# New Generalization of Geodesic Convex Function

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

- 1.
- additive if$$\phi ({u}_{1}+{v}_{1},{u}_{2}+{v}_{2})=\phi ({u}_{1},{u}_{2})+\phi ({v}_{1},{v}_{2}),\forall {u}_{1},{u}_{2},{v}_{1},{v}_{2}\in \mathbb{R}.$$
- 2.
- non-negatively homogeneous if$$\phi (t{u}_{1},t{u}_{2})=t\phi ({u}_{1},{u}_{2}),\forall {u}_{1},{u}_{2}\in \mathbb{R},t\u2a7e0.$$
- 3.
- non-negatively linear if φ is both non-negatively homogeneous and additive.

**Definition**

**3.**

**Definition**

**4.**

- 1.
- an E-convex function, if$$h(tE\left({u}_{1}\right)+(1-t)E\left({u}_{2}\right)\le th\left(E\left({u}_{1}\right)\right)+(1-t)h\left(E\left({u}_{2}\right)\right),$$
- 2.
- a ${\phi}_{E}$-convex function, if$$h(tE\left({u}_{1}\right)+(1-t)E\left({u}_{2}\right)\le h\left(E\left({u}_{2}\right)\right)+t\phi (h\left(E\left({u}_{1}\right)\right),h\left(E\left({u}_{2}\right)\right),$$

**Definition**

**5.**

**Definition**

**6.**

**Remark**

**1.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

- 1.
- geodesic E-convex if U is a geodesic E-convex set and$$h\left({\gamma}_{E\left({\mu}_{1}\right),E\left({\mu}_{2}\right)}\left(t\right)\right)\le th\left(E\left({\mu}_{1}\right)\right)+(1-t)h\left(E\left({\mu}_{2}\right)\right),\forall {\mu}_{1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mu}_{2}\in U,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\in [0,1].$$
- 2.
- geodesic φ-convex if U is a t-convex set and$$h\left({\gamma}_{{\mu}_{1},{\mu}_{2}}\left(t\right)\right)\le h\left({\mu}_{2}\right)+t\phi \left(h\left({\mu}_{1}\right),h\left({\mu}_{2}\right)\right),\forall {\mu}_{1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mu}_{2}\in U,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\in [0,1].$$

## 3. Some Properties of ${\mathit{\phi}}_{\mathit{E}}$-Convex Functions

**Example**

**1.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Properties of Geodesic ${\mathit{\phi}}_{\mathit{E}}$-Convex Functions

**Definition**

**10.**

**Remark**

**2.**

**Example**

**2.**

**Theorem**

**3.**

**Proof.**

**Proposition**

**1.**

- 1.
- If $h:B\u27f6\mathbb{R}$ is a geodesic ${\phi}_{E}$-convex function, where φ is non-negative linear, then $xh:B\u27f6\mathbb{R},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall x\u2a7e0$ is also geodesic ${\phi}_{E}$-convex.
- 2.
- Let ${h}_{i}:B\u27f6\mathbb{R},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=1,2$ be two geodesic ${\phi}_{E}$-convex functions, where φ is additive, then ${h}_{1}+{h}_{2}$ is also a geodesic ${\phi}_{E}$-convex function.

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Definition**

**11.**

**Remark**

**3.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Proposition**

**3.**

**Proposition**

**4.**

**Theorem**

**10.**

**Proof.**

## 5. ${\mathit{\phi}}_{\mathit{E}}$-Epigraphs

**Definition**

**12.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

**Corollary**

**1.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Almutairi, O.B.; Saleh, W. New Generalization of Geodesic Convex Function. *Axioms* **2023**, *12*, 319.
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Almutairi OB, Saleh W. New Generalization of Geodesic Convex Function. *Axioms*. 2023; 12(4):319.
https://doi.org/10.3390/axioms12040319

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Almutairi, Ohud Bulayhan, and Wedad Saleh. 2023. "New Generalization of Geodesic Convex Function" *Axioms* 12, no. 4: 319.
https://doi.org/10.3390/axioms12040319