.**

**Remark**

**2.**

- (i)
- $a>0$ and $c\ge a$, or
- (ii)
- $a\le 0$ and $c\ge 1+{(1+{a}^{2})}^{\frac{1}{2}}$, for $z\in U$.

**Corollary**

**2.**

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

**Corollary**

**3.**

**Proof.**

**Remark**

**4.**

- (i)
- $a>-1$and$c\ge a$, or
- (ii)
- $a\le -1$and$c\ge \sqrt{1+{(a+1)}^{2}}$.

**Corollary**

**4.**

**Proof.**

**Remark**

**5.**

**Theorem**

**3.**

**Proof.**

**Remark**

**6.**

**Example**

**1.**

## 3. Discussion

## Funding

## Conflicts of Interest

## References

- De Branges, L. A proof of the Bieberbach conjecture. Acta Math.
**1985**, 154, 137–152. [Google Scholar] [CrossRef] - Merkes, E.P.; Scott, W.T. Starlike hypergeometric functions. Proc. Am. Math. Soc.
**1961**, 12, 885–888. [Google Scholar] [CrossRef] - Kreyszig, E.; Todd, J. The radius of univalence of the error function. Numer. Math.
**1959**, 1, 78–89. [Google Scholar] [CrossRef] - Kreyszig, E.; Todd, J. On the radius of univalence of the function exp z
^{2}${\int}_{0}^{z}$ exp(−t^{2})dt. Pac. J. Math.**1959**, 9, 123–127. [Google Scholar] [CrossRef] [Green Version] - Miller, S.S.; Mocanu, P.T. Univalence of Gaussian and confluent hypergeometric Functions. Proc. Am. Math. Soc.
**1990**, 110, 333–342. [Google Scholar] [CrossRef] - Ruscheweyh, S.; Singh, V. On the order of starlikeness of hypergeometric functions. J. Math. Anal. App.
**1986**, 113, 1–11. [Google Scholar] [CrossRef] [Green Version] - Ponnusamy, S.; Vuorinen, M. Univalence and convexity properties for confluent hypergeometric functions. Complex Var. Theory Appl.
**1998**, 36, 73–97. [Google Scholar] [CrossRef] - Ponnusamy, S.; Rønning, F. Starlikeness properties for convolutions involving hypergeometric series. Ann. Univ. Mariae Curie-Sklodowska
**1998**, 52, 141–155. [Google Scholar] - Ponnusamy, S. Close-to-convexity properties of Gaussian hypergeometric functions. J. Comput. Appl. Math.
**1998**, 88, 327–337. [Google Scholar] [CrossRef] [Green Version] - Chaudhry, M.A.; Qadir, A.; Srivastava, H.M.; Paris, R.B. Extended hypergeometric and confluent hypergeometric functions. App. Math. Comput.
**2004**, 159, 589–602. [Google Scholar] [CrossRef] - Porwal, S.; Kumar, S. Confluent hypergeometric distribution and its applications on certain classes of univalent functions. Afrika Matematika
**2017**, 28, 1–8. [Google Scholar] [CrossRef] - Luo, X.-D.; Lin, W.-C. On the Nevanlinna characteristic of confluent hypergeometric functions. Complex Var. Elliptic Equ.
**2020**, 65, 200–214. [Google Scholar] [CrossRef] - Ghanim, F.; Al-Janaby, H.F. An analytical study on Mittag-Leffler-confluent hypergeometric functions with fractional integral operator. Math. Methods Appl. Sci.
**2020**. [Google Scholar] [CrossRef] - Miller, S.S.; Mocanu, P.T. Differential Subordinations. Theory and Applications; Marcel Dekker, Inc.: New York, NY, USA; Basel, Switzerland, 2000. [Google Scholar]
- Miller, S.S.; Mocanu, P.T. Second order differential inequalities in the complex plane. J. Math. Anal. Appl.
**1978**, 65, 289–305. [Google Scholar] [CrossRef] [Green Version] - Miller, S.S.; Mocanu, P.T. Differential subordinations and univalent functions. Mich. Math. J.
**1981**, 28, 157–172. [Google Scholar] [CrossRef] - Pommerenke, C. Univalent Functions; Vandenhoeck and Ruprecht: Göttingen, Germany, 1975. [Google Scholar]
- Mocanu, P.T.; Bulboacă, T.; Sălăgean, Ş.G. Geometric Theory of Analytic Functions; Casa Cărţii de Ştiinţă: Cluj-Napoca, Romanian, 1999. [Google Scholar]

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**MDPI and ACS Style**

Oros, G.I.
New Conditions for Univalence of Confluent Hypergeometric Function. *Symmetry* **2021**, *13*, 82.
https://doi.org/10.3390/sym13010082

**AMA Style**

Oros GI.
New Conditions for Univalence of Confluent Hypergeometric Function. *Symmetry*. 2021; 13(1):82.
https://doi.org/10.3390/sym13010082

**Chicago/Turabian Style**

Oros, Georgia Irina.
2021. "New Conditions for Univalence of Confluent Hypergeometric Function" *Symmetry* 13, no. 1: 82.
https://doi.org/10.3390/sym13010082