On Properties of Geodesic -Preinvex Functions

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


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 x − y in the definition of convex function to any function η x, y . 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. Rapcsák 5 and Udrişte 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 In this setting, the linear space has been replaced by Riemannian manifold and the line segment by a geodesic.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.

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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 η-preinvex functions and their relationships with η-invex functions. The relationship between strictly geodesic η-preinvex function and geodesic η-preinvex function is investigated. We also prove that the composite function is η-invex on the geodesic invex set.
The geodesic η-pre-pseudo-invex p.p.i and geodesic η-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.

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 M, g is a complete n-dimensional Riemannian manifold. A subset A of M is called totally convex, if A contains every geodesic γ x,y of M, whose end points x and y belong to A 6 .
Let M be an n-dimensional differentiable manifold and T p M be the tangent space to M at p. Also, assume that TM p∈M T p M is the tangent bundle of M. For any > 0, So, The length of a piecewise C 1 curve γ : a, b → M is defined by For any two points p, q ∈ M, we define d p, q : inf L γ : γ is a piecewise C 1 curve joining p to q .

2.6
Then d is a distance which induces the original topology on M. We know that on every Riemannian manifold there exists exactly one covariant derivation called Levi-Civita connection denoted by ∇ X Y for any vector fields X, Y ∈ M.
Definition 2.4. A geodesic is a C ∞ smooth path γ whose tangent is parallel along the path γ, that is, γ satisfies the equation ∇ dγ t /dt dγ t /dt 0. Any path γ joining p and q in M such that L γ d p, q is a geodesic and it is called a minimal geodesic. In other words, a C ∞ curve γ : I → M whose acceleration vector field vanishes identically is called geodesic 6 .
We consider now an application η : M × M → TM such that η p, q ∈ T q M for every q ∈ M and any p ∈ M. For a differentiable function f : M → R, Pini 7 defined invexity as follows.
If f is a differentiable map from the manifold M to the manifold N, we will denote by df x the differential of f at x.
Mititelu 8 generalized the above definition as follows.

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In all these definitions, df y η x, y df y η x, y .

2.10
If M, g is a Riemannian manifold and f is a differentiable map from M to N, then where grad f y is the gradient of f at the point y. Barani  Differentiable convex functions on an open convex subset A 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 x, y there exists a unique geodesic γ x,y t joining them. In this case a function which is geodetically convex is also geodesic invex with respect to η x, y γ x,y 0 .

Some Properties of Geodesic η-Preinvex Functions
The definition of preinvex function on R n is given in 12 . See also 13, 14 for the properties of preinvex functions. Barani  for every x, y ∈ S, t ∈ 0, 1 . If the above inequality is strict then, f is said to be strictly geodesic η-preinvex on S. Proof. Since f is geodesic η-preinvex function, we have Since g is an increasing and convex function, we get Hence, g f is geodesic η-preinvex on S. Proof. Since f is geodesic η-preinvex function, we have Since g is the strictly increasing and convex function, we get Hence, g f is strictly geodesic η-preinvex on S.

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Similarly, we can prove the following results.
is geodesic η-preinvex function on S.
Now, we prove the following proposition, which guarantees that a differentiable and geodesic η-preinvex function f is η-invex. for every x, y ∈ S, then f is η-invex on S.
Proof. We have Since f is differentiable on S, taking the limit as n → ∞ on both sides, we get Therefore, 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 η-invex function on S is geodesic η-preinvex on S if η satisfies the condition C in 9 .
It is revealed in the following proposition that like convex functions, η-invex functions are transformed into η-invex functions by a suitable class of monotone functions.

Proposition 3.7. Let φ : R → R be a monotone increasing differentiable convex function. If f is η-invex on geodesic invex set S, then the composite function φ f is η-invex.
Proof. Using the fact that φ x h ≥ φ x φ x h for every x, h ∈ R, we have

Properties of Generalized Geodesic η-Preinvex Functions
In 15 , Pini introduced the notion of η-pre-pseudo-invex and η-pre-quasi-invex functions on an invex set. We extend these notions to geodesic η-pre-pseudo-invexity and geodesic η-prequasi-invexity on a geodesic invex set S by replacing the line segment with the geodesic. Let f be a function defined on a geodesic invex subset S of a Riemannian manifold M with respect to η : M × M → TM. Proof.
If f x < f y for every t ∈ 0, 1 and f is geodesic η-preinvex, then where b x, y f y − f x > 0. ii the set of all strict global minimum points is geodesic invex set.
Proof. i Let y be a strict local minimum which is not global; then there exists a point x * ∈ S, such that, f x * < f y . Since, f is geodesic η-pre-quasi-invex, we have f γ x * ,y t ≤ f y , which contradicts the hypothesis that y is a strict local minimum.
ii If f has no minimum value in S, then the set of minimum points is empty and hence geodesic invex. If f has the minimum point α on S, then the set of minimum points is S ∩ C α , which is geodesic invex.
The geodesic η-pre-quasi-invexity is preserved under composition with nondecreasing function φ : R → R as can be seen below. Proof. Given that f is a geodesic η-pre-quasi-invex function and φ is a nondecreasing function. Then, we have which shows that the composite function φ f is geodesic η-pre-quasi-invex.

Proposition 4.8.
If the function f is geodesic η-preinvex on S, then f is geodesic η-pre-quasi-invex on S.
Proof. Let f be geodesic η-preinvex function on S. Then for every x, y ∈ S and t ∈ 0, 1 , it follows that f γ x,y t ≤ tf x 1 − t f y ≤ t max{f x , f y } 1 − t max{f x , f y } max{f x , f y },

4.9
which shows that f is geodesic η-pre-quasi-invex on S. Proposition 4.9. If f : S → R is geodesic η-pre-pseudo-invex on S then f is geodesic η-pre-quasiinvex on S.
Proof. Let f x < f y . Since f is geodesic η-pre-pseudo-invex function on S, for all x, y ∈ S, and for all t ∈ 0, 1 , we have f γ x,y t ≤ f y t t − 1 b x, y < f y max{f x , f y }.