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Article

A Study of Geodesic (E, F)-Preinvex Functions on Riemannian Manifolds

1
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2
Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(6), 896; https://doi.org/10.3390/math13060896
Submission received: 28 January 2025 / Revised: 26 February 2025 / Accepted: 2 March 2025 / Published: 7 March 2025
(This article belongs to the Section C: Mathematical Analysis)

Abstract

:
In this manuscript, we define the ( E , F ) -invex set, ( E , F ) -invex functions, and ( E , F ) -preinvex functions on Euclidean space, i.e., simply vector space. We extend these concepts on the Riemannian manifold. We also detail the fundamental properties of ( E , F ) -preinvex functions and provide some examples that illustrate the concepts well. We have established a relation between ( E , F ) -invex and ( E , F ) -preinvex functions on Riemannian manifolds. We introduce the conditions A and define the ( E , F ) -proximal sub-gradient. ( E , F ) -preinvex functions are also used to demonstrate their applicability in optimization problems. In the last, we establish the points of extrema of a non-smooth ( E , F ) -preinvex functions on ( E , F ) -invex subset of the Riemannian manifolds by using the ( E , F ) -proximal sub-gradient.
MSC:
52A20; 52A41; 53C20; 53C22

1. Introduction

One of the most crucial aspects of mathematical programming is the search for convexity and generalized convexity because these concepts play a vital role in optimization theory. Several notable generalizations of convexity have been made over the recent few decades (see [1,2,3,4,5,6,7,8]). Youness provided the concept of E-convex sets and E-convex functions. He also explored the E-convex programming problems in his paper [9]. Yang and Chen found some results to be incorrect in the paper written by [9]. Then, they modified those results (see [10,11]). Jian [12] came up with the concept of ( E , F ) -convex sets and functions with exciting applications regarding generalizations.
Recently, Wedad Saleh explored the concept of ( E , F ) -convex sets and functions on the Riemannian manifold and developed some fruitful results in his paper [13].
One can find the clear idea and development of the geodesic theory of convexity in the books authored by Udriste [14] and Rapcsák [15]. And, for more details on E-convex sets and functions, we refer [13,16], from where the idea of ( E , F ) -convex sets and functions was mainly generated. Convexity and its generalizations on the Riemannian manifolds have received more attention from researchers during the recent few decades, and a vast amount of literature has been developed on the subject (see [17,18,19,20,21,22,23,24,25,26]).
Motivated by the above works and results, particularly in [12,13], we provide the definitions of ( E , F ) -invex sets, ( E , F ) -invex functions and ( E , F ) -preinvex functions. We explore this concept on Riemannian manifolds, which is the generalization of ( E , F ) -convexity sets and functions. We establish a relation between ( E , F ) -invex and ( E , F ) -preinvex functions on Riemannian manifolds. We introduce the conditions A and define the ( E , F ) -proximal sub-gradient. To explore and demonstrate its applicability on problems of optimization, ( E , F ) -preinvexity is applied. In the last section of the present paper, we have obtained extremum points for a non-smooth ( E , F ) -preinvex function defined on the ( E , F ) -invex subset of the Riemannian manifold, which became possible due to use of the ( E , F ) -proximal sub-gradient.

2. Preliminaries

Definition 1
([13]). Let A be a non-empty subset of R n , and let E , F : A R n be two maps. If r 1 , s 1 A and μ [ 0 , 1 ] , F ( s 1 ) + μ ( E ( r 1 ) F ( s 1 ) ) A , then A is known as an ( E , F ) -convex set.
In this section, we recall some well-known definitions on Riemannian manifolds.
Let O be a smooth Riemannian manifold ( C ), which means that , s , the metric on the tangent space T s O × T s O of manifold O , induces a norm , s . If z and w are two points on O and β : [ c , d ] O is a piecewise smooth curve connecting β ( c ) = z to β ( d ) = w , its length L ( β ) is given by
L ( β ) = c d β ( t ) β ( t ) d t .
Now, we define
d ( z , w ) = inf { L ( γ ) : β   i s   a   p i e c e w i s e   C 1   c u r v e   c o n n e c t i n g   z   a n d   w }
for any z , w O . The original topology on O is then induced by a distance d. A Levi-Civita connection, denoted by A B , for any vector fields A , B T O also known as a covariant derivative, is known to exist uniquely on every Riemannian manifold. We also recall that a geodesic is a C smooth path β whose tangent is parallel along the path β, i.e., β satisfies the equation d β ( t ) d t d β ( t ) d t = 0 . Any path β connecting z and w in O , such that L ( β ) = d ( z , w ) , is a geodesic, and it is called a minimizing geodesic. Recall that for a given curve β : J O , a point t o J , and a vector u o T β ( t o ) O , there exists exactly one parallel vector field U ( t ) along β ( t ) s.t. U ( t o ) = u o . Moreover, the mapping defined by u o U ( t ) is a linear isometry between the tangent spaces T β ( t o ) and T β ( t ) , for each t J . We denote this mapping by P t o , β t and we call it the parallel translation from T β ( t o ) O to T β ( t ) O along the curve β. A simply connected complete Riemannian manifold with non-positive sectional curvature is called a Hadamard manifold.

3. ( E , F ) -Invex Sets and ( E , F ) -Preinvex Functions

Definition 2.
Let E , F : A R n be functions, defined on non-empty subset A of R n , and let G : R n × R n R n be another function. If r 1 , s 1 A and μ [ 0 , 1 ] , F ( s 1 ) + μ G ( E ( r 1 ) , F ( s 1 ) ) A , then A is known as an ( E , F ) -invex set.
Example 1.
Let A = [ 0 , 4 ] [ 6 , 8 ] be a non-empty subset of R , and let E , F : A R be two maps defined as
F ( s 1 ) = 1 2 if 1 s 1 3 1 , if s 1 < 1 0 , if s 1 > 3 , E ( r 1 ) = 3 , if 1 r 1 3 2 , if r 1 < 1 9 , if r 1 > 3 ,
and let be G : R × R R a bi-function defined as G ( E ( r 1 ) , F ( s 1 ) ) = F ( s 1 ) .
Now, for r 1 , s 1 > 3 and μ = 1 , we have F ( s 1 ) + μ ( E ( r 1 ) F ( s 1 ) ) = 9 A .
Then, it is obvious that the set A is not a ( E , F ) -convex set with respect to the maps E and F. Now,
F ( s 1 ) + μ G ( E ( r 1 ) , F ( s 1 ) ) = F ( s 1 ) + μ F ( s 1 ) = ( 1 + μ ) F ( s 1 ) .
It is clear that r 1 , s 1 and μ [ 0 , 1 ] . Thus, we have F ( s 1 ) + μ G ( E ( r 1 ) , F ( s 1 ) ) A . Hence, the set A is an ( E , F ) -invex set.
Remark 1.
If we take E = F in the Definition (2), then the set A becomes an E-invex set.
Remark 2.
If we take G ( E ( r 1 ) , F ( s 1 ) ) = E ( r 1 ) F ( s 1 ) in the Definition (2), then the set A becomes an ( E , F ) -convex set.
Remark 3.
If we take the E = F identity map in the Definition (2), then the set A becomes an invex set.
Remark 4.
If we take E and F to be the identity maps on A and G ( E ( r 1 ) , F ( s 1 ) ) = E ( r 1 ) F ( s 1 ) in the Definition (2), then the set A becomes a convex set.
Definition 3.
A real-valued function H : A R defined on the ( E , F ) -invex set A is known as an ( E , F ) -preinvex function if
H ( F ( s 1 ) + μ G ( E ( r 1 ) , F ( s 1 ) ) ) μ H ( E ( r 1 ) ) + ( 1 μ ) H ( F ( s 1 ) )
holds for r 1 , s 1 A and μ [ 0 , 1 ] .
Theorem 1.
Let A be a non-empty ( E , F ) -invex subset of R n . Let functions H i , i = 1 , 2 , . . , n be ( E , F ) -preinvex functions on the ( E , F ) -invex set A and α i 0 , i = 1 , 2 , . . . , n . Then, i = 1 n α i H i is also an ( E , F ) -preinvex function on the ( E , F ) -invex set A .
Definition 4.
Let H be a real-valued differentiable function on the open interval A of R n . The function H is known as an ( E , F ) -invex on A if there we have the maps E , F : A R n and a bi-function G : R n × R n R n such that the inequality
H ( E ( r 1 ) ) H ( F ( s 1 ) ) G ( E ( r 1 ) , F ( s 1 ) ) T H ( F ( s 1 ) )
holds for each r 1 , s 1 A and μ [ 0 , 1 ] .
Now, we will explore all the above definitions in Riemannian manifolds.

4. Geodesic ( E , F ) -Invex Sets and ( E , F ) -Preinvex Functions on Riemannian Manifolds

Definition 5.
Let B be a non-empty subset of the Riemannian manifold O , and let E , F : B O , G : O × O T O be functions such that for every E ( r 1 ) , F ( s 1 ) O , G ( E ( r 1 ) , F ( s 1 ) ) T F ( s 1 ) O . Then, the subset B of O is said to be geodesic ( E , F ) -invex if, for each E ( r 1 ) , F ( s 1 ) B , there exists exactly one geodesic β E ( r 1 ) , F ( s 1 ) : [ 0 , 1 ] O such that
β E ( r 1 ) , F ( s 1 ) ( 0 ) = F ( s 1 ) , β E ( r 1 ) , F ( s 1 ) ( 0 ) = G ( E ( r 1 ) , F ( s 1 ) ) , β E ( r 1 ) , F ( s 1 ) ( s ) B , s [ 0 , 1 ] .
Now, we define the ( E , F ) -invex function on an open geodesic ( E , F ) -invex subset of a Riemannian manifold.
Definition 6.
Let B be a non-empty open subset of the Riemannian manifold O , with Riemannian metric g and B being a geodesic invex set. A function H : B R is said to be ( E , F ) -invex on B if the following inequality holds:
H ( E ( r 1 ) ) H ( F ( s 1 ) ) H F ( s 1 ) , G ( E ( r 1 ) , F ( s 1 ) ) E ( r 1 ) , F ( s 1 ) B .
Definition 7.
Let B be a non-empty geodesic ( E , F ) -invex subset of the Riemannian manifold O . A function H : B R is said to be ( E , F ) -preinvex if, for each E ( r 1 ) , F ( s 1 ) B and s [ 0 , 1 ] ,
H ( β E ( r 1 ) , F ( s 1 ) ) ( s ) s H ( E ( r 1 ) ) + ( 1 s ) H ( F ( s 1 ) )
holds, where β E ( r 1 ) , F ( s 1 ) is the unique geodesic in Definition 5. If the above inequality (3) is strict for E ( r 1 ) F ( s 1 ) and s ( 0 , 1 ) , then H is said to be a strict ( E , F ) -preinvex function.
Example 2.
Let O = [ 8 , 8 ] be a non-empty subset of R , and let E , F : O R be two maps defined as
E ( r 1 ) = 3 r 1 a n d F ( s 1 ) = s 1 ,
and let G : R × R R , a point-to-point bi-function, be defined as
G ( E ( r 1 ) , F ( s 1 ) ) = E ( r 1 ) 3 F ( s 1 ) .
Now, consider the geodesic β as
β E ( r 1 ) , F ( s 1 ) ( s ) = F ( s 1 ) + s E ( r 1 ) 3 F ( s 1 ) = s 1 + s ( r 1 s 1 ) .
It can be easily seen that
β E ( r 1 ) , F ( s 1 ) ( 0 ) = F ( s 1 ) , β E ( r 1 ) , F ( s 1 ) ( 0 ) = G ( E ( r 1 ) , F ( s 1 ) ) , β E ( r 1 ) , F ( s 1 ) ( s ) O , s [ 0 , 1 ] .
So, O is a geodesic ( E , F ) -invex set. Now, we define a function H : O R as H ( r 1 ) = r 1 + r 1 2 . Then, H is an ( E , F ) -preinvex function on the set O .
Proposition 1.
Let B be a non-empty geodesic ( E , F ) -invex subset of the Riemannian manifold O . Consider the function H : B R , an ( E , F ) -preinvex. Thus, we have
(a) Every lower level set of H defined by
P ( H , t ) : = { E ( r 1 ) B | H ( E ( r 1 ) ) t }
is a geodesic ( E , F ) -invex set.
(b) An optimization problem is presented as follows:
m i n H ( w ) s u b j e c t to w B
where the set L of the solution of the problem (4) is a geodesic ( E , F ) -invex set. Also, if H is strictly an ( E , F ) -preinvex function, then O contains (at most) one point.
Proof. 
( a ) Take E ( r 1 ) , F ( s 1 ) P ( H , t ) . Then, we have H ( E ( r 1 ) ) t a n d H ( F ( s 1 ) ) t . As B is a geodesic ( E , F ) -invex set, there exists exactly one geodesic β E ( r 1 ) , F ( s 1 ) : [ 0 , 1 ] O such that
β E ( r 1 ) , F ( s 1 ) ( 0 ) = F ( s 1 ) , β E ( r 1 ) , F ( s 1 ) ( 0 ) = G ( E ( r 1 ) , F ( s 1 ) ) , β E ( r 1 ) , F ( s 1 ) ( s ) B , s [ 0 , 1 ] .
Since H is ( E , F ) -preinvex, then, by definition, we have
H ( β E ( r 1 ) , F ( s 1 ) ) ( s ) s H ( E ( r 1 ) ) + ( 1 s ) H ( F ( s 1 ) ) s t + ( 1 s ) t = t .
Hence, β E ( r 1 ) , F ( s 1 ) ( s ) P ( H , t ) for any s [ 0 , 1 ] .
(b) If we take β : = inf E ( r 1 ) B H ( E ( r 1 ) ) , then it is clear that L = t > γ P ( H , t ) . Therefore, L is an intersection of a geodesic ( E , F ) -invex set. Hence, it is a geodesic ( E , F ) -invex set as well.
Now, if H is strictly an ( E , F ) -preinvex function, then, for the geodesic ( E , F ) -invexity of L, there exists exactly one geodesic γ E ( r 1 ) , F ( s 1 ) : [ 0 , 1 ] O such that
γ E ( r 1 ) , F ( s 1 ) ( 0 ) = F ( s 1 ) , γ E ( r 1 ) , F ( s 1 ) ( 0 ) = G ( E ( r 1 ) , F ( s 1 ) ) , γ E ( r 1 ) , F ( s 1 ) ( s ) B , s [ 0 , 1 ] .
By the strictly ( E , F ) -preinvexity of H , we have
β = H ( γ E ( r 1 ) , F ( s 1 ) ) ( s ) < s H ( E ( r 1 ) ) + ( 1 s ) H ( F ( s 1 ) ) = s β + ( 1 s ) β = β
which is not possible. □
Theorem 2.
Let B be a non-empty open geodesic ( E , F ) -invex subset of the Riemannian manifold O . Consider that H : B O is a differentiable and ( E , F ) -preinvex function. Then, H is an ( E , F ) -invex function.
Proof. 
Since B is a geodesic ( E , F ) -invex set of the Riemannian manifold, there exists exactly one geodesic β E ( r 1 ) , F ( s 1 ) : [ 0 , 1 ] O for every E ( r 1 ) , F ( s 1 ) B , such that
β E ( r 1 ) , F ( s 1 ) ( 0 ) = F ( s 1 ) , β E ( r 1 ) , F ( s 1 ) ( 0 ) = G ( E ( r 1 ) , F ( s 1 ) ) , β E ( r 1 ) , F ( s 1 ) ( s ) B , s [ 0 , 1 ] .
Now, as H is the ( E , F ) -preinvex for s ( 0 , 1 ) , we obtain
H ( β E ( r 1 ) , F ( s 1 ) ) ( s ) s H ( E ( r 1 ) ) + ( 1 s ) H ( F ( s 1 ) ) .
This may also be written as
H ( β E ( r 1 ) , F ( s 1 ) ) ( s ) H ( F ( s 1 ) ) s [ H ( E ( r 1 ) ) H ( F ( s 1 ) ) ] ,
we divide both sides by s, and we obtain
H ( β E ( r 1 ) , F ( s 1 ) ) ( s ) H ( F ( s 1 ) ) s [ H ( E ( r 1 ) ) H ( F ( s 1 ) ) ] .
Taking the limit as s 0 , we have
H ( β E ( r 1 ) , F ( s 1 ) ) ( 0 ) , β E ( r 1 ) , F ( s 1 ) ( 0 ) H ( E ( r 1 ) ) H ( F ( s 1 ) ) .
Therefore,
H F ( s 1 ) , G ( E ( r 1 ) , F ( s 1 ) ) H ( E ( r 1 ) ) H ( F ( s 1 ) ) .
Hence, H is an ( E , F ) -invex function. □
Definition 8.
Let the functions E , F : O O be defined on a Riemannian manifold O . Then, we say that the function G : O × O T O satisfies the condition A , if, for every E ( r 1 ) , F ( s 1 ) O and for the geodesic β : [ 0 , 1 ] O satisfying β E ( r 1 ) , F ( s 1 ) ( 0 ) = F ( s 1 ) , β E ( r 1 ) , F ( s 1 ) ( 0 ) = G ( E ( r 1 ) , F ( s 1 ) ) , we have
( A 1 ) P s , β o [ G ( F ( s 1 ) , β ( s ) ) ] = s G ( E ( r 1 ) , F ( s 1 ) )
( A 2 ) P s , β o [ G ( E ( r 1 ) , β ( s ) ) ] = ( 1 s ) G ( E ( r 1 ) , F ( s 1 ) )
for all s [ 0 , 1 ] .
Theorem 3.
Let B be a non-empty open geodesic ( E , F ) -invex subset of the Riemannian manifold O . Suppose that the function H : B R is differentiable. If H is an ( E , F ) -invex on B and H satisfies the condition A , then H is an ( E , F ) -preinvex function on B .
Proof. 
Since B is a geodesic ( E , F ) -invex set of the Riemannian manifold, then, for every E ( r 1 ) , F ( s 1 ) B , there exists exactly one geodesic β E ( r 1 ) , F ( s 1 ) : [ 0 , 1 ] O such that
β E ( r 1 ) , F ( s 1 ) ( 0 ) = F ( s 1 ) , β E ( r 1 ) , F ( s 1 ) ( 0 ) = G ( E ( r 1 ) , F ( s 1 ) ) , β E ( r 1 ) , F ( s 1 ) ( s ) B , s [ 0 , 1 ] .
Now, fix s [ 0 , 1 ] and set F ( s 1 ¯ ) : = β E ( r 1 ) , F ( s 1 ) ( s ) . Then, we have
H ( E ( r 1 ) ) H ( F ( s 1 ¯ ) ) H F ( s 1 ¯ ) , G ( E ( r 1 ) , F ( s 1 ¯ ) )
H ( E ( s 1 ) ) H ( F ( s 1 ¯ ) ) H F ( s 1 ¯ ) , G ( E ( s 1 ) , F ( s 1 ¯ ) ) .
Multiplying Inequality (5) by s and Inequality (6) by 1 s , respectively, and then carrying out addition, will lead to
s H ( E ( r 1 ) ) + ( 1 s ) H ( F ( s 1 ) ) H ( F ( s 1 ¯ ) ) H F ( s 1 ¯ ) , s G ( E ( r 1 ) , F ( s 1 ¯ ) + ( 1 s ) G ( E ( r 1 ) , ( F ( s 1 ¯ ) ) .
From Condition A , we obtain
s G ( E ( r 1 ) , F ( s 1 ¯ ) ) + ( 1 s ) G ( E ( r 1 ) , ( F ( s 1 ¯ ) ) = s ( 1 s ) P s , β o [ G ( E ( r 1 ) , ( F ( s 1 ) ) ] + ( 1 s ) ( s ) [ G ( E ( r 1 ) , ( F ( s 1 ) ) ] = 0 .
Therefore, we obtain
s H ( E ( r 1 ) ) + ( 1 s ) H ( F ( s 1 ) ) H ( F ( s 1 ¯ ) ) .
Hence, H is an ( E , F ) -preinvex function on the ( E , F ) -invex set O . □
Theorem 4.
Let B be a non-empty open geodesic ( E , F ) -invex subset of the Riemannian manifold O . Suppose that the function H : B R is an ( E , F ) -preinvex. If F ( r ¯ 1 ) B is a local optimal solution to the problem
m i n i m i z e H ( F ( r 1 ) ) s u b j e c t to F ( r 1 ) B ,
then F ( r ¯ 1 ) is a global minimum in Problem ( 7 ) .
Proof. 
Consider a local minimum F ( r ¯ 1 ) . Then, there exists a neighborhood C ϵ ( F ( r ¯ 1 ) ) such that
H ( F ( r ¯ 1 ) ) H ( E ( r 1 ) ) , E ( r 1 ) B C ϵ ( F ( r ¯ 1 ) ) .
Conversely, we assume that F ( r ¯ 1 ) is not a global minimum of H . Then, there exists an element E ( s 1 * ) B s.t.
H ( E ( r 1 * ) ) < H ( F ( r ¯ 1 ) ) .
As B is a geodesic ( E , F ) -invex set, there exists exactly one geodesic β E ( r 1 * ) , F ( r 1 ¯ ) : [ 0 , 1 ] O such that
β E ( r 1 * ) , F ( r 1 ¯ ) ( 0 ) = F ( r 1 ¯ ) , β E ( r 1 * ) , F ( r 1 ¯ ) ( 0 ) = G ( E ( r 1 * ) , F ( r 1 ¯ ) ) , β E ( r 1 * ) , F ( r 1 ¯ ) ( s ) B , s [ 0 , 1 ] .
Now, if we choose ϵ > 0 to be small enough s.t. d ( β E ( r 1 * ) , F ( r 1 ¯ ) ( s ) , F ( r 1 ¯ ) ) < ϵ , then β E ( r 1 * ) , F ( r 1 ¯ ) ( s ) C ϵ ( F ( r 1 ¯ ) ) . As H represents ( E , F ) -preinvexity, we have
H ( β E ( r 1 * ) , F ( r 1 ¯ ) ( s ) ) s H ( E ( r 1 * ) ) + ( 1 s ) H ( F ( r 1 ¯ ) < H ( F ( r 1 ¯ ) ) , s ( 0 , 1 ) .
Therefore, for each β E ( r 1 * ) , F ( r 1 ¯ ) ( s ) B C ϵ ( F ( r 1 ¯ ) ) , which is a contradiction to Inequality (8), our result is proved. □
Definition 9.
Let the functions E , F : O O be defined on Riemannian manifold O , and let function H : O ( , ] be a lower semi-continuous function. An element σ in T F ( s 1 ) O is an ( E , F ) -proximal sub-gradient of H at F ( s 1 ) d o m ( H ) , if the numbers μ > 0 and λ > 0 s.t.
H ( E ( r 1 ) ) H ( F ( s 1 ) ) + σ , exp F ( s 1 ) 1 E ( r 1 ) F ( s 1 ) λ d ( E ( r 1 ) , F ( s 1 ) ) 2 ,
E ( r 1 ) N ( F ( s 1 ) , μ ) , and where d o m ( H ) : = { E ( r 1 ) O : H ( E ( r 1 ) ) < } . The set of all ( E , F ) -proximal sub-gradients of H at F ( s 1 ) O is denoted by ( E , F ) H ( F ( s 1 ) ) and is called the ( E , F ) -proximal sub-differential of H at F ( s 1 ) .
Theorem 5.
Let B be a non-empty open geodesic ( E , F ) -invex subset of the Cartan–Hadamard manifold O with G ( E ( r 1 ) , F ( s 1 ) ) 0 for E ( r 1 ) F ( s 1 ) . Also, let H : B ( , ] be a lower semi-continuous ( E , F ) -preinvex function. Assume the F ( s 1 ) domain to be ( H ) and σ ( E , F ) H ( F ( s 1 ) ) . Then, there exists a real number μ s.t.
H ( E ( r 1 ) ) H ( F ( s 1 ) ) + σ , G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) , E ( r 1 ) B N ( F ( s 1 ) , μ )
Proof. 
Using the definition of the ( E , F ) -proximal sub-gradient ( E , F ) H ( F ( s 1 ) ) , the numbers μ > 0 and λ > 0 exist s.t.
H ( E ( r 1 ) ) H ( F ( s 1 ) ) + σ , exp F ( s 1 ) 1 E ( r 1 ) F ( s 1 ) λ d ( E ( r 1 ) , F ( s 1 ) ) 2 ,
E ( r 1 ) N ( F ( s 1 ) , μ ) , specifically for E ( r 1 ) B N ( F ( s 1 ) , μ ) . As B is a geodesic ( E , F ) -invex set, there exists exactly one geodesic β E ( r 1 * ) , F ( r 1 ¯ ) : [ 0 , 1 ] O such that
β E ( r 1 ) , F ( s 1 ) ( 0 ) = F ( s 1 ) , β E ( r 1 ) , F ( s 1 ) ( 0 ) = G ( E ( r 1 ) , F ( s 1 ) ) , β E ( r 1 ) , F ( s 1 ) ( s ) B , s [ 0 , 1 ] .
As O is a Cartan–Hadamard manifold, β E ( r 1 ) , F ( s 1 ) ( s ) = exp F ( s 1 ) ( s G ( E ( r 1 ) , F ( s 1 ) ) ) for any s [ 0 , 1 ] (see [24] p. 776). Now, take r 1 = μ G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) . Therefore, β E ( r 1 ) , F ( s 1 ) ( s ) = exp F ( s 1 ) ( s G ( E ( r 1 ) , F ( s 1 ) ) ) B N ( F ( s 1 ) , μ ) , s ( 0 , r 1 ) .
As H is geodesic ( E , F ) -preinvex, we have
H ( exp F ( s 1 ) ( s G ( E ( r 1 ) , F ( s 1 ) ) ) ) s H ( E ( r 1 ) ) + ( 1 s ) H ( F ( s 1 ) ) , s ( 0 , r 1 )
Combining Inequalities (10) and (11), we have
H ( exp F ( s 1 ) ( s G ( E ( r 1 ) , F ( s 1 ) ) ) H ( F ( s 1 ) ) λ d ( exp F ( s 1 ) ( s G ( E ( r 1 ) , F ( s 1 ) ) ) , F ( s 1 ) ) 2 + σ , exp F ( s 1 ) 1 exp F ( s 1 ) ( s G ( E ( r 1 ) , F ( s 1 ) ) ) F ( s 1 ) = H ( F ( s 1 ) ) + σ , s G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) λ d ( exp F ( s 1 ) ( s G ( E ( r 1 ) , F ( s 1 ) ) , F ( s 1 ) ) 2
E ( r 1 ) in N ( F ( s 1 ) , μ ) . As O can be a Cartan–Hadamard manifold, we have
d ( exp F ( s 1 ) ( s G ( E ( r 1 ) , F ( s 1 ) ) , F ( s 1 ) ) 2 = s G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) 2 = s 2 G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) 2 .
By combining Inequalities (11) and (12) with Equation (13), we have
s H ( E ( r 1 ) ) + ( 1 s ) H ( F ( s 1 ) ) H ( exp F ( s 1 ) ( s G ( E ( r 1 ) , F ( s 1 ) ) ) H ( F ( s 1 ) ) + σ , s G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) λ s 2 G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) 2 .
s [ H ( E ( r 1 ) ) H ( F ( s 1 ) ] s σ , G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) λ s 2 G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) 2
H ( E ( r 1 ) ) H ( F ( s 1 ) σ , G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) λ s G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) 2 .
Now, taking limit lim s 0 on both sides, we have
H ( E ( r 1 ) ) H ( F ( s 1 ) σ , G ( E ( r 1 ) , F ( s 1 ) ) F ( s 1 ) , E ( r 1 ) B N ( F ( s 1 ) , μ ) .
Therefore, the proof is completed. □

Author Contributions

The authors contributed almost equally to this work. Moreover, Conceptualization, by E.A., M.B. and M.A.; methodology by E.A. and M.A.; software by E.A. and M.A.; validation by E.A. and M.B.; formal analysis, M.B. and M.A.; investigation by E.A. and M.A.; resources, M.B.; data curation, by E.A., M.B. and M.A.; writing-original draft preparation by E.A., M.B. and M.A.; writing—review and editing by E.A. and M.A.; visualization, by E.A. and M.B.; supervision by M.A.; project administration by M.B. and M.A.; funding acquisition by M.B. All authors have read and agreed to the published version of the manuscript.

Funding

This research work was funded by Umm Al-Qura University, Saudi Arabia, under grant number 25UQU4330007GSSR03.

Data Availability Statement

Observational data used in this paper were quoted from the cited works.

Acknowledgments

The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia, for funding this research work through grant number 25UQU4330007GSSR03.

Conflicts of Interest

The authors declare no conflicts of interest.

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MDPI and ACS Style

Akhter, E.; Bilal, M.; Ali, M. A Study of Geodesic (E, F)-Preinvex Functions on Riemannian Manifolds. Mathematics 2025, 13, 896. https://doi.org/10.3390/math13060896

AMA Style

Akhter E, Bilal M, Ali M. A Study of Geodesic (E, F)-Preinvex Functions on Riemannian Manifolds. Mathematics. 2025; 13(6):896. https://doi.org/10.3390/math13060896

Chicago/Turabian Style

Akhter, Ehtesham, Mohd Bilal, and Musavvir Ali. 2025. "A Study of Geodesic (E, F)-Preinvex Functions on Riemannian Manifolds" Mathematics 13, no. 6: 896. https://doi.org/10.3390/math13060896

APA Style

Akhter, E., Bilal, M., & Ali, M. (2025). A Study of Geodesic (E, F)-Preinvex Functions on Riemannian Manifolds. Mathematics, 13(6), 896. https://doi.org/10.3390/math13060896

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