International Journal of Analysis and Applications

Volume 17, Number 3 (2019), 388-395

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-388

SOME PROPERTIES OF GEODESIC STRONGLY E-B-VEX FUNCTIONS

WEDAD SALEH∗

Department of Mathematics, Taibah University, Al- Medina, Saudi Arabia

∗Corresponding author: wed 10 777@hotmail.com

Abstract. Geodesic E-b-vex sets and geodesic E-b-vex functions on a Riemannian manifold are extended to

the so called geodesic strongly E-b-vex sets and geodesic strongly E-b-vex functions. Some basic properties

of geodesic strongly E-b-vex sets are also studied.

1. Introduction

Convexity and its generalizations play an important role in optimization theory, convex anlysis and

Minkowski space [3, 4, 6, 9, 10].

Youness [17] defined E-convex sets and E-convex functions by relaxing the definitions of convex sets

and convex functions, which have some important applications in various branches of mathematical sci-

ences [1, 12, 13]. Also, Youness [18] extended the definitions of E-convex sets and E-convex functions to

strongly E-convex sets and strongly E-convex functions. The B-vex functions which shares many properties

with convex functions was introduced by Bector and Singh [2]. Some reserchers studied some new gen-

eralizations of convex functions by relaxing definitions of E-convex functions and B-vex functions such as

E-B-vex functions [15] and strongly E-B-vex functions [19]. Also, generalization of convexity on Riemannian

manifolds were presented in ( [5], [8] , [14], [16]).

Received 2019-03-08; accepted 2019-04-05; published 2019-05-01.

2010 Mathematics Subject Classification. 52A20, 52A41, 53C20, 53C22.

Key words and phrases. geodesic E-convex sets; geodesic E-convex functions; Riemannian manifolds.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

388

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-388


Int. J. Anal. Appl. 17 (3) (2019) 389

In this paper, a new class of sets on Riemannian manifolds, called geodesic strongly E-b-vex sets, and

a new class of functions defined on them, called geodesic strongly E-convex functions, have been proposed.

Also, some of their properties have been discussed. This paper divides into three sections. In section 2,

some of definities and properties which will be used throughout this work are presented that can be found in

many books on differential geometry such as [16] . In section 3, a geodedic strongly E-b-vex set and geodesic

strongly E-b-vex function are studied with some of their properties.

2. Preliminaries

Now, let ℵ is a C∞ n-dimensional Riemannian manifold, also µ1,µ2 ∈ℵ and δ : [0, 1] −→ℵ be a geodesic

joining the points µ1 and µ2 , which means that δµ1,µ2 (0) = µ2 and δµ1,µ2 (1) = µ1.

Strongly E-convex sets (SEC) and strongly E-convex (SEC) functions were introduced in [18] such as:

Definition 2.1. (1) A subset Ω ⊆ Rn is strongly E-convex (SEC) set if there is a map ε: Rn −→ Rn

such that

δ(αµ1 + ε(µ1)) + (1 − δ)(αµ2 + ε(µ2)) ∈ B

for each µ1,µ2 ∈ Ω,α ∈ [0, 1] and δ ∈ [0, 1].

(2) A function g : Ω ⊆ Rn −→ R is strongly E-convex (SEC) function on Ω if there is a map ε: Rn −→ Rn

such that Ω is a SEC set and

g(δ(αµ1 + ε(µ1)) + (1 −δ)(αµ2 + ε(µ2))) ≤ δg(ε(µ1)) + (1 − δ)g(ε(µ2)),

∀µ1,µ2 ∈ Ω,α ∈ [0, 1] and δ ∈ [0, 1].

Definition 2.2. [5]

(1) Considering ε: ℵ −→ ℵ is a map. A subset Ω ⊂ ℵ is geodesic E-convex iff there exists a unique

geodesic ηε(µ1),ε(µ2)(δ) of length d(µ1,µ2), which belongs to Ω, ∀µ1,µ2 ∈ Ω and δ ∈ [0, 1].

(2) A functin g : Ω ⊆ℵ−→ R where Ω is a GEC set in ℵ is geodesic E-convex if

g(ηε(µ1),ε(µ2)(δ)) ≤ δg(ε(µ1)) + (1 − δ)g(ε(µ2)),

∀µ1,µ2 ∈ Ω and δ ∈ [0, 1].

Definition 2.3. [7]

(1) Considering ε: ℵ −→ ℵ is a map. A subset Ω ⊂ ℵ is geodesic strongly E-convex(GSEC) iff there

exists a unique geodesic

ηαµ1+ε(µ1),αµ2+ε(µ2)(δ) of length d(µ1,µ2), which belongs to Ω, ∀µ1,µ2 ∈ Ω, α ∈ [0, 1] and δ ∈ [0, 1]

and .



Int. J. Anal. Appl. 17 (3) (2019) 390

(2) A functin g : Ω ⊆ ℵ −→ R, where Ω is a GSEC set in ℵ, is geodesic strongly E-convex (GSEC)

funtion if

g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δ)) ≤ δg(ε(µ1)) + (1 −δ)g(ε(µ2)),

∀µ1,µ2 ∈ Ω and δ ∈ [0, 1].

3. Geodesic Strongly E-b-vex Sets and Geodesic Strongly E-b-vex Functions

In this part of work, a geodesic strongly E-b-vex (GSE-b-vex) set and a geodesic strongly E-b-convex

(GSE-b-vex) function in a Riemannian manifold ℵ are given and some of their properties are discussed.

Definition 3.1. A subset Ω of ℵ is called a geodesic strongly E-b-vex (GSE-b-vex) iff there exists a unique

geodesic ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) of length d(µ1,µ2), which belongs to Ω, ∀µ1,µ2 ∈ Ω,α ∈ [0, 1] and δ ∈ [0, 1].

Remark 3.1. (1) Every GSE-b-vex set is a GSEC set when b(µ1,µ2,δ) = 1.

(2) Every GSE-b-vex set is a GE-b-vex set when α = 0.

(3) When

ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) = δb (αµ1 + ε(µ1)) + (1 −δb) (αµ2 + ε(µ2)) ,

then we have strongly E-B-vex set.

Now, some propertie of GSE-b-vex sets are propoed.

Proposition 3.1. Every convex set Ω ⊂ℵ is a GSE-b-vex set.

The proof of the above proposition is direct that by taking ε: ℵ−→ℵ as the identity map,b(µ1,µ2,δ) = 1

and α = 0.

Proposition 3.2. Let Ω ⊂ℵ be a GSE-b-vex set, then ε(Ω) ⊆ Ω.

Proof. Since Ω is a GSE-b-vex set, then

ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) ∈ Ω,

µ1,µ2 ∈ Ω, α ∈ [0, 1] and δ ∈ [0, 1]. Let δb = 1 and α = 0, then ηε(µ1),ε(µ2) = ε(µ2) ∈ Ω, then ε(Ω) ⊆ Ω. �

Theorem 3.1. Suppose that a set {Ωj}j=1,2,··· ,n is an arbitrary collection of GSE-v-vex subsets of ℵ, then

∩j=1,2,··· ,nΩi is a GSE-b-vex set.

Proof. Considering {Ωj}j=1,2,··· ,n is a collection of GSE-b-vex subsets of Ω. If ∩j=1,2,··· ,nΩj is an empty set,

then the result is obvious. Assume that µ1,µ2 ∈∩j=1,2,··· ,nΩj, then µ1,µ2 ∈ Ωj. Hence, ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) ∈

Ωj,∀α ∈ [0, 1] and δ ∈ [0, 1]. Hence, ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) ∈∩j=1,2,··· ,nΩj,∀α ∈ [0, 1] and δ ∈ [0, 1]. �



Int. J. Anal. Appl. 17 (3) (2019) 391

Remark 3.2. However, the above theorem is not true for the union of GSE-b-vex sets.

Now, we introduce the definition of a geodesic E-b-vex (GSE-b-vex) function on ℵ.

Definition 3.2. Assume that Ω ⊂ℵ is a GSE-b-vex set . A function g : Ω −→ R is called a geodesic strongly

E-b-vex (GSE-b-vex) if

g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ γg(ε(µ1)) + (1 −γ)g(ε(µ2)), (3.1)

∀µ1,µ2 ∈ Ω,α ∈ [0, 1] and δ ∈ [0, 1].

If the inequality (3.1) is strict, then g is called a strictly GSE-b-vex function.

Example 3.1. Assume that g : R −→ R such that g(µ) = −|µ|. Aslo, assume that ε: R −→ R is defined as

ε(µ) = αµ where 0 < α ≤ 1,∀µ ∈ R and the geodesic η is given as

ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)

=




1
2α

[αµ2 + ε(µ2) + δb(αµ1 + ε(µ1) −αµ2 −ε(µ2))] ; µ1µ2 ≥ 0,

1
2α

[αµ2 + ε(µ2) + δb(αµ2 + ε(µ2) −αµ1 −ε(µ1))] ; µ1µ2 < 0

=



µ2 + δb(µ1 −µ2) ; µ1µ2 ≥ 0,

µ2 + δb(µ2 −µ1) ; µ1µ2 < 0,

then g is GSE-b-vex function.

Proposition 3.3. Let g : Ω −→ R be a GSE-b-vex function on a GSE-b-vex set Ω ×ℵ, then g(αµ + ε(µ)) ≤

g(ε(µ)), µ ∈ Ω and α ∈ [0, 1].

Proof. Since g is GSE-b-vex function on GSE-b-vex set Ω, then

g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbg(ε(µ1)) + (1 −δb)g(ε(µ2)),

then for δb = 1, we have

g(αµ1 + ε(µ1)) ≤ g(ε(µ1)).

�

Theorem 3.2. If g1 : Ω −→ R is a GSE-b-vex function on a GSE-b-vex set Ω ⊂ ℵ and g2 : U −→ R is

a non-decreasing convex function such that rang(g1) ⊂ U, then the composite function g2og1 is GSE-b-vex

function on Ω.



Int. J. Anal. Appl. 17 (3) (2019) 392

Proof. By using the hypothesis, we can write all x1,x2 ∈ B,α ∈ [0, 1] and γ ∈ [0, 1],

g1(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbg1(ε(µ1)) + (1 −δb)g1(ε(µ2)),

∀µ1,µ2 ∈ Ω,α ∈ [0, 1] and δ ∈ [0, 1] and since g2 is a non-decreasing convex function, then we get

g2og1(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) = g2
(
g2(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb))

)
≤ g2 (δbg1(ε(µ1)) + (1 −δb)g1(ε(µ2)))

≤ δbg2 (g1(ε(µ1))) + (1 − δb)g2 (g1(ε(µ2)))

= δb(g2og1)(ε(µ1)) + (1 −δb)(g2og1)(ε(µ2))

hence, g2og1 is GSE-b-vex on Ω. Moreover, g2og1 is a strictly GSE-b-vex function if g2 is a strictly non-

decreasing convex function. �

Theorem 3.3. Considering gi : Ω −→ R, i = 1, 2, ...,n are GSE-b-vex functions. Then, the function

g =

n∑
i=1

ξigi

is also GSE-b-vex geodesic on Ω, ∀ξi ∈ R,ξi ≥ 0.

Proof. Since gi, i = 1, 2, ...,n are GSE-b-vex functions, then

gi(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbgi(ε(µ1)) + (1 − δb)gi(ε(µ2)),

∀µ1,µ2 ∈ Ω,α ∈ [0, 1] and δ ∈ [0, 1], Hence,

ξigi(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbξigi(ε(µ1)) + (1 − δb)ξigi(ε(µ2)).

This implies to,

g(ηαµ1+ε(x1),αµ2+ε(µ2)(δb)) =

n∑
i=1

ξigi(ηαµ1+ε(x1),αµ2+ε(µ2)(δb))

≤ δb
n∑
i=1

ξigi(ε(µ1)) + (1 − δb)
n∑
i=1

ξigi(ε(µ2))

= δbg(ε(µ1)) + (1 − δb)g(ε(µ2)).

Then g is GSE-b-vex function. �

Next, we show that a funcion is GSE-b-vex iff its epigraph is a GSE-b-vex set.

Definition 3.3. Assume that Ω ⊂ ℵ× R,E : ℵ −→ ℵ, b : Ω × Ω × [0, 1] −→ R+ and F : R −→ R. A set Ω

is called a geodesic strongly E ×F -convex ( GSE ×F -b-vex ) if

(
ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbF(ξ1) + (1 −δb)F(ξ2)

)
∈ Ω,

∀(µ1,ξ1), (µ2,ξ2) ∈ Ω, α ∈ [0, 1] and γ ∈ [0, 1].



Int. J. Anal. Appl. 17 (3) (2019) 393

Remark 3.3. From Definition 3.3, we have found Ω ⊆ℵ is a GSE-b-vex set iff Ω×R is a GSE×F -b-vex set.

Now , the epigraph of a function g : Ω ⊂ℵ−→ R is given as

E(g) = {(µ,a) : µ ∈ Ω,a ∈ R,g(µ) ≤ a} . (3.2)

Theorem 3.4. Suppose that Ω ⊆ℵ is a GSE-b-vex set, g : Ω −→ R is a function and F : R −→ R is a map

such that F(g(µ)+a) = g(ε(µ))+a, ∀a ∈ R, a > 0. Then, g is a GSE-b-vex on Ω iff E(g) is a GSE×F -b-vex

on Ω ×R.

Proof. Let (µ1,a1), (µ2,a2) ∈ E(g). Since Ω is GSE-b-vex, then

ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) ∈ Ω,

∀α ∈ [0, 1] and δ ∈ [0, 1]. When α = 0 and δb = 1, we have ε(µ1) ∈ Ω also, when α = 0 and δb = 0 we get

ε(µ2) ∈ Ω. Assume that F(a1) and F(a2) where g(ε(µ1)) ≤ F(a1) and g(ε(µ2)) ≤ F(a2). Then

(ε(µ1),F(a1)), (ε(µ2),F(a2)) ∈ E(g).

Considering g is a GSE-b-vex on Ω, then

g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbg(ε(µ1)) + (1 −δb)g(ε(µ2))

≤ δbF(a1) + (1 − δb)F(a2).

This is leading to,
(
ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbF(a1) + (1 − δb)F(a2)

)
∈ E(g), which means that E(g) is

GSE × É-b-vex on Ω ×R.

Conversely, let us take E(g) is GSE × É-b-vex on Ω × R. Assume that µ1,µ2 ∈ Ω,α ∈ [0, 1] and δ ∈ [0, 1],

then (µ1,g(µ1)) ∈ E(g) and (µ2,g(µ2)) ∈ E(g).

In addition,
(
ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbF(g(µ1)) + (1 − δb)F(g(µ2))

)
∈ E(g) =⇒

g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbF(g(µ1)) + (1 −δb)F(g(µ2))

= δbg(ε(µ1)) + (1 − δb)g(ε(µ2)).

Hence, the result. �

Theorem 3.5. Let {Ωj}j=1,··· ,n be a family of GSE×F -b-vex sets. Then ∩j=1,··· ,nΩj is also GSE×F -b-vex

set.

Proof. Let (µ1,a1), (µ2,a2) ∈∩j=1,··· ,nΩj, then (µ1,a1), (µ2,a2) ∈ Ωj, ∀j. =⇒

(
ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbF(a1) + (1 − δb)F(a2)

)
∈ Ωj,



Int. J. Anal. Appl. 17 (3) (2019) 394

∀α ∈ [0, 1] and δ ∈ [0, 1] . Hence,

(
ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbF(a1) + (1 − δb)F(a2)

)
∈∩j=1,··· ,nΩj,

∀α ∈ [0, 1] and δ ∈ [0, 1]. This shows that , ∩j=1,··· ,nΩj is GSE ×F-b-vex set. �

Theorem 3.6. Suppose that G : R −→ R such that G(g(x) + µ) = g(ε(x)) + µ,∀µ ∈ R,µ > 0. Let {gi}i∈I
be a family of real valued functions that is defined on a GSE-b-vex set Ω and bounded from above. Then,

g(x) = supi∈Igi(x),x ∈ Ω is GSE-b-vex on Ω.

Proof. Let gi, i ∈ I be a GSE-b-vex function on Ω, then

E(gi) = {(x,µ) : x ∈ Ω,µ ∈ R,gi(x) ≤ µ}

are GSE ×F-b-vex on Ω ×R. Hence,

∩i∈IE(gi) = {(x,µ) : x ∈ Ω,µ ∈ R,gi(x) ≤ µ,i ∈ I}

= {(x,µ) : x ∈ Ω,µ ∈ R,g(x) ≤ µ}

is GSE ×F-b-vex set. Then, by Theorem 3.4 g is a GSE-b-vex function. �

References

[1] I. A. Abou-Tair and W. T. Sulaiman, Inequalities via convex functions, Int. J.Math. Sci. 22(1999), 543-546.

[2] C. R. Bector and C. Singh, B-vex functions, J. Optim. Theory Appl. 71(2) (1991), 237-253.

[3] V. Boltyanski, H. Martini and P.S. Soltan, Excursions Into Combinatorial Geometry, Springer, Berlin, 1997.

[4] L. Danzer, B. Grünbaum and V. Klee, Helly’s theorem and its relatives. In: V. Klee (ed.) Convexity. Proc. Sympos. Pure

Math., vol.7, pp.101-180. Amer. Math. Soc., Providence, 1963.

[5] A. Iqbal, S. Ali and I. Ahmad, On geodesic E-convex sets, geodesic E-convex functions and E-epigraphs, J. Optim. Theory

Appl. 55(1)(2012), 239-251.

[6] M. A. Jiménez, G. R. Garzón and A. R. Lizana, Optimality conditions in vector optimization. Bentham Science Publishers,

2010.

[7] A. Kiliçman and W. Saleh, On geodesic strongly E-convex sets and geodesic strongly E-convex functions, J. Inequal. Appl.

2015 (2015), 297.

[8] A. Kiliçman and W. Saleh, On properties of geodesic semilocal E-preinvex functions, J. Inequal. Appl. 2018 (2018), 353.

[9] H.Martini and K.J. Swanepoel, Generalized Convexity notions and Combinatorial Geometry, Gongr. Numer. 164 (2003),

65-93.

[10] H. Martini and K.J. Swanepoel, The geometry of minkowski spaces- a survey, Part II. Expo. Math. 22 (2004), 14-93.

[11] F. Mirzapour, A. Mirzapour and M. Meghdadi, Generalization of some important theorems to E-midconvex functions,

Appl. Math. Lett. 24(8) (2011),1384-1388.

[12] M. A. Noor, Fuzzy preinvex functions, Fuzzy Sets Syst. 64(1994), 95-104.

[13] M. A. Noor, K. I. Noor and M. U. Awan, Generalized convexity and integral inequalities, Appl. Math. Inf. Sci. 9(1)(2015),

233-243.



Int. J. Anal. Appl. 17 (3) (2019) 395

[14] T. Rapcsak, Smooth Nonlinear Optimizatio in Rn, Kluwer Academic, 1997.

[15] Y. R. Syau, Lixing Jia, and E. Stanley Lee, Generalizations of E-convex and B-vex functions, Comput. Math. Appl. 58(4)

(2009), 711-716.

[16] C. Udrist, Convex Funcions and Optimization Methods on Riemannian Manifolds, Kluwer Academic, 1994.

[17] E. A. Youness, On E-convex sets, E-convex functions and E-convex programming, J. Optim.Theory Appl. 102 (1999),

439-450.

[18] E. A. Youness and Tarek Emam, Strongly E-convex sets and strongly E-convex functions, J. Interdisciplinary Math.

8(1)(2005), 107-117.

[19] G.Y. Wang, Some Properties of strongly E-B-vex functions, Sustainable Development-Special track within SCET

2012(2012), 247.


	1. Introduction
	2. Preliminaries
	3. Geodesic Strongly E-b-vex Sets and Geodesic Strongly E-b-vex Functions 
	References