EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 860-869 ISSN 1307-5543 – ejpam.com Published by New York Business Global Strongly Geodesic Log-Preinvex Functions Wedad Saleh1,∗, Abdelghani Lakhdari2, Badreddine Meftah3 1Department of Mathematics, Taibah University, Al- Medina 42353, Saudi Arabia. 2Department CPST, National Higher School of Technology and Engineering, Annaba 23005, Algeria. 3Department of Mathematics, University 8 may 1945 Guelma, Algeria. Abstract. In this article, we delve into the intriguing concept of strongly geodesic log-preinvex functions in Riemannian manifolds. We present essential preliminaries and fundamental results that shed light on this specialized area of study. By examining the properties and implications of these functions, we aim to contribute to the growing body of knowledge in convexity theory within the context of Riemannian manifolds. 2020 Mathematics Subject Classifications: 52A20, 52A41, 53C20, 53C22 Key Words and Phrases: log-convex functions, geodesic convex functions, geodesic convex sets, Riemannian manifolds. 1. Introduction and Preliminaries Let ϑ ⊆ R be an interval. A function ξ : ϑ −→ R is said to be strongly convex with modulus ε > 0 if ξ(ςu1+(1−ς)u2) ≤ ςξ(u1)+(1−ς)ξ(u2)−ες(1−ς)(u1−u2) 2, ∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. (1) The concept of strongly convex functions, initially introduced by Polyak (1966) [13], holds substantial relevance in the fields of optimization theory and mathematical eco- nomics. An extensive exploration of their properties and applications is well-documented across various studies, including those by Angulo et al. [1], Awan et al. [2], and Merentes et al. [6], among others. The notion of convexity has been expanded to encompass strong convexity of order n on Rn, as defined by Lin et al. [5]: A function ξ, defined on a subset ϑ of the real numbers (R), is termed strongly convex of order n if it satisfies the following condition for all µ1 and µ2 within ϑ and for ς in the range [0, 1]: There exists a positive constant ε > 0 such that: ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5103 Email addresses: wlehabi@taibahu.edu.sa (W. Saleh), a.lakhdari@ensti-annaba.dz (A. Lakhdari), badrimeftah@yahoo.fr (B. Meftah) https://www.ejpam.com 860 © 2024 EJPAM All rights reserved. W. Saleh, A. Lakhdari, B. Meftah / Eur. J. Pure Appl. Math, 17 (2) (2024), 860-869 861 ξ(ςµ1 + (1− ς)µ2) ≤ ςξ(µ1) + (1− ς)ξ(µ2)− ες(1− ς)∥µ1 − µ2∥n. This condition characterizes the strong convexity property of the function ξ in the context of mathematical analysis. Additional applications, numerical techniques, and variational-like inequalities for con- vex functions are discussed in [10, 11]. It is noteworthy that log-convex functions, as opposed to convex functions, have been shown to yield more precise results and inequali- ties. Numerous aspects related to exponentially preinvex functions and their variants are introduced in [8, 9]. Let (N, ξ) be a complete m-dimensional Riemannian manifold equipped with a Rie- mannian connection ∇. Consider a piecewise C1 path γ : [µ1, µ2] −→ ϑ connecting a1 to a2, where γ(µ1) = a2 and γ(µ2) = a1. The length of γ is defined as: L(γ) = ∫ u2 u1 ∥γ́(λ)∥γ(λ)dλ. For any two points a1 and a2 in N , we introduce the following metric: d(a1, a2) = inf { L(γ) : γ is a piecewise C1 path connecting a1 to a2 } . This metric d induces the original topology on N . In every Riemannian manifold, there exists a uniquely determined Riemannian connec- tion known as the Levi-Civita connection, denoted by ∇XY , for any vector fields X and Y in ϑ. Furthermore, a smooth path γ is considered a geodesic if and only if its tangent vector is a parallel vector field along the path γ, i.e., γ satisfies the equation ∇γ′γ′ = 0. Any path γ that connects µ1 and µ2 in N such that L(γ) = d(µ1, µ2) is a geodesic and is referred to as a minimal geodesic. Let N be a C∞ complete n-dimensional Riemannian manifold with metric g and Levi- Civita connection∇. Additionally, consider the points µ1 and µ2 inN , and let γ : [0, 1] −→ N be a geodesic connecting µ1 and µ2, i.e., γµ1,µ2(0) = µ2 and γµ1,µ2(1) = µ1. Definition 1. [3]. Let a set ϑ ⊂ N be geodesic invex w.r.t. η : N × N −→ TN . A function ξ : ϑ −→ Ris said to be geodesic preinvex w.r.t.η iff ξ(γµ1,µ2) ≤ (1− ς)ξ(µ1) + ςξ(µ2), ∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. The strongly geodesic convexity of order n on a Riemannian manifold is elaborated in [4]. Definition 2. Suppose ϑ ⊆ N is a geodesically convex subset of N . A function ξ : ϑ −→ R is termed strongly geodesically convex of order n > 0 on ϑ if there exists a positive constant ε > 0 such that for all µ1 and µ2 in ϑ and for t in the interval [0, 1], the following inequality holds: ξ(γµ1,µ2(ς)) ≤ ςξ(µ1) + (1− ς)ξ(µ2)− ες(1− ς)∥γ′µ1,µ2 (ς)∥n. W. Saleh, A. Lakhdari, B. Meftah / Eur. J. Pure Appl. Math, 17 (2) (2024), 860-869 862 Pini [12] conducted an investigation into various properties of invex functions on Rie- mannian manifolds, while Mititelu [7] explored its generalization. Noor and Noor [9] introduced a novel concept known as exponentially preinvex functions. Subsequently, a multitude of papers have emerged in the literature, delving into the realm of (generalized) convexity on Riemannian manifolds, we refer readers to [14–16]. In this article, we present some introductory concepts and fundamental results per- taining to strongly geodesic log-preinvex functions in Riemannian manifolds. 2. Main results In this article, we introduce an innovative concept of generalized convexity in the context of Riemannian manifolds. Specifically, we define the concept of strongly geodesic log-preinvex functions, which serves as a comprehensive generalization and extension of various previously introduced notions of generalized convexity in the existing literature. Definition 3. A function ξ : ϑ −→ R∗ + is considered strongly geodesic log-preinvex w.r.t a bifunction η if there exists a constant ε ≥ 0 such that: log ξ(γµ1,µ2(ς)) ≤ (1− ς) log ξ(µ1) + ς log ξ(µ2)− ες(1− ς)∥η(µ2, µ1)(ς)∥2 (2) for all µ1, µ2 ∈ ϑ and ς ∈ [0, 1]. Definition 4. A function ξ : ϑ −→ R∗ + is considered to be strongly geodesic log-quasi preinvex w.r.t the bifunction η if: log ξ(γµ1,µ2(ς)) ≤ max {log ξ(µ1), log ξ(µ2)}−ες(1−ς)∥η(µ2, µ1)(ς)∥2, ∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. Definition 5. A function ξ : ϑ −→ R∗ + is said to be first kind of strongly geodesic log- preinvex w.r.t. any arbitrary bifunction η , if log ξ(γµ1,µ2(ς)) ≤ (log ξ(µ1)) 1−ς(log ξ(µ2)) t − ες(1− ς)∥η(µ2, µ1)(ς)∥2,∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. From the above defintions, we have (i) log ξ(γµ1,µ2(ς)) ≤ (log ξ(µ1)) 1−ς(log ξ(µ2)) t − ες(1− ς)∥η(µ2, µ1)(ς)∥2 ≤ (1− ς) log ξ(µ1) + ς log ξ(µ2)− ες(1− ς)∥η(µ2, µ1)(ς)∥2,∀ ≤ max {log ξ(µ1), log ξ(µ2)} − ες(1− ς)∥η(µ2, µ1)(ς)∥2, ∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. This demonstrates that every first kind of strongly geodesic log-preinvex function is indeed a strongly geodesic log-preinvex function, and a strongly geodesic log-preinvex function can be considered a strongly geodesic log-quasipreinvex function. However, it’s important to note that the converse is not necessarily true. W. Saleh, A. Lakhdari, B. Meftah / Eur. J. Pure Appl. Math, 17 (2) (2024), 860-869 863 (ii) If ε = 0, then (a) Strongly geodesic log-preinvex function is called geodesic log-preinvex function. (b) Strongly geodeic log-quasi preinvex is called geodesic log-quasi preinvex. (c) The first kind of strongly geodesic log-preinvex is called the first kind of geodesic log-preinvex. (iii) If ς = 1, then Definitions 3 and 5 will be become ξ(µ1) ≤ ξ(µ2), ∀µ1, µ2 ∈ ϑ. Definition 6. A function ξ : ϑ −→ R∗ + is said to be a strongly affine geodesic log-preinvex w.r.t. the bifunction η , if there exists a constant ε ≥ 0, such that log ξ(γµ1,µ2(ς)) = (1− ς) log ξ(µ1) + ς log ξ(µ2)− ες(1− ς)∥η(µ2, µ1)(ς)∥2, (3) ∀µ1, µ2 ∈ ϑ and ς ∈ [0, 1]. Example 1. Assume that map η : R× R −→ R is defined as η(µ1, µ2) = { 0 ifµ1 = µ2, 1−mu1 ifµ1 ̸= µ2, also, γµ1,µ2(ς) = { µ2 ifµ1 = µ2, µ2 + ς(1− µ1) ifµ1 ̸= µ2, Assume that ξ : R+ −→ R, where ξ(µ) = expµ, then ξ is strongly geodesic log-preinvex w.r.t. the bifunction η. Example 2. In this example, we give some new parallegram low of uniformly Banach spaces involving the notion of strongly affine geodesic log-preinvex w.r.t. the bifunction η Assume that η(µ1, µ2) = { 0 ifµ1 = µ2, mu1 −mu2 ifµ1 ̸= µ2, also, γµ1,µ2(ς) = { µ2 ifµ1 = µ2, µ2 + ς(µ1 − µ2) ifµ1 ̸= µ2, Frome equality (3), we get ∥ log ξ(µ2 + ς(µ1 − µ2))∥2 = (1− ς)∥ log ξ(µ1)∥2 + ς∥ log ξ(µ2)∥2 − ες(1− ς)∥µ2 − µ1∥2, (4) ∀µ1, µ2 ∈ ϑ and ς ∈ [0, 1]. Taking ς = 1 2 in (4), we get ∥ log ξ(µ1 + µ2 2 )∥2 + ε 4 ∥µ2 − µ1∥2 = 1 2 ( ∥ log ξ(µ1)∥2 + ∥ log ξ(µ2)∥2 ) , (5) ∀µ1, µ2 ∈ ϑ. Which ξ is called the log-paralleogram for the inner product spaces. By putting log ξ(µ1) = ∥µ1∥2 in (5), we have the paralleogram for the inner product spaces. W. Saleh, A. Lakhdari, B. Meftah / Eur. J. Pure Appl. Math, 17 (2) (2024), 860-869 864 3. Some Aspects of Geodesic Log-Preinvexity Properties In this section, we examine fundamental properties of geodesic log-preinvex functions. Theorem 1. If ξ is a strongly geodesic log-preinvex fun ction, then any point that serves as a local minimum is also considered a global minimum. Proof. Considering that the function ξ is geodesic log-preinvex and possesses a local minimum at µ1 ∈ ϑ. Let’s assume the contrary, which is, ξ(µ2) < ξ(µ1) for some µ2 ∈ ϑ. Since ξ is a geodesic log-preinvex function, we have: log ξ(γµ1,µ2(ς)) < (1− ς) log ξ(µ1) + ς log ξ(µ2), for ς ∈ (0, 1). Thus, we can derive: log ξ(γµ1,µ2(ς))− log ξ(µ1) < t (log ξ(µ2)− log ξ(µ1)) < 0, (6) From which it follows that: log ξ(γµ1,µ2(ς)) < log ξ(µ1), for arbitrary small ς > 0. This contradicts the fact that the function has a minimum at the point µ1. Theorem 2. A positive function F is considered to be geodesic log-preinvex if and only if the set epi(F) = {(u, v) : u ∈ U, logF(u) ≤ v, v ∈ R} is a geodesic invex set. Proof. Let F be a geodesic log-preinvex function. Let (u1, v1), (u2, v2) ∈ epi(F). Then logF(u1) ≤ v1 and logF(u2) ≤ v2. Thus LogF(γu1,u2(ς)) ≤ (1− ς)logF(u1) + ςlogF(u2) ≤ (1− ς)v1 + ςv2.∀ς ∈ [0, 1]. That means (γu1,u2(ς), (1− ς)v1 + ςv2) ∈ epi(F). This epi(F) is geodesic invex set. Conversely, suppose epi(F) is a geodesic invex set. Let u, v ∈ U . Then, we have (u, logF(u)) ∈ epi(F) and (v, logF(v)) ∈ epi(F). Since epi(F) is a geodesic invex set, it follows that: (γu,v(ς), (1− ς) logF(u) + ς logF(v)) ∈ epi(F), which implies: logF(γu,v(ς)) ≤ (1− ς) logF(u) + ς logF(v). This demonstrates that F is a geodesic log-preinvex function. W. Saleh, A. Lakhdari, B. Meftah / Eur. J. Pure Appl. Math, 17 (2) (2024), 860-869 865 4. Proporties of Strongly Geodesic Log-Preinvex Functions In this section, we delve into fundamental properties of functions exhibiting strongly geodesic log-preinvexity. Theorem 3. Assume that the function ξ is differentiable on ϑ◦ . If ξ is geodesic log- preinvex, then log ξ(µ2)− log ξ(µ1) ≥ dξµ1η(µ2, µ1) ξ(µ1) + ε∥η(µ2, µ1)(ς)∥2 (7) Proof. Assume that the function ξ is strongly geodesic log-preinvex, one has that log ξ(γµ1,µ2(ς)) ≤ (1− ς) log ξ(µ1) + ς log ξ(µ2)− ες(1− ς)∥η(µ2, µ1)(ς)∥2, hence log ξ(µ2)− log ξ(µ1) ≥ log ξ(γµ1,µ2(ς))− log ξ(µ1) t + ε∥η(µ2, µ1)(ς)∥2. Upon approaching the limit as ς tends towards zero in the preceding inequality, we obtain: log ξ(µ2)− log ξ(µ1) ≥ dµ1ξγµ1,µ2 ξ(u) + ε∥η(µ2, µ1)(ς)∥2. Which (7), the required result. Remark 1. Form (7), we have ξ(µ2) ≥ ξ(µ1) exp { dµ1ξγµ1,µ2 ξ(µ1) + ε∥η(µ2, µ1)(ς)∥2 } ,∀µ1, µ2 ∈ ϑ. By interchanging the roles of µ1 and µ2 in the inequality above, we also obtain: ξ(µ1) ≥ ξ(µ2) exp { dµ2ξγµ1,µ2 ξ(µ2) + ε∥η(µ2, µ1)(ς)∥2 } ,∀µ1, µ2 ∈ ϑ. Therefore, we can deduce the subsequent inequality: ξ(µ1) + ξ(µ2) ≥ ξ(µ2) exp { dµ2ξγµ1,µ2 ξ(µ2) + ε∥η(µ2, µ1)(ς)∥2 } +ξ(µ1) exp { dµ1ξγµ1,µ2 ξ(µ1) + ε∥η(µ2, µ1)(ς)∥2 } ,∀µ1, µ2 ∈ ϑ. The last Theorem allows us to introduce the concept of geodesic log-monotone opera- tors, which appears to be a novel addition to the field. Definition 7. (i) The differential f ′ is considered to be strongly geodesic log-monotone , if dfµ1η(µ2, µ1) f(µ1) + dfµ2η(µ1, µ2) f(v) ≤ −ε { ∥η(µ2, µ1)(ς)∥2 + ∥η(µ1, µ2)(ς)∥2 } , ∀µ1, µ2 ∈ ϑ. W. Saleh, A. Lakhdari, B. Meftah / Eur. J. Pure Appl. Math, 17 (2) (2024), 860-869 866 (ii) The differential f ′ is considered to be geodesic log-monotone , if dfµ1η(µ2, µ1) f(µ1) + dfµ2η(µ1, µ2) f(µ2) ≤ 0, ∀µ1, µ2 ∈ ϑ. (iii) The differential f ′ is considered to be geodesic log- pseudo-monotone , if dfµ1η(v, µ1) f(µ1) ≥ 0 =⇒ −dfvη(µ1, v) f(v) ≥ 0∀µ1, v ∈ ϑ. From these definitions, we can deduce that strongly geodesic log-monotonicity entails geodesic log-monotonicity, which, in turn, implies geodesic log-pseudo-monotonicity. It’s crucial to emphasize that the reverse may not always hold true. Theorem 4. Let f be differentiable strongly geodesic log-preinvex function on the geodesic set. If (7) holds, then f ′ satisfies dfµ1η(µ2, µ1) f(µ1) + dfµ2η(µ1, v) f(µ2) ≤ −ε { ∥η(µ2, µ1)(ς)∥2 + ∥η(µ1, µ2)(ς)∥2 } , ∀µ1, µ2 ∈ ϑ. (8) Proof. log f(µ2)− log f(µ1) ≥ dfµ1η(µ2, µ1) f(µ1) + ε∥η(µ2, µ1)(ς)∥2. (9) By swapping the roles of µ1 and µ2 in inequality (9), we obtain: log f(µ1)− log f(µ2) ≥ dfµ2η(µ1, µ2) f(µ2) + ε∥η(µ1, µ2)(ς)∥2. (10) Adding (9) and (10), we have dfµ1η(µ2, µ1) f(µ1) + dfµ2η(µ1, µ2) f(µ2) ≤ −ε { ∥η(µ2, µ1)(ς)∥2 + ∥η(µ1, µ2)(ς)∥2 } , ∀µ1, µ2 ∈ ϑ, (11) demonstrating that the derivative f ′ is strongly geodesic log-monotone. Definition 8. The function f is considered to be sharply geodesic log-pseudo preinvex, if dfµ1η(µ2, µ1) f(µ1) ≥ 0 =⇒ f(µ2) ≥ log f(γµ1,µ2(ς)),∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. Theorem 5. Assume that f is a sharply geodesic log-pseudo preinvex function on A. Then dfµ2η(µ1, µ2) f(µ2) ≥ 0, ∀µ1, µ2 ∈ ϑ. Proof. Let f be a sharply geodesic log-pseudo preinvex function on ϑ, then f(µ2) ≥ log f(γµ1,µ2(ς)), ∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. Hence, by taking ς −→ 0, we have the result. W. Saleh, A. Lakhdari, B. Meftah / Eur. J. Pure Appl. Math, 17 (2) (2024), 860-869 867 Definition 9. A function ξ is considered geodesic log-pseudo preinvex w.r.t. a strictly positive bifunction β , such that log ξ(µ2) < log ξ(µ1) =⇒ log ξ(γµ1,µ2) < log ξ(µ1) + ς(ς − 1)β(µ2, µ1), ∀µ1, µ2 ∈ ϑ and ς ∈ [0, 1]. Theorem 6. If ξ is a strongly geodesic log-preinvex function such that log ξ(µ2) < log ξ(µ1), then ξ is strongly geodesic log-pseudo preinex. Proof. Since log ξ(µ2) < log ξ(µ1) and ξ is a strongly geodesic log-preinvex function, then ∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1], we have log ξ(γµ1,µ2(ς)) ≤ log ξ(µ1) + ς (log ξ(µ2)− log ξ(µ1))− ες(1− ς)∥η(µ2, µ1)(ς)∥2 < log ξ(µ1) + ς(1− ς) (log ξ(µ2)− log ξ(µ1))− ες(1− ς)∥η(µ2, µ1)(ς)∥2 = log ξ(µ1) + ς(ς − 1) (log ξ(µ1)− log ξ(µ2))− ες(1− ς)∥η(µ2, µ1)(ς)∥2 < log ξ(µ1) + ς(1− ς)β(µ2, µ1)− ες(1− ς)∥η(µ2, µ1)(ς)∥2, where β(µ2, µ1) = log ξ(µ1)− log ξ(v) > 0. This demonstrates that ξ is a strongly geodesic log-preinvex function. Now, we demonstrate that the subtraction of a strongly geodesic log-preinvex func- tion and an affine geodesic strongly log-preinvex function results in another geodesic log- preinvex function. Theorem 7. Let f be affine strongly geodesic log-preinvex function. If F is a strongly geodesic log-preinvex , then F − f is a geodesic log-preinvex function. Proof. Assume that f is an affine strongly geodesic log-preinvex function.Then log f(γµ1,µ2(ς)) = (1−ς) log f(µ1)+ς log f(µ2)−ες(1−ς)∥η(µ2, µ1)(ς)∥2, ∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. (12) From the strongly geodesic log-preivexity of F , we have logF(γµ1,µ2(ς)) = (1−ς) logF(µ1)+ς logF(µ2)−ες(1−ς)∥η(µ2, µ1)(ς)∥2,∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. (13) From (1) and (12), we have logF(γµ1,µ2(ς))− log f(γµ1,µ2(ς)) ≤ (1− ς)(logF(µ1)− log(µ1)) + ς(logF(µ2)− log f(µ2)). (14) logF(γµ1,µ2(ς))−log f(γµ1,µ2(ς)) ≤ (1−ς)(logF(µ1)−log f(µ1))+ς(logF(µ2)−log f(µ2)). This demonstrates that F − f is a geodesic log-preinvex function. REFERENCES 868 Remark 2. We observe that if a strictly positive function F is strongly geodesic log- preinvex, then the following inequality holds for all µ1, µ2 ∈ ϑ and ς ∈ [0, 1], known as the Wright strongly geodesic Log-preinvex function: logF(γµ1,µ2(ς))+logF(γµ2,µ1(ς)) ≤ logF(µ1)+logF(µ2)−2ς(1−ς)∥η(µ2, µ1)(ς)∥2. (15) From (15), we can deduce the following: F(γµ1,µ2(ς))F(γµ2,µ1(ς)) = logF(γµ1,µ2(ς)) + logF(γµ2,µ1(ς)) ≤ logF(µ1) + logF(µ2) = logF(µ1)F(µ2), ∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1]. This implies that F(γµ1,µ2(ς))F(γµ2,µ1(ς)) ≤ F(µ1)F(µ2), ∀µ1, µ2 ∈ ϑ, ς ∈ [0, 1], which demonstrates that every strictly positive function F is multiplicative Wright Strongly Geodesic Log-Preinvex. Funding: The authors confirm that they did not receive any financial support for this article. Conflict of Interest: The authors affirm the absence of any conflicts of interest. Data and Material Availability: Data and materials are not applicable to this study. 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