New Generalization of Geodesic Convex Function

Abstract

As a generalization of a geodesic function, this paper introduces the notion of the geodesic ${\phi }_E$-convex function. Some properties of the ${\phi }_E$-convex function and geodesic ${\phi }_E$-convex function are established. The concepts of a geodesic ${\phi }_E$-convex set and ${\phi }_E$-epigraph are also given. The characterization of geodesic ${\phi }_E$-convex functions in terms of their ${\phi }_E$-epigraphs, are also obtained.

1. Introduction

Convexity is an essential concept in pure and applied mathematics, serving as a potent instrument for analyzing functions and sets, establishing inequalities, and modeling and solving real-world problems. This concept is crucial for estimating integrals and establishing bounds in numerous mathematical fields and beyond [1,2,3,4,5,6,7]. Thus, the convex function can be defined as follows:

A function $h:U\subseteq \mathbb{R}⟶\mathbb{R}$ is convex if

$$h(\eta u_1+(1-\eta )u_2)\le \eta h\left(u_1\right)+(1-\eta )h\left(u_2\right),\forall u_1,u_2\in U,\eta \in [0,1].$$

(1)

If the inequality sign in (1) is reversed, then h is called a concave function on the set U.

For example, in economics, for a production function $u=h\left(L\right)$, the concavity of h is expressed by saying that h exhibits diminishing returns. If h is convex, then it exhibits increasing returns. On the other hand, many new problems in applied mathematics are encountered where the notion of convexity is not enough to describe them, in order to reach favorable results. For this reason, the concept of convexity has been extended and generalized in several studies, see [8,9,10,11,12,13]. Curvature and torsion of Riemannian manifolds lead to a high level of nonlinearity when examining the convexity of such manifolds. A geodesic, is a locally length-minimizing curve, and the notion of a geodesic convex function occurs naturally in a complete Riemannian manifold, which has been studied in [14,15]. The geodesic bifurcation has equally been studied by many authors [16,17].

In 1999, an important generalization of the convex function, called the E-convex function, was defined by Youness [18]. This type of function has some applications in various branches of mathematical sciences [19,20]. On the other hand, Yang [21] showed that some results given by Youness [18] seem to be incorrect. Following these developments, Duca and Lupşa [22] fixed the mistakes in both Youness [18] and Yang [21]. Therefore, Chen [23] extended E-convexity to a semi E-convexity and discussed some of its properties. For more results on the E-convex function and semi E-convex function, one should consult the following references [22,24,25,26,27]. The geodesic convexity involving sets was first studied by [28], who extended the existing concept of geodesic convexity defined by [29]. Geodesic E-convex sets and geodesic E-convex functions on Riemannian manifolds, are a new class of convex sets and functions, that Iqbal et al. introduced and researched in [26], these were extended to geodesic strongly E-convex sets and geodesic strongly E-convex functions in 2015, by Adem and Saleh [30]. In addition, Iqbal et al. [25] introduced geodesic semi E-convex functions. Following these developments, Adem and Saleh [4] introduced geodesic semi E-b-vex (GSEB) functions, of which some properties were discussed.

Other developments include the work of Eshaghi Gordji et al. [31], who introduced the notion of a $\phi$-convex function, in 2016. They equally studied Jensen and Hermite–Hadamard type inequalities related to this function. Moreover, the notion of ${\phi }_E$-convex functions was defined as the generalization of $\phi$-convex functions. Absos et al. further introduced the notion of a geodesic $\phi$-convex function, through which some basic properties of this function were studied [32].

The structure of this article is as follows. Basic information about convex functions and convex sets is covered in Section 2. The evaluation of the properties of ${\phi }_E$-convex functions is covered in Section 3. In Section 4, we discuss a new class of functions on Riemannian manifolds, called the geodesic ${\phi }_E$-convex function. Some of the properties of this function are also studied. In Section 5, the characterization of geodesic ${\phi }_E$-convex functions, through their corresponding ${\phi }_E$-epigraphs, is reported.

2. Preliminaries

This section provides some definitions and properties that can later be used in the study, to report our results. Several definitions and properties of real number sets and the Riemannian manifold can be found in many different geometry books and papers [15]. Throughout this paper, we consider an interval $U=[u_1,u_2]$ in $\mathbb{R}$ and $\phi :\mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ is a bifunction.

Definition 1.

Ref. [31]. A function $h:U⟶\mathbb{R}$ is called φ-convex if

$$h(t{u}_1+(1-t)u_2)\le h\left(u_2\right)+t\phi (h\left(u_1\right),h\left(u_2\right)),$$

(2)

for all $u_1,u_2\in U,t\in [0,1]$

In the above definition, if $\phi (h\left(u_1\right),h\left(u_2\right))=h\left(u_1\right)-h\left(u_2\right)$, then inequality (2) becomes inequality (1).

Definition 2.

Ref. [31]. The function $\phi :\mathbb{R}\times \mathbb{R}⟶\mathbb{R}$, is called

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

$$\phi (t{u}_1,t{u}_2)=t\phi (u_1,u_2),\forall u_1,u_2\in \mathbb{R},t⩾0.$$

3.

non-negatively linear if φ is both non-negatively homogeneous and additive.

Definition 3.

Ref. [18]. A set $U\subset {\mathbb{R}}^n$, is said to be an E-convex set if there is a mapping $E:{\mathbb{R}}^n⟶{\mathbb{R}}^n$, such that

$$t\left(E\left(u_1\right)\right)+(1-t)E\left(u_2\right)\in U,\forall u_1,u_2\in U,t\in [0,1]$$

Definition 4.

Refs. [18,31]. Consider $U\subset {\mathbb{R}}^n$ to be an E-convex set, then the function $h:U⟶\mathbb{R}$ is said to be

1.

an E-convex function, if

$$h(tE\left(u_1\right)+(1-t)E\left(u_2\right)\le th\left(E\left(u_1\right)\right)+(1-t)h\left(E\left(u_2\right)\right),$$

(3)

$\forall u_1,u_2\in U,t\in [0,1]$

2.

a ${\phi }_E$-convex function, if

$$h(tE\left(u_1\right)+(1-t)E\left(u_2\right)\le h\left(E\left(u_2\right)\right)+t\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right),$$

(4)

$\forall u_1,u_2\in U,t\in [0,1]$

If $\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right)=h\left(E\left(u_1\right)\right)-h\left(E\left(u_2\right)\right)$ in inequality (4), then we obtain the E-convex function.

Now, let $(N,g)$ be a complete m-dimensional Riemannian manifold, with Riemannian connection ▽. If $a_1$ and $a_2$ are two points on N, and $\gamma :[{\mu }_1,{\mu }_2]⟶N$ is a piecewise smooth curve joining $\gamma \left({\mu }_1\right)=a_1$ to $\gamma \left({\mu }_2\right)=a_2$ and its length, $L\left(\gamma \right)$, is defined by

$$L\left(\gamma \right)={\int }_{{\mu }_1}^{{\mu }_2}\parallel \frac{d\gamma \left(\lambda \right)}{d\lambda }\parallel d\lambda .$$

For any two points $a_1,a_2\in N$, we define $d(a_1,a_2)=inf\left\{L\left(\gamma \right):\gamma \right.$ a piecewise smooth curve connecting the points $\left.a_1\mathrm{to}{a}_2\right\}.$

Then d is a metric, which induces the original topology on N.

For every Riemannian manifold, there is a unique determined Riemannian connection, called a Levi–Civita connection, denoted by ${▽}_{X}Y$, for any vector fields $X,Y\in N$. In addition, a smooth path $\gamma$, is a geodesic if and only if its tangent vector is a parallel vector field along the path $\gamma$, i.e., $\gamma$ satisfies the equation ${▽}_{\frac{d\gamma \left(t\right)}{dt}}\frac{d\gamma \left(t\right)}{dt}=0$. Any path $\gamma$ joining ${\mu }_1$ and ${\mu }_2$ in N, such that $L\left(\gamma \right)=d({\mu }_1,{\mu }_2)$, is a geodesic and is called a minimal geodesic. Let N be a $C^{\infty }$ complete n-dimensional Riemannian manifold, with metric g and Levi–Civita connection ▽. Moreover, considering that the points ${\mu }_1,{\mu }_2\in N$ and $\gamma :[0,1]⟶N$ is a geodesic joining ${\mu }_1,{\mu }_2$, i.e., ${\gamma }_{{\mu }_1,{\mu }_2}\left(0\right)={\mu }_2$ and ${\gamma }_{{\mu }_1,{\mu }_2}\left(1\right)={\mu }_1$.

Definition 5.

Ref. [33]. Assume that $N_1,N_2$ are smooth manifolds. A map $h:N_1⟶N_2$ is a diffeomorphism if it is smooth, bijective, and the inverse $h^{-1}$ is smooth.

Definition 6.

Ref. [15]. A subset $U\subseteq N$, is called t-convex if and only if U contains every geodesic ${\gamma }_{{\mu }_1,{\mu }_2}$ of N whose endpoints ${\mu }_1$ and ${\mu }_2$ are in U.

Remark 1.

If $U_1$ and $U_2$ are t-convex sets, then $U_1\cap U_2$ is a t-convex set, but $U_1\cup U_2$ is not necessarily a t-convex set.

Definition 7.

Ref. [15]. A function $h:U\subset N⟶\mathbb{R}$ is called geodesic convex if and only if for all geodesic arcs ${\gamma }_{{\mu }_1,{\mu }_2}$, then

$$h\left({\gamma }_{{\mu }_1,{\mu }_2}\left(t\right)\right)\le th\left({\mu }_1\right)+(1-t)h\left({\mu }_2\right)$$

for each ${\mu }_1,{\mu }_2\in U$ and $t\in [0,1]$.

Definition 8.

Ref. [26]. A set $U\subset N$, is geodesic E-convex, where $E:N⟶N$, if and only if there exists a unique geodesic ${\gamma }_{E\left({\mu }_1\right), E\left({\mu }_2\right)}\left(t\right)$ of length $d({\mu }_1,{\mu }_2)$, which belongs to U for every ${\mu }_1,{\mu }_2\in U$ and $t\in [0,1]$.

Definition 9.

Refs. [26,32]. A function $h:U⟶\mathbb{R}$ is said to be

1.

geodesic E-convex if U is a geodesic E-convex set and

$$h\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\left(t\right)\right)\le th\left(E\left({\mu }_1\right)\right)+(1-t)h\left(E\left({\mu }_2\right)\right),\forall {\mu }_1,{\mu }_2\in U,t\in [0,1].$$

2.

geodesic φ-convex if U is a t-convex set and

$$h\left({\gamma }_{{\mu }_1,{\mu }_2}\left(t\right)\right)\le h\left({\mu }_2\right)+t\phi \left(h\left({\mu }_1\right),h\left({\mu }_2\right)\right),\forall {\mu }_1,{\mu }_2\in U,t\in [0,1].$$

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

This part of the work deals with some properties of ${\phi }_E$-convex functions. Considering that $h:B⟶\mathbb{R}$ is a ${\phi }_E$-convex function and $E:\mathbb{R}⟶\mathbb{R}$, we present the following. For any two points $E\left({\mu }_1\right),E\left({\mu }_2\right)\in B$ with $E\left({\mu }_1\right)<E\left({\mu }_2\right)$ and for each point $E\left(\mu \right)\in (E\left({\mu }_1\right),E\left({\mu }_2\right))$ can be expressed as

$$E\left(\mu \right)=tE\left({\mu }_1\right)+(1-t)E\left({\mu }_1\right),t=\frac{E\left({\mu }_2\right)-E\left(\mu \right)}{E\left({\mu }_2\right)-E\left({\mu }_1\right)}.$$

Also, since a function h is ${\phi }_E$-convex function if

$$h\left(E\left(\mu \right)\right)\le h\left(E\left({\mu }_2\right)\right)+\frac{E\left({\mu }_2\right)-E\left(\mu \right)}{E\left({\mu }_2\right)-E\left({\mu }_1\right)}\phi (h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right)),$$

then

$$\begin{array}{c}\hfill \frac{h\left(E\left({\mu }_2\right)\right)-h\left(E\left(\mu \right)\right)}{E\left({\mu }_2\right)-E\left(\mu \right)}⩾\frac{\phi (h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right))}{E\left({\mu }_1\right)-E\left({\mu }_2\right)},\end{array}$$

(5)

$\forall E\left(\mu \right)\in \left(E\left({\mu }_2\right),E\left({\mu }_1\right)\right)$.

Hence, we can say that a function h is a ${\phi }_E$-convex function if it satisfies the inequality (5).

The next example shows that a ${\phi }_E$-convex function is not necessarily a $\phi$-convex function.

Example 1.

Consider

$$\begin{array}{ccc}\hfill h\left(u_1\right)& =& \left\{\begin{array}{c}1;u_1⩾0,\hfill \\ -u_1^2;u_1<0,\hfill \end{array}\right.\hfill \end{array}$$

with $E\left(u_1\right)=-a$ where $a\in {\mathbb{R}}^{+}$ and $\phi (u_1,u_2)=u_1-2u_2$. Then $h(tE\left(u_1\right)+(1-t)E\left(u_2\right))=-a^2$ while $h\left(E\left(u_2\right)\right)+t\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right))=(t-1)a^2$, which means that h is a ${\phi }_E$-convex function. On the other hand, if we take $u_1>0$ and $u_2>0$, then h is not a φ-convex function.

Theorem 1.

If $h:B\subset E\left(\mathbb{R}\right)⟶\mathbb{R}$ is differentiable and a ${\phi }_E$-convex function in B, and $h\left(E\left(u_1\right)\right)\ne h\left(E\left(u_2\right)\right)$, then there are $E\left(\alpha \right),E\left(\beta \right)\in (E\left(u_2\right),E\left(u_1\right))\subset B$, such that

$$h^{\prime }\left(E\left(\alpha \right)\right)⩾\frac{\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right))}{h\left(E\left(u_1\right)\right)-h\left(E\left(u_2\right)\right)}{h}^{\prime }\left(E\left(\beta \right)\right)⩾h^{\prime }\left(E\left(\beta \right)\right).$$

Proof. 

Since h is a ${\phi }_E$-convex function, then

$$\begin{array}{ccc}\hfill \frac{h\left(E\left(u_2\right)\right)-h\left(E\left(u\right)\right)}{E\left(u_2\right)-E\left(u\right)}& ⩾& \frac{\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right))}{E\left(u_1\right)-E\left(u_2\right)}\hfill \\ & & =\frac{\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right))}{h\left(E\left(u_1\right)\right)-h\left(E\left(u_2\right)\right)}\times \frac{h\left(E\left(u_1\right)\right)-h\left(E\left(u_2\right)\right)}{E\left(u_1\right)-E\left(u_2\right)}.\hfill \end{array}$$

(6)

Now, applying the mean value theorem, then the inequality (6) can be written as

$$\begin{array}{c}\hfill h^{\prime }\left(E\left(\alpha \right)\right)⩾\frac{\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right))}{h\left(E\left(u_1\right)\right)-h\left(E\left(u_2\right)\right)}{h}^{\prime }\left(E\left(\beta \right)\right),\end{array}$$

(7)

for some $E\left(\alpha \right)\in (E\left(u_1\right),E\left(u\right))\subset (E\left(u_1\right),E\left(u_2\right))$ and $E\left(\beta \right)\in (E\left(u_1\right),E\left(u_2\right))$.

Since $\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right))⩾h\left(E\left(u_1\right)\right)-h\left(E\left(u_2\right)\right)$, then the inequality (7) yields

$$\begin{array}{c}\hfill h^{\prime }\left(E\left(\alpha \right)\right)⩾\frac{\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right))}{h\left(E\left(u_1\right)\right)-h\left(E\left(u_2\right)\right)}{h}^{\prime }\left(E\left(\beta \right)\right)⩾h^{\prime }\left(E\left(\beta \right)\right).\end{array}$$

□

Theorem 2.

Assume that $h:B⟶\mathbb{R}$ is a differentiable ${\phi }_E$-convex function. Then, for all $E\left({\mu }_i\right)\in B,i=1,2,3$, such that $E\left({\mu }_1\right)<E\left({\mu }_2\right)<E\left({\mu }_3\right)$, the following inequality holds

$$h^{\prime }\left(E\left({\mu }_2\right)\right)+h^{\prime }\left(E\left({\mu }_3\right)\right)\le \frac{\phi (h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right))+\phi (h\left(E\left({\mu }_2\right)\right),h\left(E\left({\mu }_3\right)\right))}{E\left({\mu }_1\right),E\left({\mu }_3\right)}.$$

Proof. 

Since h is ${\phi }_E$-convex in each interval $W_1=[E\left({\mu }_1\right),E\left({\mu }_2\right)]$ and $W_2=[E\left({\mu }_2\right),E\left({\mu }_3\right)]$, hence

$$\begin{array}{c}\hfill h(tE\left({\mu }_1\right)+(1-t)E\left({\mu }_2\right))\le h\left(E\left({\mu }_2\right)\right)+t\phi (h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right))\end{array}$$

(8)

and

$$\begin{array}{c}\hfill h(tE\left({\mu }_2\right)+(1-t)E\left({\mu }_3\right))\le h\left(E\left({\mu }_3\right)\right)+t\phi (h\left(E\left({\mu }_2\right)\right),h\left(E\left({\mu }_3\right)\right)).\end{array}$$

(9)

From inequalities (8) and (9), we get

$$\begin{array}{ccc}& & \frac{h(tE\left({\mu }_1\right)+(1-t)E\left({\mu }_2\right))-h\left(E\left({\mu }_2\right)\right)+h(tE\left({\mu }_2\right)+(1-t)E\left({\mu }_3\right))-h\left(E\left({\mu }_3\right)\right)}{t}\hfill \\ \hfill & & \le \phi (h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right))+\phi (h\left(E\left({\mu }_2\right)\right),h\left(E\left({\mu }_3\right)\right)).\hfill \end{array}$$

Now, setting $t⟶0$, we get

$$\begin{array}{ccc}\hfill & & h^{\prime }\left(E\left({\mu }_2\right)\right)(E\left({\mu }_1\right)-E\left({\mu }_2\right))+h^{\prime }\left(E\left({\mu }_3\right)\right)(E\left({\mu }_2\right)-E\left({\mu }_3\right))\hfill \\ & & \le \phi (h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right))+\phi (h\left(E\left({\mu }_2\right)\right),h\left(E\left({\mu }_3\right)\right)).\hfill \end{array}$$

(10)

Also, $E\left({\mu }_3\right)>E\left({\mu }_2\right)$ and $E\left({\mu }_2\right)>E\left({\mu }_1\right)$, which means that $E\left({\mu }_1\right)-E\left({\mu }_3\right)<E\left({\mu }_1\right)-E\left({\mu }_2\right)$ and $E\left({\mu }_1\right)-E\left({\mu }_3\right)<E\left({\mu }_2\right)-E\left({\mu }_3\right)$, then

$$\begin{array}{ccc}\hfill & & (E\left({\mu }_1\right)-E\left({\mu }_3\right))(h^{\prime }\left(E\left({\mu }_2\right)\right)+h^{\prime }\left(E\left({\mu }_3\right)\right))\hfill \\ \hfill & & \le (E\left({\mu }_1\right)-E\left({\mu }_2\right))h^{\prime }\left(E\left({\mu }_2\right)\right)+(E\left({\mu }_2\right)-E\left({\mu }_3\right))h^{\prime }\left(E\left({\mu }_3\right)\right).\hfill \end{array}$$

(11)

Hence, from inequalities (10) and (11), we get the required result. □

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

This section makes the assumption that ${\mu }_1,{\mu }_2\in N$ and $\gamma :[0,1]⟶N$ is a geodesic joining ${\mu }_1,{\mu }_2$, i.e., ${\gamma }_{{\mu }_1,{\mu }_2}\left(0\right)=\mu 2$ and ${\gamma }_{{\mu }_1,{\mu }_2}\left(1\right)={\mu }_1$, and E is a mapping, such that $E:N⟶N$, where N is a $C^{\infty }$ complete n-dimensional Riemannian manifold, with Riemannian connection ▽. In addition, we define the geodesic ${\phi }_E$-convex function in N and examine some of its characteristics.

Definition 10.

A function $h:B⟶\mathbb{R}$ is geodesic ${\phi }_E$-convex if B is also a geodesic E-convex set and

$$h\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\left(t\right)\right)\le h\left(E\left({\mu }_2\right)\right)+t\phi \left(h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right)\right),$$

for all ${\mu }_1,{\mu }_2\in B,t\in [0,1].$

If the above inequality strictly holds for all ${\mu }_1,{\mu }_2\in B,E\left({\mu }_1\right)\ne E\left({\mu }_2\right),t\in [0,1]$, then h is called a strictly geodesic ${\phi }_E$-convex function.

Remark 2.

If E is identity mapped in the above definition, then we have a geodesic φ-convex function. Moreover, if

$$\phi \left(h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right)\right)=\left(h\left(E\left({\mu }_1\right)\right)-h\left(E\left({\mu }_2\right)\right)\right),$$

then we have a geodesic E-convex function.

Example 2.

This example shows that the geodesic ${\phi }_E$-convex function on N does not necessarily have to be geodesic convex. Let $N=\mathbb{R}\times {\mathbb{S}}^1$ and $h:N⟶\mathbb{R}$ is defined as $h(\mu ,a)={\mu }^3$, then h is not geodesic convex in N. Now, by taking a function $\phi :\mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ as $\phi ({\mu }_1,{\mu }_2)={\mu }_1^3-{\mu }_2^3$ and $E:\mathbb{R}⟶{\mathbb{R}}^{+}$, then for any two points (${\mu }_1,a$) and (${\mu }_2,b$), the geodesic joining them is a portion of a helix of the form $\gamma \left(t\right)=(t{\mu }_1+(1-t){\mu }_2,{exp}^{i[t{\omega }_1+(1-t){\omega }_2]})$ for $t\in [0,1]$ and ${exp}^{i{\omega }_1}=a,{exp}^{i{\omega }_2}=b$ for ${\omega }_1,{\omega }_2\in [0,2\pi ]$. Hence,

$$\begin{array}{ccc}\hfill h\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\right)& =& {(tE\left({\mu }_1\right)+(1-t)E\left({\mu }_2\right))}^3\hfill \\ & =& t^3{(E\left({\mu }_1\right)-E\left({\mu }_2\right))}^3+t^2(3E^2\left({\mu }_1\right)E\left({\mu }_2\right)-6E\left({\mu }_1\right)E^2\left({\mu }_2\right))\hfill \\ & +& 3E^3\left({\mu }_2\right)+t[3E\left({\mu }_1\right)E^2\left({\mu }_2\right)-3E^3\left({\mu }_2\right)]+E^3\left({\mu }_2\right)\hfill \\ & \le & E^3\left({\mu }_2\right)+t[E^3\left({\mu }_1\right)-E^3\left({\mu }_2\right)]\hfill \\ & =& h(E\left({\mu }_2\right),b)+t\phi (h(E\left({\mu }_1\right),a),hE\left({\mu }_2\right),b).\hfill \end{array}$$

(12)

Then h is a geodesic ${\phi }_E$-convex function.

Theorem 3.

Considering that $B\subset N$ is an E-convex set, then a function $h:B⟶\mathbb{R}$ is a geodesic ${\phi }_E$-convex if and only if for each $u_1,u_2\in B$ the function $K=h\circ {\gamma }_{E\left(u_1\right),E\left(u_2\right)}$ is ${\phi }_E$-convex on $[0,1]$.

Proof. 

Let K be ${\phi }_E$-convex on $[0,1]$, then

$$\begin{array}{c}\hfill K(tE\left({\mu }_1\right)+(1-t)E\left({\mu }_2\right))\le K\left(E\left({\mu }_2\right)\right)+t\phi (K\left(E\left({\mu }_1\right)\right),K\left(E\left({\mu }_2\right)\right))\end{array}$$

(13)

holds.

Also, let $E\left({\mu }_1\right)=1,E\left({\mu }_2\right)=0,$ then $K\left(t\right)\le K\left(0\right)+t\phi \left(K\right(1),K(0\left)\right).$ Hence

$$h\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\left(t\right)\right)\le h\left(E\left({\mu }_2\right)\right)+t\phi \left(h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right)\right).$$

Conversely, assume that h is a geodesic ${\phi }_E$-convex function. By restricting the domain of ${\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}$ to $[{\eta }_1,{\eta }_2]$, and hence the parametrized form of this restriction can be rewritten as

$$\alpha \left(t\right)={\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}(tE\left({\mu }_1\right)+(1-t)E\left({\mu }_2\right))$$

$$\alpha \left(0\right)={\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\left(E\left({\mu }_2\right)\right).$$

Since $h\left(\alpha \right(t\left)\right)\le h\left(\alpha \right(0\left)\right)+t\phi \left(h\right(\alpha \left(1\right)),h(\alpha \left(0\right)\left)\right).$ That means

$$\begin{array}{c}\hfill h\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}(tE\left({\mu }_1\right)+(1-t)E\left({\mu }_2\right))\right)\\ \hfill \le h\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\left(E\left({\mu }_2\right)\right)\right)+t\phi (h\left(\gamma \left(E\left({\mu }_1\right)\right)\right),h\left(\gamma \left(E\left({\mu }_2\right)\right)\right))\end{array}$$

It follows that

$$\begin{array}{c}\hfill K(tE\left({\mu }_1\right)+(1-t)E\left({\mu }_2\right)))\\ \hfill \le K\left(E\left({\mu }_2\right)\right)+t\phi (K\left(E\left({\mu }_1\right)\right),K\left(E\left({\mu }_2\right)\right))\end{array}$$

Hence, K is ${\phi }_E$-convex on $[0,1]$. □

Proposition 1.

1.

If $h:B⟶\mathbb{R}$ is a geodesic ${\phi }_E$-convex function, where φ is non-negative linear, then $xh:B⟶\mathbb{R},\forall x⩾0$ is also geodesic ${\phi }_E$-convex.

2.

Let $h_i:B⟶\mathbb{R},i=1,2$ 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.

Suppose that $B\subset N$ is a geodesic E-convex set, $h_1:B⟶\mathbb{R}$ is a geodesic E-convex function, and $h_2:U⟶\mathbb{R}$ is a non-decreasing ${\phi }_E$-convex function, such that $Rang\left(h_1\right)\subseteq U$. Then, $h_{2}o{h}_1$ is also a geodesic ${\phi }_E$-convex.

Proof. 

The above theorem can be proved in the following way

$$\begin{array}{ccc}\hfill h_2\circ h_1\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\right)& =& h_2\left(h_1\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\right)\right)\hfill \\ & \le & h_2\left(h_1\left(E\left({\mu }_2\right)\right)+t\phi \left(h_1\left(E\left({\mu }_1\right)\right),h_1\left(E\left({\mu }_2\right)\right)\right)\right)\hfill \\ & \le & h_2\left(h_1\left(E\left({\mu }_2\right)\right)\right)+t\phi \left(h_2\left(h_1\left(E\left({\mu }_1\right)\right)\right),h_2\left(h_1\left(E\left({\mu }_2\right)\right)\right)\right)\hfill \\ & =& h_{2}o{h}_1\left(E\left({\mu }_2\right)\right)+t\phi \left(h_{2}o{h}_1\left(E\left({\mu }_1\right)\right),h_{2}o{h}_1\left(E\left({\mu }_2\right)\right)\right).\hfill \end{array}$$

Thus, $h_2\circ h_1$ is a geodesic ${\phi }_E$-convex function. □

Theorem 5.

Suppose that, $h_i:B\subset N⟶\mathbb{R},i=1,2,\cdots ,n$ are geodesic ${\phi }_E$-convex functions and φ is non-negatively linear. Then the function $h={\sum }_{i=1}^n{x}_i{h}_i$ is also a geodesic ${\phi }_E$-convex, for all $x_i\in \mathbb{R}$ and $x_i⩾0$.

Proof. 

Considering ${\mu }_1,{\mu }_2\in B$, and since $h_i,i=1,2,\cdots ,n$ are geodesic ${\phi }_E$-convex functions, then

$$\begin{array}{c}\hfill h_i\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\right)\le h_i\left(E\left({\mu }_2\right)\right)+t\phi \left(h_i\left(E\left({\mu }_1\right)\right),h_i\left(E\left({\mu }_2\right)\right)\right).\end{array}$$

Also,

$$\begin{array}{c}\hfill x_i{h}_i\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\right)\le x_i{h}_i\left(E\left({\mu }_2\right)\right)+t\phi \left(x_i{h}_i\left(E\left({\mu }_1\right)\right),x_i{h}_i\left(E\left({\mu }_2\right)\right)\right).\end{array}$$

Hence,

$$\begin{array}{c}\hfill \sum _{i=1}^n{x}_i{h}_i\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\right)\le \sum _{i=1}^n\left[x_i{h}_i\left(E\left({\mu }_2\right)\right)+t\phi \left(x_i{h}_i\left(E\left({\mu }_1\right)\right),x_i{h}_i\left(E\left({\mu }_2\right)\right)\right)\right],\end{array}$$

which means that

$$\begin{array}{c}\hfill h\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\right)\le h\left(E\left({\mu }_2\right)\right)+t\phi \left(h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right)\right).\end{array}$$

□

Now we consider that $N_1$ and $N_2$ are two complete Riemannian manifolds, and ▽ is the Levi–Civita connection on $N_1$. If $H:N_1⟶N_2$ is a diffeomorphism, then $H\ast ▽={▽}^{*}$ is an affine connection of $N_2$. Moreover, let $\gamma$ be a geodesic in $(N_1,▽)$, then $Ho\gamma$ is also a geodesic in $(N_2,{▽}^{*})$, see [15].

Theorem 6.

Suppose that $h:B⟶\mathbb{R}$ is a geodesic ${\phi }_E$-convex function and $H:N_1⟶N_2$, then a sufficient condition for $ho{H}^{-1}:H\left(B\right)⟶\mathbb{R}$ to be a geodesic ${\phi }_E$-convex function, is H must be a diffeomorphism.

Proof. 

Assume that ${\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}$ is a geodesic joining $E\left({\mu }_1\right)$ and $E\left({\mu }_2\right)$, where ${\mu }_1,{\mu }_2\in B$. Since H is a diffeomorphism, then $H\left(B\right)$ is totally geodesic, and $Ho{\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}$ is geodesic joining $H\left(E\left({\mu }_1\right)\right)$ and $H\left(E\left({\mu }_2\right)\right)$. Then

$$\begin{array}{ccc}& & \left(ho{H}^{-1}\right)\left(Ho{\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\left(t\right)\right)\hfill \\ \hfill & & =h\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\left(t\right)\right)\hfill \\ \hfill & & \le h\left(E\left({\mu }_2\right)\right)+t\phi \left(h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right)\right)\hfill \\ \hfill & & =\left(ho{H}^{-}\right)\left(H\left(E\left({\mu }_2\right)\right)\right)+t\phi \left(\left(ho{H}^{-}\right)\left(E\left({\mu }_1\right)\right),\left(ho{H}^{-}\right)\left(E\left({\mu }_2\right)\right)\right).\hfill \end{array}$$

□

Theorem 7.

Assume that $h:B⟶\mathbb{R}$ is a geodesic ${\phi }_E$-convex function, and φ bounded from above on $h\left(B\right)\times h\left(B\right)$, with an upper bound k. Then h is continuous on $Int\left(B\right)$.

Proof. 

Assume that $E\left({\mu }^{*}\right)\in Int\left(B\right)$, then there exists an open ball $B(E\left({\mu }^{*}\right),r)\subset Int\left(B\right)$ for some $r>0$. Let us choose s, where $(0<s<r)$, such that the closed ball $\overline{B}(E\left({\mu }^{*}\right),s+\epsilon )\subset B(E\left({\mu }^{*}\right),r)$ for some arbitrary small $\epsilon >0$. Choose any $E\left({\mu }_1\right),E\left({\mu }_2\right)\in \overline{B}(E\left({\mu }^{*}\right),s)$. Put $E\left({\mu }_3\right)=E\left({\mu }_2\right)+\frac{\epsilon }{\parallel {\mu }_2-{\mu }_1\parallel }(E\left({\mu }_2\right)-E\left({\mu }_1\right))$ and $t=\frac{\parallel {\mu }_2-{\mu }_1\parallel }{\epsilon +\parallel {\mu }_2-{\mu }_1\parallel }$. Then it is obvious that $E\left({\mu }_3\right)\in \overline{B}(E\left({\mu }^{*}\right),s+\epsilon )$ and $E\left({\mu }_2\right)=tE\left({\mu }_3\right)+(1-t)E\left({\mu }_1\right)$. Thus,

$$\begin{array}{c}\hfill h\left(E\left({\mu }_2\right)\right)\le h\left(E\left({\mu }_1\right)\right)+t\phi (h\left(E\left({\mu }_3\right)\right),h\left(E\left({\mu }_1\right)\right))\le h\left(E\left({\mu }_1\right)\right)+tk.\end{array}$$

Then, the above inequality can be written as

$$h\left(E\left({\mu }_2\right)\right)-h\left(E\left({\mu }_1\right)\right)\le tk\le \frac{\parallel {\mu }_2-{\mu }_1\parallel }{\epsilon }k=L\parallel E\left({\mu }_2\right)-E\left({\mu }_3\right)\parallel ,$$

where $L=\frac{k}{\epsilon }$.

Moreover,

$$h\left(E\left({\mu }_1\right)\right)-h\left(E\left({\mu }_2\right)\right)\le L\parallel E\left({\mu }_2\right)-E\left({\mu }_3\right)\parallel ,$$

Then

$$\left|h\left(E\left({\mu }_1\right)\right)-h\left(E\left({\mu }_2\right)\right)\right.∥\le L\parallel E\left({\mu }_2\right)-E\left({\mu }_3\right)\parallel ,$$

and since $\overline{B}(E\left({\mu }^{*}\right),s)$ is arbitrary, then h is continuous on $Int\left(B\right)$. □

Definition 11.

A bifunction $\phi :{\mathbb{R}}^2⟶\mathbb{R}$, is called sequentially upper bounded with respect to E if

$$\underset{i}{sup}\phi \left(E\left(u_i\right)-E\left(v_i\right)\right)\le \phi \left(\underset{i}{sup}E\left(u_i\right),\underset{i}{sup}E\left(v_i\right)\right)$$

for any two bounded real sequences $\left\{E\left(u_i\right)\right\},\left\{E\left(v_i\right)\right\}$.

Remark 3.

If E is an identity mapping in Definition 11, then a bifunction $\phi :{\mathbb{R}}^2⟶\mathbb{R}$ is called sequentially upper bounded [32].

Proposition 2.

Suppose that $B\subset N$ is a geodesic E-convex set, and ${\left\{h_i\right\}}_{i\in \mathbb{N}}$ are a non-empty family of geodesic ${\phi }_E$-convex functions on B, where ${\phi }_E$ is sequentially upper bounded with respect to E. If ${sup}_i{h}_i\left(u\right)$ exist for each $u\in B$, then $h\left(u\right)={sup}_i{h}_i\left(u\right)$ are also geodesic ${\phi }_E$-convex functions.

Proof. 

Let $E\left(u_1\right),E\left(u_2\right)\in B$, then

$$\begin{array}{ccc}\hfill h\left({\gamma }_{E\left(u_1\right),E\left(u_2\right)}\left(t\right)\right)& =& \underset{i}{sup}{h}_i\left({\gamma }_{E\left(u_1\right),E\left(u_2\right)}\left(t\right)\right)\hfill \\ & \le & \underset{i}{sup}{h}_i\left(E\left(u_2\right)\right)+t\underset{i}{sup}\phi \left(h_i\left(E\left(u_1\right)\right),h_i\left(E\left(u_2\right)\right)\right)\hfill \\ & \le & \underset{i}{sup}{h}_i\left(E\left(u_2\right)\right)+t\phi \left(\underset{i}{sup}{h}_i\left(E\left(u_1\right)\right),\underset{i}{sup}{h}_i\left(E\left(u_2\right)\right)\right)\hfill \\ & \le & h\left(E\left(u_2\right)\right)+t\phi \left(h\left(E\left(u_1\right)\right),\left(E\left(u_2\right)\right)\right).\hfill \end{array}$$

This implies that h is a geodesic ${\phi }_E$-convex function. □

Theorem 8.

The function $h:C⟶\mathbb{R}$ is geodesic ${\phi }_E$-convex, where C is a geodesic E-convex set. The inequality $\phi \left(h\right(E\left(\mu \right),h\left(E\left({\mu }^{*}\right)\right)⩾0,\forall E\left(\mu \right)\in C$ is necessary for h to have a local minimum at $E\left({\mu }^{*}\right)\in Int\left(C\right)$.

Proof. 

Due to the fact that C is a geodesic E-convex set and $E\left({\mu }^{*}\right)\in Int\left(C\right)$, then $B(E\left({\mu }^{*}\right),S)\subset C$ for some $S>0$. Let $E\left(\mu \right)\in C$, then

$$h\left({\gamma }_{E\left(\mu \right),E\left({\mu }^{*}\right)}\left(t\right)\right)\le h\left(E\left({\mu }^{*}\right)\right)+t\phi \left(h\left(E\left(\mu \right)\right),\left(E\left({\mu }^{*}\right)\right)\right).$$

Since h attains its local minimum at $E\left({\mu }^{*}\right)$, then

$$h\left(E\left({\mu }^{*}\right)\right)\le h\left({\gamma }_{E\left(\mu \right),E\left({\mu }^{*}\right)}\left(\zeta \right)\right),$$

(14)

where $\zeta \in \left(\left.0,1\right]\right.$ such that $h\left({\gamma }_{E\left(\mu \right),E\left({\mu }^{*}\right)}\left(t\right)\right)\in B(E\left({\mu }^{*}\right),S),$ for all $t\in [0,\zeta ]$.

Also,

$$\begin{array}{c}\hfill h\left({\gamma }_{E\left(\mu \right),E\left({\mu }^{*}\right)}\left(\zeta \right)\right)\le h\left(E\left({\mu }^{*}\right)\right)+\zeta \phi (h\left(E\left(\mu \right)\right),h\left(E\left({\mu }^{*}\right)\right),\end{array}$$

(15)

then from (14) and (15), we obtain $\phi (h\left(E\left(\mu \right)\right),h\left(E\left({\mu }^{*}\right)\right)⩾0$, for all $E\left(\mu \right)\in C$. □

Theorem 9.

The function $h:B⟶\mathbb{R}$ is geodesic ${\phi }_E$-convex, where B is a geodesic E-convex set and φ is bounded from above on $h\left(B\right)\times h\left(B\right)$, with an upper bound K, with respect to E. Then h is continuous on $Int\left(B\right)$.

Proof. 

Assume that $E\left(u\right)\in Int\left(B\right)$ and $(U,\psi )$, is a chart containing $E\left(u\right)$. Since $\psi$ is a diffeomorphism, and by using Theorems 6 and 7, we get $ho{\psi }^{-1}:\psi (U\cap Int\left(B\right))⟶\mathbb{R}$ as also geodesic ${\phi }_E$-convex and then it is continuous. Hence, $h=ho{\psi }^{-1}o\psi :(U\cap Int\left(B\right))⟶\mathbb{R}$ is continuous.

Also, since $E\left(u\right)$ is arbitrary, then h is continuous on $Int\left(B\right)$. □

From the definition of geodesic ${\phi }_E$-convex, we obtain the following proposition.

Proposition 3.

Assume that $\left\{{\phi }^i:i\in \mathbb{N}\right\}$ is a collection of bifunctions, such that $h:B⟶\mathbb{R}$ is a geodesic ${\phi }_E^i$-convex function for each i. If ${\phi }^i⟶\phi$ as $i⟶\infty$, then h is also a geodesic ${\phi }_E$-convex function.

As a special case in the above proposition, we have the following proposition.

Proposition 4.

Assume that $\left\{{\phi }^i:i\in \mathbb{N}\right\}$ is a collection of bifunctions, such that $h:B⟶\mathbb{R}$ is a geodesic ${\phi }_E^{*}$-convex function, where ${\phi }_E^{*}={\sum }_{l=1}^i{\phi }_E^l$. If ${\phi }_E^{*}$ converges to ${\phi }_E$, then h is also a geodesic ${\phi }_E$-convex function.

Theorem 10.

Consider $h:B⟶\mathbb{R}$ to be strictly geodesic ${\phi }_E$-convex, where B is a geodesic E-convex set, φ is an antisymmetric function with respect to E $\dot{\gamma }$ and stands for the derivative of γ with respect t. Then

$$d{h}_{E\left({\mu }_1\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\ne d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)},$$

for all $E\left({\mu }_1\right),E\left({\mu }_2\right)\in B$ and $E\left({\mu }_1\right)\ne E\left({\mu }_2\right)$.

Proof. 

Since ${\gamma }_{E\left({\mu }_2\right),E\left({\mu }_1\right)}\left(t\right)={\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}(1-t),\forall t\in [0,1]$, then

$$d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_2\right),E\left({\mu }_1\right)}=-d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)}.$$

By contradiction, let

$$d{h}_{E\left({\mu }_1\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)}=d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)},$$

but if h is a geodesic ${\phi }_E$-convex function, then

$$d{h}_{E\left({\mu }_1\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)}<\phi (h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right).$$

(16)

Also,

$$d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_2\right),E\left({\mu }_1\right)}<\phi (h\left(E\left({\mu }_2\right)\right),h\left(E\left({\mu }_1\right)\right).$$

On the other hand,

$$d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_2\right),E\left({\mu }_1\right)}=-d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)},$$

then

$$\begin{array}{c}\hfill -d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)}<\phi (h\left(E\left({\mu }_2\right)\right),h\left(E\left({\mu }_1\right)\right).\end{array}$$

(17)

Moreover, since $\phi$ is an antisymmetry function, then (17) becomes

$$d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)}>\phi (h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right),$$

hence,

$$d{h}_{E\left({\mu }_1\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)}>\phi (h\left(E\left({\mu }_1\right)\right),h\left(E\left({\mu }_2\right)\right).$$

(18)

From (16) and (18), we obtain a contradiction, then $d{h}_{E\left({\mu }_1\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\ne d{h}_{E\left({\mu }_2\right)}{\dot{\gamma }}_{E\left({\mu }_1\right),E\left({\mu }_2\right)}.$ □

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

In this section, ${\phi }_E$-epigraphs are introduced on complete Riemannian manifolds, and a characterization of geodesic ${\phi }_E$-convex functions in terms of their ${\phi }_E$-epigraphs is obtained.

Definition 12.

A set $B\subset N\times \mathbb{R}$ is called a geodesic ${\phi }_E$-convex set if

$$\left({\gamma }_{E\left(u_1\right),E\left(u_2\right)}\left(t\right),v_2+t\phi (v_1,v_2)\right)\in B,$$

for all $(u_i,v_i)\in B,t\in [0,1].$

Therefore, a ${\phi }_E$- epigraph of function h is defined by

$$ep{i}_{{\phi }_E}\left(h\right)=\left\{(u,v)\in E\left(N\right)\times \mathbb{R}:h\left(u\right)\le v\right\}.$$

Theorem 11.

Consider $B\subset N$ to be a geodesic E-convex set, and φ is non-decreasing. The set $ep{i}_{{\phi }_E}\left(h\right)$ is geodesic ${\phi }_E$-convex, if and only if $h:B⟶\mathbb{R}$ is a geodesic ${\phi }_E$-convex function.

Proof. 

Let $u_1,u_2\in B$ and $t\in [0,1]$, and since B is an E-convex set, then $E\left(u_1\right),E\left(u_2\right)\in E\left(B\right)\subseteq B$. Hence,

$$\left(\left(E\left(u_1\right),h\left(E\left(u_1\right)\right)\right),\left(E\left(u_2\right),h\left(E\left(u_2\right)\right)\right)\right)\in ep{i}_{{\phi }_E}\left(h\right).$$

Due to the fact that $ep{i}_{{\phi }_E}\left(h\right)$ is a geodesic ${\phi }_E$-convex set, then

$$\left({\gamma }_{E\left(u_1\right),E\left(u_2\right)}\left(t\right),h\left(E\left(u_2\right)\right)+t\phi \left(h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right)\right)\right)\in ep{i}_{{\phi }_E}\left(h\right).$$

This implies that $h\left({\gamma }_{E\left(u_1\right),E\left(u_2\right)}\left(t\right)\right)\le h\left(E\left(u_2\right)\right)+t\phi (h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right))$. Consequently, h is a geodesic ${\phi }_E$-convex function.

Now, let us consider that $(u_1^{*},v_1),(u_2^{*},v_2)\in ep{i}_{{\phi }_E}\left(h\right)$, then $u_1^{*},u_1^{*}\in E\left(B\right)$, which means that there are $u_1,u_2\in B$ such that $E\left(u_1\right)=u_1^{*}$ and $E\left(u_2\right)=u_2^{*}$. Hence, $h\left(E\left(u_1\right)\right)\le v_1,h\left(E\left(u_2\right)\right)\le v_2$ and, since h is a geodesic ${\phi }_E$-convex function, then

$$\begin{array}{ccc}\hfill h\left({\gamma }_{E\left(u_1\right),E\left(u_2\right)}\left(t\right)\right)& \le & h\left(E\left(u_2\right)\right)+t\phi \left(h\left(E\left(u_1\right)\right),h\left(E\left(u_2\right)\right)\right)\hfill \\ & \le & v_2+t\phi (v_1,v_2),\hfill \end{array}$$

which implies that $\left({\gamma }_{E\left(u_1\right),E\left(u_2\right)}\left(t\right),v_2+t\phi (v_1,v_2)\right)\in ep{i}_{{\phi }_E}\left(h\right)$, for all $t\in [0,1]$.

That is, $ep{i}_{{\phi }_E}\left(h\right)$ is a geodesic ${\phi }_E$-convex set. □

Theorem 12.

Consider $\left\{B_i,i\in I\right\}$ to be a family of geodesic ${\phi }_E$-convex sets, then $B={\cap }_{i\in I}{B}_i$ is also a geodesic ${\phi }_E$-convex set.

Proof. 

Let $({\mu }_1,{\nu }_1),({\mu }_2,{\nu }_2)\in {\cap }_{i\in I}{B}_i$, then $({\mu }_1,{\nu }_1),({\mu }_2,{\nu }_2)\in B_i$, for all $i\in I$. Hence,

$$\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\left(t\right),{\nu }_2+t\phi ({\nu }_1,{\nu }_2)\right)\in B_i$$

Then,

$$\left({\gamma }_{E\left({\mu }_1\right),E\left({\mu }_2\right)}\left(t\right),{\nu }_2+t\phi ({\nu }_1,{\nu }_2)\right)\in {\cap }_{i\in I}{B}_i$$

for all $t\in [0,1].$

This implies ${\cap }_{i\in I}{B}_i$ is a geodesic ${\phi }_E$-convex function. □

By using the above theorem, we can obtain the following corollary

Corollary 1.

Let $\left\{h_i,i\in I\right\}$ be a family of geodesic ${\phi }_E$-convex functions defined on a geodesic E-convex set $B\subset N$, which is bounded above, and φ is non-decreasing. If the E-epigraphs $ep{i}_{{\phi }_E}\left(h_i\right)$ are geodesic ${\phi }_E$-convex sets, then $h={sup}_{i\in I}{h}_i$ is also a geodesic ${\phi }_E$-convex function on B.

6. Conclusions

Some important properties of geodesic ${\phi }_E$-convex functions are established in this study. A new class of function on Riemannian manifold—together with its properties—is also studied here. We also reported how the characterization of the geodesic ${\phi }_E$-convex function can be obtained through their ${\phi }_E$-epigraph counterparts. The results presented in this paper can be used for future research on the Riemannian manifold. The ideas and techniques of this paper may motivate further research, for example, in fractional manifolds.