#### Abstract

The present paper deals with the properties of geodesic -preinvex functions and their relationships with -invex functions and strictly geodesic -preinvex functions. The geodesic -pre-pseudo-invex and geodesic -pre-quasi-invex functions on the geodesic invex set are introduced and some of their properties are discussed.

#### 1. Introduction

In the recent years, several generalizations have been developed for the classical properties of convexity. This concept in linear topological spaces relies on the possibility of connecting any two points of the space by the line segment between them. In 1981, Hanson [1] introduced the concept of invexity by generalizing the difference in the definition of convex function to any function . Hanson's initial results inspired a great deal of subsequent work, which has greatly expanded the role and applications of invexity in nonlinear optimization and other branches of pure and applied sciences.

Ben-Israel and Mond [2] introduced a new generalization of convex sets and convex functions. Craven [3] called them invex sets and preinvex functions, respectively. Jeyakumar [4] studied the properties of preinvex functions and their role in optimization and mathematical programming.

In general, a manifold is not a linear space, but the extension of concepts and techniques from linear spaces to Riemannian manifold are natural. RapcsΓ‘k [5] and UdriΕte [6] considered a generalization of convexity, called geodesic convexity and extended many results of convex analysis and optimization theory to Riemannian manifolds. In this setting, the linear space has been replaced by Riemannian manifold and the line segment by a geodesic. For more details, readers may see [5, 6] and the references cited therein.

The notion of invex functions on Riemannian manifold was introduced in [7], however its generalization has been investigated by Mititelu [8]. Barani and Pouryayevali [9] introduced the geodesic invex set, geodesic invex function and geodesic preinvex function on Riemannian manifold with respect to particular maps, and studied the relations between them.

In this paper, we discuss various concepts, definitions and properties for the functions on Riemannian manifold. The notion of invexity and its generalization on Riemannian manifold is recalled in Section 2. In Section 3, we discuss some properties of geodesic -preinvex functions and their relationships with -invex functions. The relationship between strictly geodesic -preinvex function and geodesic -preinvex function is investigated. We also prove that the composite function is -invex on the geodesic invex set.

The geodesic -pre-pseudo-invex (p.p.i) and geodesic -pre-quasi-invex (p.q.i) functions are introduced and their properties are also discussed in Section 4. The results obtained here generalize those which are present in the literature.

#### 2. Preliminaries

In this section, we recall some definitions and known results about Riemannian manifolds, which will be used throughout the paper. For the standard material on differential geometry, one can consult [10].

Suppose that is a complete -dimensional Riemannian manifold. A subset of is called totally convex, if contains every geodesic of , whose end points and belong to [6].

Definition 2.1. Let be a complete -dimensional Riemannian manifold and be a totally convex set in . A function is said to be geodesic convex if for all geodesic arcs and all , one hasorif is a differentiable function.

Let be an -dimensional differentiable manifold and be the tangent space to at . Also, assume that is the tangent bundle of . For any , .

Definition 2.2 (see [7, 8]). A differentiable curve is called a differentiable application.
Let . Then tangent vector to the curve at is .
Assume that is another differentiable manifold and is a differentiable application.

Definition 2.3 (see [8]). The linear application defined by , where and is called the differential of at the point . ButSo, The length of a piecewise curve is defined byFor any two points , we defineThen is a distance which induces the original topology on . We know that on every Riemannian manifold there exists exactly one covariant derivation called Levi-Civita connection denoted by for any vector fields .

Definition 2.4. A geodesic is a smooth path whose tangent is parallel along the path , that is, satisfies the equation . Any path joining and in such that is a geodesic and it is called a minimal geodesic.
In other words, a curve whose acceleration vector field vanishes identically is called geodesic [6].

We consider now an application such that for every and any . For a differentiable function , Pini [7] defined invexity as follows.

Definition 2.5. The differentiable function is said to be -invex on if for any ,
If is a differentiable map from the manifold to the manifold , we will denote by the differential of at .

Mititelu [8] generalized the above definition as follows.

Definition 2.6. The differentiable function is said to be -pseudoinvex on if for any ,

Definition 2.7. The differentiable function is said to be -quasiinvex on if for any ,

Remark 2.8. If is a differentiable and -invex function defined on and for all , then is -pseudoinvex on .

In all these definitions,If is a Riemannian manifold and is a differentiable map from to , thenwhere is the gradient of at the point .

Barani and Pouryayevali [9] defined the geodesic invex set and the invexity of a function on an open geodesic invex subset of a Riemannian manifold.

Definition 2.9. Let be a Riemannian manifold and let be a function such that for every , . A nonempty subset of is said to be geodesic invex set with respect to if for every there exists a unique geodesic such that

Definition 2.10. Let be a Riemannian manifold, let be an open subset of which is geodesic invex with respect to , and let be a real differentiable function defined on . Then, is said to be -invex on if for every , Differentiable convex functions (on an open convex subset ) are invex. Also, the wider class of geodetically convex functions on manifolds is included in the class of invex functions, under some additional hypotheses on manifold [11]. In particular, this is true if the manifold has the property that given any two points there exists a unique geodesic joining them. In this case a function which is geodetically convex is also geodesic invex with respect to .

#### 3. Some Properties of Geodesic -Preinvex Functions

The definition of preinvex function on is given in [12]. See also [13, 14] for the properties of preinvex functions. Barani and Pouryayevali [9] extended this notion to Riemannian manifolds.

Definition 3.1 (see [9]). Let be a Riemannian manifold and let be an open subset of which is geodesic invex with respect to . Then, is said to be geodesic -preinvex on iffor every . If the above inequality is strict then, is said to be strictly geodesic -preinvex on .

Theorem 3.2. Let be a Riemannian manifold and let be an open subset of which is geodesic invex with respect to . Let be a geodesic -preinvex function and let be an increasing convex function such that range . Then the composite function is geodesic -preinvex on .

Proof. Since is geodesic -preinvex function, we haveSince is an increasing and convex function, we get Hence, is geodesic -preinvex on .

Theorem 3.3. Let be a Riemannian manifold and let be an open subset of which is geodesic invex with respect to . Let be a geodesic -preinvex function and let be a strictly increasing convex function such that range . Then the composite function is strictly geodesic -preinvex on .

Proof. Since is geodesic -preinvex function, we haveSince is the strictly increasing and convex function, we get Or Hence, is strictly geodesic -preinvex on .

Similarly, we can prove the following results.

Theorem 3.4. Let be a Riemannian manifold and let be an open subset of which is geodesic invex with respect to . Let be a geodesic -preinvex function and let be increasing and strictly convex function such that range . Then the composite function is strictly geodesic -preinvex on .

Theorem 3.5. Let be an open subset of which is geodesic invex with respect to . Suppose , , be geodesic -preinvex. Thenis geodesic -preinvex function on .

Now, we prove the following proposition, which guarantees that a differentiable and geodesic -preinvex function is -invex.

Proposition 3.6. Let be a complete manifold and which is geodesic invex with respect to . Let be a differentiable function and there exists a sequence of positive real numbers such that as andfor every , then is -invex on .

Proof. We haveSince is differentiable on , taking the limit as on both sides, we getTherefore,Hence, the result.

It is to be noted that the converse of above proposition is not true in general. However, Barani and Pouryayevali [9] proved that a -invex function on is geodesic -preinvex on if satisfies the condition in [9].

It is revealed in the following proposition that like convex functions, -invex functions are transformed into -invex functions by a suitable class of monotone functions.

Proposition 3.7. Let be a monotone increasing differentiable convex function. If is -invex on geodesic invex set , then the composite function is -invex.

Proof. Using the fact that for every , we have Hence is -invex on .

#### 4. Properties of Generalized Geodesic -Preinvex Functions

In [15], Pini introduced the notion of -pre-pseudo-invex and -pre-quasi-invex functions on an invex set. We extend these notions to geodesic -pre-pseudo-invexity and geodesic -pre-quasi-invexity on a geodesic invex set by replacing the line segment with the geodesic.

Let be a function defined on a geodesic invex subset of a Riemannian manifold with respect to .

Definition 4.1. Function is said to be geodesic -pre-pseudo-invex (p.p.i) on if there exist a geodesic and a strictly positive function such thatfor every and .

Theorem 4.2. Let be a Riemannian manifold and be an open subset of which is geodesic invex with respect to . If is geodesic -preinvex, then is geodesic -pre-pseudo-invex for the same geodesic.

Proof. If for every and is geodesic -preinvex, then where .

Theorem 4.3. Let be a geodesic -pre-pseudo-invex function on and let be strictly increasing convex function such that range . Then, the composite function is geodesic -pre-pseudo-invex on .

Proof. Since is geodesic -pre-pseudo-invex function on , we havefor every and , where is strict positive function.
Since is strictly increasing convex function, we get for every and , where is strict positive function. Which shows that is a geodesic -pre-pseudo-invex function on .

Definition 4.4. Function is said to be geodesic -pre-quasi-invex (p.q.i) on iffor all and for every .

Now, we characterize geodesic -pre-quasi-invex function in terms of its lower level sets.

Theorem 4.5. Let be a geodesic invex subset of and . Then, is geodesic -pre-quasi-invex on if and only if its lower level sets are geodesic invex.

Proof. Suppose that is geodesic -pre-quasi-invex function on and is the subset of . If is empty, the result is trivial. If is neither empty nor the whole set , take any two points and in . We have to show that the geodesic is contained in . Since is geodesic -pre-quasi-invex function, we havefor all and for every . Hence is geodesic invex.
Conversely, suppose that for every real number the set is geodesic invex. Take any two points and suppose that . Consider the lower level set . Since is geodesic invex, the geodesic is contained in . Thus,for every . The proof is complete.

Proposition 4.6. Let be a geodesic -pre-quasi-invex function on . Then,
(i)every strict local minimum of is also a strict global minimum;(ii)the set of all strict global minimum points is geodesic invex set.

Proof. Let be a strict local minimum which is not global; then there exists a point , such that, . Since, is geodesic -pre-quasi-invex, we have , which contradicts the hypothesis that is a strict local minimum.
If has no minimum value in , then the set of minimum points is empty and hence geodesic invex. If has the minimum point on , then the set of minimum points is , which is geodesic invex.
The geodesic -pre-quasi-invexity is preserved under composition with nondecreasing function as can be seen below.

Proposition 4.7. Let be a geodesic -pre-quasi-invex function and let be a nondecreasing function. Then, is geodesic -pre-quasi-invex.

Proof. Given that is a geodesic -pre-quasi-invex function and is a nondecreasing function. Then, we have which shows that the composite function is geodesic -pre-quasi-invex.

Proposition 4.8. If the function is geodesic -preinvex on , then is geodesic -pre-quasi-invex on .

Proof. Let be geodesic -preinvex function on . Then for every and , it follows that which shows that is geodesic -pre-quasi-invex on .

Proposition 4.9. If is geodesic -pre-pseudo-invex on then is geodesic -pre-quasi-invex on .

Proof. Let . Since is geodesic -pre-pseudo-invex function on for all , and for all , we have Hence, is geodesic -pre-quasi-invex on .

#### Acknowledgment

The authors are highly thankful to anonymous referees for their valuable suggestions/comments which have improved the paper.