# Third-Order Differential Subordinations Using Fractional Integral of Gaussian Hypergeometric Function

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## Abstract

**:**

## 1. Introduction

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

**Definition 5.**

**Remark 1.**

**Definition 6.**

**Definition 7.**

**Definition 8.**

**Lemma 1.**

- (i)
- $\mathrm{Re}\frac{{\zeta}_{0}{q}^{\u2033}\left({\zeta}_{0}\right)}{{q}^{\prime}\left({\zeta}_{0}\right)}\ge 0$ and $\left|{\displaystyle \frac{z{p}^{\prime}\left(z\right)}{{q}^{\prime}\left(\zeta \right)}}\right|\le n$;
- (ii)
- ${z}_{0}{p}^{\prime}\left({z}_{0}\right)=n{\zeta}_{0}{q}^{\prime}\left({\zeta}_{0}\right)$;
- (iii)
- $\mathrm{Re}\left({\displaystyle \frac{{z}_{0}{p}^{\u2033}\left({z}_{0}\right)}{{p}^{\prime}\left({z}_{0}\right)}+1}\right)\ge n\mathrm{Re}\left({\displaystyle \frac{{\zeta}_{0}{q}^{\u2033}\left({\zeta}_{0}\right)}{{q}^{\prime}\left({\zeta}_{0}\right)}+1}\right)$;
- (iv)
- $\mathrm{Re}\frac{{z}_{0}^{2}{p}^{\u2034}\left({z}_{0}\right)}{{p}^{\prime}\left({z}_{0}\right)}\ge {n}^{2}\mathrm{Re}\frac{{\zeta}_{0}{q}^{\u2034}\left({\zeta}_{0}\right)}{{q}^{\prime}\left({\zeta}_{0}\right)}$.

## 2. Main Results

**Theorem 1.**

- (p)
- $\psi \in {\psi}_{n}[h,{q}_{\rho}]$ for a certain $\rho \in (0,1)$; or
- (pp)
- there exists ${\rho}_{0}\in (0,1)$ such that $\psi \in {\psi}_{n}[{h}_{\rho},{q}_{\rho}]$, for all $\rho \in ({\rho}_{0},1)$.

**Proof.**

**Theorem 2.**

- (r)
- $q\in Q$ and $\psi \in {\rho}_{2}[h,q]$;
- (rr)
- $q\in S$ and $\psi \in {\psi}_{2}[h,{q}_{\rho}]$ for a certain $\rho \in (0,1)$;
- (rrr)
- $q\in S$ and there exists ${\rho}_{0}\in (0,1)$ such that $\psi \in {\psi}_{2}[{h}_{\rho},{q}_{\rho}]$ for all $\rho \in ({\rho}_{0},1)$.

**Proof.**

**Theorem 3.**

- (r)
- $q\in Q$ and $\psi \in {\psi}_{n}[h,q]$;
- (rr)
- $q\in S$ and $\psi \in {\psi}_{n}[h,{q}_{\rho}]$ for a certain $\rho \in (0,1)$;
- (rrr)
- $q\in S$ and there exists ${\rho}_{0}\in (0,1)$ such that $\psi \in {\psi}_{n}[{h}_{\rho},{q}_{\rho}]$, for all $\rho \in ({\rho}_{0},1)$.

**Proof.**

**Theorem 4.**

- (i)
- $Q\in {S}^{*}$;
- (ii)
- $\mathrm{Re}\frac{z{h}^{\prime}\left(z\right)}{Q\left(z\right)}>0,\phantom{\rule{4pt}{0ex}}z\in U$. If function $p\left(z\right)\in H\left(U\right)$ is given by (7) with $p\left(0\right)=q\left(0\right)=0$;
- (iii)
- and it satisfies the conditions $\mathrm{Re}\frac{\zeta {q}^{\u2033}\left(\zeta \right)}{{q}^{\prime}\left(\zeta \right)}>0$ and $\left|{\displaystyle \frac{z{p}^{\prime}\left(z\right)}{{q}^{\prime}\left(\zeta \right)}}\right|\le n$, where $z\in U$, $\zeta \in \partial U\backslash E\left(q\right)$, then

**Proof.**

**Remark 2.**

**Corollary 1.**

- (i)
- $Q\in {S}^{*}$;
- (ii)
- $\mathrm{Re}\frac{z{h}^{\prime}\left(z\right)}{Q\left(z\right)}>0,\phantom{\rule{4pt}{0ex}}z\in U$. If function ${D}_{z}^{-\lambda}F(a,b,c;z)\in H\left(U\right)$ given by (7) with ${D}_{z}^{-\lambda}F(a,b,c;0)=q\left(0\right)=0$, ${D}_{z}^{-\lambda}F\left(U\right)\subset D$, satisfies:
- (iii)
- $\mathrm{Re}\frac{\zeta {q}^{\u2033}\left(\zeta \right)}{{q}^{\prime}\left(\zeta \right)}=\mathrm{Re}\frac{2\zeta}{2\zeta +1}=2\xb7\frac{2+cosa}{5+4cosa}>0$ and $\left|{\displaystyle \frac{z{\left({D}_{z}^{-\lambda}F(a,b,c;z)\right)}^{\prime}}{2\zeta +1}}\right|\le n$, where $z\in U$, $\zeta \in \partial U\backslash E\left(q\right)$, then

**Proof.**

**Example 1.**

- (i)
- $Q\in {S}^{*}$,
- (ii)
- $\mathrm{Re}\frac{z{h}^{\prime}\left(z\right)}{Q\left(z\right)}>0,\phantom{\rule{4pt}{0ex}}z\in U$. If the function ${D}_{z}^{-1}F(-2,1+i,1-i;z)\in H\left(U\right)$, ${D}_{z}^{-1}F(-2,1+i,1-i;0)=q\left(0\right)=0$ satisfies the conditions
- (iii)
- $\mathrm{Re}\frac{\zeta {q}^{\u2033}\left(\zeta \right)}{{q}^{\prime}\left(\zeta \right)}=\mathrm{Re}\frac{-2\zeta}{1-2\zeta}>0$ and

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Oros, G.I.; Oros, G.; Preluca, L.F.
Third-Order Differential Subordinations Using Fractional Integral of Gaussian Hypergeometric Function. *Axioms* **2023**, *12*, 133.
https://doi.org/10.3390/axioms12020133

**AMA Style**

Oros GI, Oros G, Preluca LF.
Third-Order Differential Subordinations Using Fractional Integral of Gaussian Hypergeometric Function. *Axioms*. 2023; 12(2):133.
https://doi.org/10.3390/axioms12020133

**Chicago/Turabian Style**

Oros, Georgia Irina, Gheorghe Oros, and Lavinia Florina Preluca.
2023. "Third-Order Differential Subordinations Using Fractional Integral of Gaussian Hypergeometric Function" *Axioms* 12, no. 2: 133.
https://doi.org/10.3390/axioms12020133