Open Access

On geodesic strongly E-convex sets and geodesic strongly E-convex functions

Journal of Inequalities and Applications20152015:297

https://doi.org/10.1186/s13660-015-0824-z

Received: 1 May 2015

Accepted: 15 September 2015

Published: 25 September 2015

Abstract

In this article, geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold are extended to the so-called geodesic strongly E-convex sets and geodesic strongly E-convex functions. Some properties of geodesic strongly E-convex sets are also discussed. The results obtained in this article may inspire future research in convex analysis and related optimization fields.

Keywords

geodesic E-convex setsgeodesic E-convex functionsRiemannian manifolds

MSC

52A2052A4153C2053C22

1 Introduction

Convexity and its generalizations play an important role in optimization theory, convex analysis, Minkowski space, and fractal mathematics [17]. In order to extend the validity of their results to large classes of optimization, these concepts have been generalized and extended in several directions using novel and innovative techniques. Youness [8] defined E-convex sets and E-convex functions, which have some important applications in various branches of mathematical sciences [911]. However, some results given by Youness [8] seem to be incorrect according to Yang [12]. Chen [13] extended E-convexity to a semi-E-convexity and discussed some of there properties. Also, Youness and Emam [14] discussed a new class functions which is called strongly E-convex functions by taking the images of two points \(x_{1} \) and \(x_{2} \) under an operator \(E\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{n} \) besides the two points themselves. Strong E-convexity was extended to a semi-strong E-convexity as well as quasi- and pseudo-semi-strong E-convexity in [15]. The authors investigated the characterization of efficient solutions for multi-objective programming problems involving semi-strong E-convexity [16].

A generalization of convexity on Riemannian manifolds was proposed by Rapcsak [17] and Udriste [18]. Moreover, Iqbal et al. [19] introduced geodesic E-convex sets and geodesic E-convex functions on Riemannian manifolds.

Motivated by earlier research works [18, 2025] and by the importance of the concepts of convexity and generalized convexity, we discuss a new class of sets on Riemannian manifolds and a new class of functions defined on them, which are called geodesic strongly E-convex sets and geodesic strongly E-convex functions, and some of their properties are presented.

2 Preliminaries

In this section, we introduce some definitions and well-known results of Riemannian manifolds, which help us throughout the article. We refer to [18] for the standard material on differential geometry.

Let N be a \(C^{\infty} \) m-dimensional Riemannian manifold, and \(T_{z}N \) be the tangent space to N at z. Also, assume that \(\mu_{z}(x_{1},x_{2}) \) is a positive inner product on the tangent space \(T_{z}N \) (\(x_{1},x_{2}\in T_{z}N \)), which is given for each point of N. Then a \(C^{\infty} \) map \(\mu\colon z\rightarrow\mu_{z} \), which assigns a positive inner product \(\mu _{z} \) to \(T_{z}N \) for each point z of N is called a Riemannian metric.

The length of a piecewise \(C^{1} \) curve \(\eta\colon [a_{1},a_{2}]\rightarrow N \) which is defined as follows:
$$L(\eta)= \int_{a_{1}}^{a_{2}} \bigl\Vert \acute{ \eta}(x)\bigr\Vert \, dx. $$
We define \(d(z_{1},z_{2})= \inf \lbrace L(\eta)\colon\eta\mbox{ is a piecewise } C^{1} \mbox{ curve joining } z_{1} \mbox{ to } z_{2} \rbrace\) for any points \(z_{1},z_{2}\in N \). Then d is a distance which induces the original topology on N. As we know on every Riemannian manifold there is a unique determined Riemannian connection, called a Levi-Civita connection, denoted by \(\bigtriangledown_{X}Y \), for any vector fields \(X,Y\in N \). Also, a smooth path η is a geodesic if and only if its tangent vector is a parallel vector field along the path η, i.e., η satisfies the equation \(\bigtriangledown_{\acute{\eta}(t)}\acute{\eta}(t)=0 \). Any path η joining \(z_{1} \) and \(z_{2} \) in N such that \(L(\eta )=d(z_{1},z_{2}) \) is a geodesic and is called a minimal geodesic.

Finally, assume that \((N,\eta) \) is a complete m-dimensional Riemannian manifold with Riemannian connection . Let \(x_{1} , x_{2} \in N \) and \(\eta\colon[0,1]\rightarrow N \) be a geodesic joining the points \(x_{1} \) and \(x_{2} \), which means that \(\eta_{x_{1},x_{2}}(0)=x_{2}\) and \(\eta_{x_{1},x_{2}}(1)=x_{1} \).

Definition 2.1

[18]

A set B in a Riemannian manifold N is called totally convex if B contains every geodesic \(\eta_{x_{1},x_{2}} \) of N whose endpoints \(x_{1} \) and \(x_{2} \) belong to B.

Note the whole of the manifold N is totally convex, and conventionally, so is the empty set. The minimal circle in a hyperboloid is totally convex, but a single point is not. Also, any proper subset of a sphere is not necessarily totally convex.

The following theorem was proved in [18].

Theorem 2.2

[18]

The intersection of any number of a totally convex sets is totally convex.

Remark 2.3

In general, the union of a totally convex set is not necessarily totally convex.

Definition 2.4

[18]

A function \(f\colon B\rightarrow\mathbb{R} \) is called a geodesic convex function on a totally convex set \(B\subset N \) if for every geodesic \(\eta_{x_{1},x_{2}} \), then
$$f\bigl(\eta_{x_{1},x_{2}}(\gamma)\bigr)\leq\gamma f(x_{1})+(1- \gamma)f(x_{2}) $$
holds for all \(x_{1},x_{2}\in B \) and \(\gamma\in[0,1] \).

In 2005, strongly E-convex sets and strongly E-convex functions were introduced by Youness and Emam [14] as follows.

Definition 2.5

[14]

  1. (1)
    A subset \(B\subseteq\mathbb{R}^{n} \) is called a strongly E-convex set if there is a map \(E\colon\mathbb{R}^{n}\rightarrow \mathbb{R}^{n} \) such that
    $$\gamma\bigl(\alpha b_{1}+E(b_{1})\bigr)+(1-\gamma) \bigl( \alpha b_{2}+E(b_{2})\bigr)\in B $$
    for each \(b_{1},b_{2}\in B\), \(\alpha\in[0,1] \) and \(\gamma\in[0,1] \).
     
  2. (2)
    A function \(f\colon B\subseteq\mathbb {R}^{n}\rightarrow\mathbb{R} \) is called a strongly E-convex function on N if there is a map \(E\colon\mathbb {R}^{n}\rightarrow\mathbb{R}^{n} \) such that B is a strongly E-convex set and
    $$f\bigl(\gamma\bigl(\alpha b_{1}+E(b_{1})\bigr)+(1-\gamma) \bigl(\alpha b_{2}+E(b_{2})\bigr)\bigr)\leq \gamma f \bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) $$
    for each \(b_{1},b_{2}\in B\), \(\alpha\in[0,1] \) and \(\gamma\in[0,1] \).
     

In 2012, the geodesic E-convex set and geodesic E-convex functions on a Riemannian manifold were introduced by Iqbal et al. [19] as follows.

Definition 2.6

[19]

  1. (1)

    Assume that \(E\colon N\rightarrow N \) is a map. A subset B in a Riemannian manifold N is called geodesic E-convex iff there exists a unique geodesic \(\eta_{E(b_{1}),E(b_{2})}(\gamma) \) of length \(d(b_{1},b_{2}) \), which belongs to B, for each \(b_{1},b_{2}\in B \) and \(\gamma\in[0,1] \).

     
  2. (2)
    A function \(f\colon B\subseteq N \rightarrow\mathbb {R}\) is called geodesic E-convex on a geodesic E-convex set B if
    $$f\bigl(\eta_{E(b_{1}),E(b_{2})}(\gamma)\bigr)\leq\gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) $$
    for all \(b_{1},b_{2}\in B \) and \(\gamma\in[0,1] \).
     

3 Geodesic strongly E-convex sets and geodesic strongly E-convex functions

In this section, we introduce a geodesic strongly E-convex (GSEC) set and a geodesic strongly E-convex (GSEC) function in a Riemannian manifold N and discuss some of their properties.

Definition 3.1

Assume that \(E\colon N\rightarrow N \) is a map. A subset B in a Riemannian manifold N is called GSEC if and only if there is a unique geodesic \(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma) \) of length \(d(b_{1},b_{2}) \), which belongs to B, \(\forall b_{1},b_{2}\in B\), \(\alpha\in[0,1] \), and \(\gamma\in [0,1] \).

Remark 3.2

  1. (1)

    Every GSEC set is a GEC set when \(\alpha=0 \).

     
  2. (2)

    A GEC set is not necessarily a GSEC set. The following example shows this statement.

     

Example 3.3

Let \(N^{2} \) be a 2-dimensional simply complete Riemannian manifold of non-positive sectional curvature, and \(B\subset N^{2} \) be an open star-shaped. Let \(E\colon N^{2}\rightarrow N^{2} \) be a map such that \(E(z)= \lbrace y\colon y\in \operatorname{ker}(B), \forall z\in B \rbrace \). Then B is GEC; on the other hand it is not GSEC.

Proposition 3.4

Every convex set \(B\subset N \) is a GSEC set.

Proof

Let us take a map \(E\colon N\rightarrow N \) such as \(E=I \) where I is the identity map and \(\alpha=0 \), then we have the required result. □

Note if we take the mapping \(E(x)=(1-\alpha)x\), \(x\in B \), then the definition of a GSE reduces to the definition of a t-convex set.

Theorem 3.5

If \(B\subset N \) is a GSEC set, then \(E(B)\subseteq B \).

Proof

Since B is a GSEC set, we have for each \(b_{1},b_{2}\in B\), \(\alpha \in[0,1] \), and \(\gamma\in[0,1] \),
$$\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\in B. $$
For \(\gamma=0 \) and \(\alpha=0 \), we have \(\eta _{E(b_{1}),E(b_{2})}(0)=E(b_{2})\in B \), then \(E(B)\subseteq B \). □

Theorem 3.6

If \(\lbrace B_{j}, j\in I \rbrace\) is an arbitrary family of GSEC subsets of N with respect to the mapping \(E\colon N\rightarrow N \), then the intersection \(\bigcap_{j\in I}B_{j} \) is a GSEC subset of N.

Proof

If \(\bigcap_{j\in I}B_{j} \) is an empty set, then it is obviously a GSEC subset of N. Assume that \(b_{1},b_{2}\in\bigcap_{j\in I} B_{j} \), then \(b_{1},b_{2} \in B_{j} \), \(\forall j\in I \). By the GSEC of \(B_{j} \), we get \(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma )\in B_{j}\), \(\forall j\in I\), \(\alpha\in[0,1] \), and \(\gamma\in[0,1] \). Hence, \(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\in \bigcap_{j\in I} B_{j}\), \(\forall\alpha\in[0,1] \) and \(\gamma\in[0,1] \). □

Remark 3.7

The above theorem is not generally true for the union of GSEC subsets of N.

Now, we extend the definition of a GEC function on a Riemannian manifold to a GSEC function on a Riemannian manifold.

Definition 3.8

A real-valued function \(f\colon B\subset N\rightarrow\mathbb{R} \) is said to be a GSEC function on a GSEC set B, if
$$f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr), $$
\(\forall b_{1},b_{2}\in B \), \(\alpha\in[0,1]\), and \(\gamma\in[0,1] \). If the above inequality is strict for all \(b_{1},b_{2}\in B\), \(\alpha b_{1}+E(b_{1})\neq\alpha b_{2}+E(b_{2})\), \(\alpha\in[0,1]\), and \(\gamma \in(0,1) \), then f is called a strictly GSEC function.

Remark 3.9

  1. (1)

    Every GSEC function is a GEC function when \(\alpha=0 \). The following example shows that a GEC function is not necessarily a GSEC function.

     

Example 3.10

Consider the function \(f\colon\mathbb{R}\rightarrow\mathbb{R} \) where \(f(b)= -|b| \) and suppose that \(E\colon\mathbb {R}\rightarrow\mathbb{R} \) is given as \(E(b)=-b \). We consider the geodesic η such that
$$\begin{aligned} \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma) =& \textstyle\begin{cases} - [\alpha b_{2}+E(b_{2}) +\gamma(\alpha b_{1}+E(b_{1})-\alpha b_{2}-E(b_{2})) ] ;& b_{1}b_{2}\geq0, \\ - [\alpha b_{2}+E(b_{2}) +\gamma(\alpha b_{2}+E(b_{2})-\alpha b_{1}-E(b_{1})) ] ;& b_{1}b_{2}< 0 \end{cases}\displaystyle \\ =& \textstyle\begin{cases} - [(\alpha-1) b_{2} +\gamma((\alpha-1) b_{1}+(1-\alpha) b_{2}) ] ;& b_{1}b_{2}\geq0, \\ - [(\alpha-1) b_{2} +\gamma ((\alpha-1) b_{2}+(1-\alpha) b_{1}) ] ;& b_{1}b_{2}< 0. \end{cases}\displaystyle \end{aligned}$$
If \(\alpha=0 \), then
$$ \eta_{E(b_{1}),E(b_{2})}(\gamma) = \textstyle\begin{cases} {[ b_{2} +\gamma( b_{1}-b_{2}) ]} ;& b_{1}b_{2}\geq0, \\ {[ b_{2} +\gamma( b_{2}- b_{1}) ]} ;& b_{1}b_{2}< 0. \end{cases} $$
If \(b_{1}, b_{2}\geq0 \), then
$$\begin{aligned} f\bigl(\eta_{E(b_{1}),E(b_{2})}(\gamma)\bigr) =& f\bigl(b_{2}+\gamma (b_{1}-b_{2})\bigr) \\ =& -\bigl[(1-\gamma)b_{2}+\gamma b_{1}\bigr]. \end{aligned}$$
On the other hand
$$ \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2}) \bigr)= \gamma f(-b_{1})+(1-\gamma )f(-b_{2}) = -\bigl[(1-\gamma)b_{2}+\gamma b_{1}\bigr]. $$
Hence, \(f(\eta_{E(b_{1}),E(b_{2})}(\gamma))\leq\gamma f(E(b_{1}))+(1-\gamma)f(E(b_{2})) \), \(\forall\gamma\in[0,1] \).

Similarly, the above inequality holds true when \(b_{1},b_{2}<0 \).

Now, let \(b_{1}<0\), \(b_{2}>0 \), then
$$\begin{aligned} f\bigl(\eta_{E(b_{1}),E(b_{2})}(\gamma)\bigr) =& f\bigl(b_{2}+ \gamma(b_{2}-b_{1})\bigr) \\ =& -\bigl[(1+\gamma)b_{2}-\gamma b_{1}\bigr]. \end{aligned}$$
On the other hand
$$ \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) = \gamma f(-b_{1})+(1-\gamma )f(-b_{2}) = \gamma b_{1}-(1-\gamma)b_{2}. $$
It follows that
$$f\bigl(\eta_{E(b_{1}),E(b_{2})}(\gamma)\bigr)\leq\gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma )f\bigl(E(b_{2})\bigr) $$
if and only if
$$-\bigl[(1+\gamma)b_{2}-\gamma b_{1}\bigr]\leq\gamma b_{1}-(1-\gamma)b_{2} $$
if and only if
$$-2\gamma b_{2}\leq0, $$
which is always true for all \(\gamma\in[0,1] \).

Similarly, \(f(\eta_{E(b_{1}),E(b_{2})}(\gamma))\leq\gamma f(E(b_{1}))+(1-\gamma)f(E(b_{2})) \), \(\forall\gamma\in[0,1] \) also holds for \(b_{1}>0 \) and \(b_{2}<0 \).

Thus, f is a GEC function on \(\mathbb{R} \), but it is not a GSEC function because if we take \(b_{1}=0\), \(b_{2}=-1 \) and \(\gamma=\frac {1}{2} \), then
$$\begin{aligned} f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr) =& f\biggl(\frac{1}{2}\alpha- \frac{1}{2}\biggr) \\ =& \frac{1}{2}\alpha-\frac {1}{2} \\ > & \frac{1}{2}f\bigl(E(0)\bigr)+\frac{1}{2}f\bigl(E(-1)\bigr) \\ =& \frac{-1}{2} ,\quad \forall\alpha\in ( 0,1 ] . \end{aligned}$$
  1. (2)

    Every g-convex function f on a convex set B is a GSEC function when \(\alpha=0 \) and E is the identity map.

     

Proposition 3.11

Assume that \(f\colon B\rightarrow\mathbb{R} \) is a GSEC function on a GSEC set \(B\subseteq N \), then \(f(\alpha b+E(b))\leq f(E(b)) \), \(\forall b\in B \) and \(\alpha\in[0,1] \).

Proof

Since \(f\colon B\rightarrow\mathbb{R} \) is a GSEC function on a GSEC set \(B\subseteq N \), then \(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma) \in B\), \(\forall b_{1},b_{2}\in B\), \(\alpha\in [0,1]\), and \(\gamma\in[0,1] \). Also,
$$f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) $$
thus, for \(\gamma=1 \), we get \(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)=\alpha b_{1}+E(b_{1}) \). Then
$$f\bigl(\alpha b_{1}+E(b_{1})\bigr)\leq f \bigl(E(b_{1})\bigr) . $$
 □

Theorem 3.12

Consider that \(B\subseteq N \) is a GSEC set and \(f_{1}\colon B\rightarrow\mathbb{R} \) is a GSEC function. If \(f_{2}\colon I\rightarrow\mathbb{R} \) is a non-decreasing convex function such that \(\operatorname{rang}(f_{1})\subset I \), then \(f_{2}\circ f_{1} \) is a GSEC function on B.

Proof

Since \(f_{1} \) is a GSEC function, for all \(b_{1},b_{2}\in B\), \(\alpha \in[0,1] \), and \(\gamma\in[0,1] \),
$$f_{1}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f_{1}\bigl(E(b_{1})\bigr)+(1-\gamma)f_{1} \bigl(E(b_{2})\bigr). $$
Since \(f_{2} \) is a non-decreasing convex function,
$$\begin{aligned}& f_{2}\circ f_{2}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma )\bigr) \\& \quad = f_{2} \bigl( f_{2}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})} (\gamma)\bigr) \bigr) \\& \quad \leq f_{2} \bigl(\gamma f_{1} \bigl(E(b_{1}) \bigr)+(1-\gamma) f_{1}\bigl(E(b_{2})\bigr) \bigr) \\& \quad \leq \gamma f_{2} \bigl( f_{1} \bigl(E(b_{1}) \bigr) \bigr) +(1-\gamma) f_{2} \bigl( f_{1} \bigl(E(b_{2})\bigr) \bigr) \\& \quad = \gamma(f_{2}\circ f_{1}) \bigl(E(b_{1})\bigr) +(1-\gamma) (f_{2}\circ f_{1}) \bigl(E(b_{2}) \bigr), \end{aligned}$$
which means that \(f_{2}\circ f_{1} \) is a GSEC function on B. Similarly, if \(f_{2} \) is a strictly non-decreasing convex function, then \(f_{2}\circ f_{1} \) is a strictly GSEC function. □

Theorem 3.13

Assume that \(B\subseteq N \) is a GSEC set and \(f_{j}\colon B\rightarrow\mathbb{R}\), \(j=1,2,\ldots,m \) are GSEC functions. Then the function
$$f=\sum_{j=1}^{m}n_{j}f_{j} $$
is GSEC on B, \(\forall n_{j}\in\mathbb{R}\), \(n_{j}\geq0 \).

Proof

Since \(f_{j}\), \(j=1,2,\ldots,m \) are GSEC functions, \(\forall b_{1},b_{2}\in B \), \(\alpha\in[0,1]\), and \(\gamma\in[0,1] \), we have
$$f_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f_{j}\bigl(E(b_{1})\bigr)+(1-\gamma)f_{j} \bigl(E(b_{2})\bigr). $$
It follows that
$$n_{j}f_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})} (\gamma )\bigr)\leq \gamma n_{j} f_{j}\bigl(E(b_{1})\bigr)+(1- \gamma)n_{j}f_{j}\bigl(E(b_{2})\bigr). $$
Then
$$\begin{aligned}& \sum_{j=1}^{m}n_{j}f_{j} \bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})} (\gamma)\bigr) \\& \quad \leq \gamma\sum _{j=1}^{m} n_{j} f_{j} \bigl(E(b_{1})\bigr)+(1-\gamma)\sum_{j=1}^{m}n_{j}f_{j} \bigl(E(b_{2})\bigr) \\& \quad = \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2}) \bigr). \end{aligned}$$
Thus, f is a GSEC function. □

Theorem 3.14

Let \(B\subseteq N \) be a GSEC set and \(\lbrace f_{j},j\in I \rbrace\) be a family of real-valued functions defined on B such that \(\sup_{j\in I}f_{j}(b) \) exists in \(\mathbb{R} \), \(\forall b\in B \). If \(f_{j}\colon B\rightarrow\mathbb{R} \), \(j\in I\) are GSEC functions on B, then the function \(f\colon B\rightarrow \mathbb{R} \), defined by \(f(b)=\sup_{j\in I}f_{j}(b)\), \(\forall b\in B \) is GSEC on B.

Proof

Since \(f_{j}\), \(j\in I \) are GSEC functions on a GSEC set B, \(\forall b_{1},b_{2}\in B \), \(\alpha\in[0,1]\), and \(\gamma\in[0,1] \), we have
$$f_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f_{j}\bigl(E(b_{1})\bigr)+(1-\gamma)f_{j} \bigl(E(b_{2})\bigr). $$
Then
$$\begin{aligned}& \sup_{j\in I}f_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma) \bigr) \\& \quad \leq \sup_{j\in I} \bigl[ \gamma f_{j} \bigl(E(b_{1})\bigr)+(1-\gamma)f_{j}\bigl(E(b_{2}) \bigr) \bigr] \\& \quad = \gamma\sup_{j\in I} f_{j}\bigl(E(b_{1}) \bigr)+(1-\gamma)\sup_{j\in I} f_{j} \bigl(E(b_{2})\bigr) \\& \quad = \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2}) \bigr). \end{aligned}$$
Hence,
$$f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr), $$
which means that f is a GSEC function on B. □

Proposition 3.15

Assume that \(h_{j}\colon N\rightarrow\mathbb{R} \), \(j=1,2,\ldots,m\) are GSEC functions on N, with respect to \(E\colon N\rightarrow N \). If \(E(B)\subseteq B \), then \(B= \lbrace b\in N\colon h_{j}(b)\leq0, j=1,2,\ldots,m \rbrace \) is a GSEC set.

Proof

Since \(h_{j}\), \(j=1,2,\ldots m \) are GSEC functions,
$$\begin{aligned} h_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr) \leq & \gamma h_{j}\bigl(E(b_{1})\bigr)+(1-\gamma)h_{j} \bigl(E(b_{2})\bigr) \\ \leq& 0, \end{aligned}$$
\(\forall b_{1},b_{2}\in B \), \(\alpha\in[0,1]\), and \(\gamma\in[0,1] \). Since \(E(B) \subseteq B \), \(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma) \in B \). Hence, B is a GSEC set. □

4 Epigraphs

Youness and Emam [14] defined a strongly \(E\times F \)-convex set where \(E\colon\mathbb{R}^{n} \rightarrow\mathbb{R}^{n}\) and \(F\colon\mathbb{R} \rightarrow\mathbb{R}\) and studied some of its properties. In this section, we generalize a strongly \(E\times F \)-convex set to a geodesic strongly \(E\times F \)-convex set on Riemannian manifolds and discuss GSEC functions in terms of their epigraphs. Furthermore, some properties of GSE sets are given.

Definition 4.1

Let \(B\subset N\times\mathbb{R} \), \(E\colon N\rightarrow N\) and \(F\colon\mathbb{R} \rightarrow\mathbb{R}\). A set B is called geodesic strongly \(E\times F \)-convex if \((b_{1},\beta _{1}),(b_{2},\beta_{2})\in B \) implies
$$\bigl( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(\beta_{1})+(1-\gamma)F( \beta_{2}) \bigr) \in B $$
for all \(\alpha\in[0,1] \) and \(\gamma\in[0,1] \).

It is not difficult to prove that \(B\subseteq N \) is a GSEC set if and only if \(B\times\mathbb{R} \) is a geodesic strongly \(E\times F \)-convex set.

An epigraph of f is given by
$$\operatorname{epi}(f)= \bigl\lbrace (b,a)\colon b\in B, a\in\mathbb{R}, f(b)\leq a \bigr\rbrace . $$
A characterization of a GSEC function in terms of its \(\operatorname{epi}(f) \) is given by the following theorem.

Theorem 4.2

Let \(E\colon N\rightarrow N \) be a map, \(B\subseteq N \) be a GSEC set, \(f\colon B\rightarrow\mathbb{R} \) be a real-valued function and \(F\colon\mathbb{R}\rightarrow\mathbb{R} \) be a map such that \(F(f(b)+a)=f(E(b))+a \), for each non-negative real number a. Then f is a GSEC function on B if and only if \(\operatorname{epi}(f) \) is geodesic strongly \(E\times F \)-convex on \(B\times\mathbb{R} \).

Proof

Assume that \((b_{1},a_{1}) ,(b_{2},a_{2})\in \operatorname{epi}(f)\). If B is a GSEC set, then \(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\in B\), \(\forall\alpha\in[0,1] \) and \(\gamma\in [0,1] \). Since \(E(b_{1})\in B \) for \(\alpha=0\), \(\gamma=1 \), also \(E(b_{2})\in B \) for \(\alpha=0\), \(\gamma=0 \), let \(F(a_{1}) \) and \(F(a_{2}) \) be such that \(f(E(b_{1}))\leq F(a_{1}) \) and \(f(E(b_{2}))\leq F(a_{2}) \). Then \((E(b_{1}),F(a_{1})),(E(b_{2}),F(a_{2}))\in \operatorname{epi}(f) \).

Let f be GSEC on B, then
$$\begin{aligned} f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr) \leq& \gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) \\ \leq& \gamma F(a_{1})+(1-\gamma)F(a_{2}). \end{aligned}$$
Thus, \(( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(a_{1})+(1-\gamma)F(a_{2}) ) \in \operatorname{epi}(f) \), then \(\operatorname{epi}(f)\) is geodesic strongly \(E\times F \)-convex on \(B\times\mathbb{R} \).
Conversely, assume that \(\operatorname{epi}(f)\) is geodesic strongly \(E\times F \)-convex on \(B\times\mathbb{R} \). Let \(b_{1},b_{2}\in B \), \(\alpha\in [0,1] \), and \(\gamma\in[0,1] \), then \((b_{1},f(b_{1}))\in \operatorname{epi}(f) \) and \((b_{2},f(b_{2}))\in \operatorname{epi}(f) \). Now, since \(\operatorname{epi}(f)\) is geodesic strongly \(E\times F \)-convex on \(B\times\mathbb{R} \), we obtain \(( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(f(b_{1}))+(1-\gamma)F(f(b_{2})) ) \in \operatorname{epi}(f) \), then
$$\begin{aligned} f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr) \leq& \gamma F\bigl(f(b_{1}) \bigr)+(1-\gamma)F\bigl(f(b_{2})\bigr) \\ =& \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2}) \bigr). \end{aligned}$$
This shows that f is a GSEC function on B. □

Theorem 4.3

Assume that \(\lbrace B_{j}, j\in I \rbrace \) is a family of geodesic strongly \(E\times F \)-convex sets. Then the intersection \(\bigcap_{j\in I}B_{j} \) is a geodesic strongly \(E\times F \)-convex set.

Proof

Assume that \((b_{1},a_{1}) ,(b_{2},a_{2})\in\bigcap_{j\in I}B_{j} \), so \(\forall j\in I \), \((b_{1},a_{1}) ,(b_{2},a_{2})\in B_{j}\). Since \(B_{j} \) is the geodesic strongly \(E\times F \)-convex sets \(\forall j\in I \), we have
$$\bigl( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(a_{1})+(1- \gamma)F(a_{2}) \bigr)\in B_{j} , $$
\(\forall\alpha\in[0,1]\) and \(\gamma\in[0,1] \). Therefore,
$$\bigl( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(a_{1})+(1- \gamma)F(a_{2}) \bigr)\in\bigcap_{j\in I}B_{j}, $$
\(\forall\alpha\in[0,1]\) and \(\gamma\in[0,1] \). Then \(\bigcap_{j\in I}B_{j} \) is a geodesic strongly \(E\times F \)-convex set. □

Theorem 4.4

Assume that \(E\colon N \rightarrow N \) and \(F\colon\mathbb {R}\rightarrow\mathbb{R} \) are two maps such that \(F(f(b)+a)=f(E(b))+a \) for each non-negative real number a. Suppose that \(\lbrace f_{j}, j\in I \rbrace \) is a family of real-valued functions defined on a GSEC set \(B\subseteq N \) which are bounded from above. If \(\operatorname{epi}(f_{j}) \) are geodesic strongly \(E\times F \)-convex sets, then the function f which is given by \(f(b)=\sup_{j\in I}f_{j}(b)\), \(\forall b\in B \), is a GSEC function on B.

Proof

If each \(f_{j}\), \(j\in I \) is a GSEC function on a GSEC geodesic set B, then
$$\operatorname{epi}(f_{j})= \bigl\lbrace (b,a)\colon b\in B, a\in \mathbb{R}, f_{j}(b)\leq a \bigr\rbrace $$
are geodesic strongly \(E\times F \)-convex on \(B\times\mathbb{R} \). Therefore,
$$\begin{aligned} \bigcap_{j\in I}\operatorname{epi}(f_{j}) =& \bigl\lbrace (b,a)\colon b\in B, a\in \mathbb{R}, f_{j}(b) \leq a, j\in I \bigr\rbrace \\ =& \bigl\lbrace (b,a)\colon b\in B, a\in\mathbb{R}, f(b)\leq a \bigr\rbrace \end{aligned}$$
is geodesic strongly \(E\times F \)-convex set. Then, according to Theorem 4.2 we see that f is a GSEC function on B. □

Declarations

Acknowledgements

The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, University Putra Malaysia

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© Kılıçman and Saleh 2015