2. Preliminaries and Definitions
Definition 2.1 (see [<xref ref-type="bibr" rid="B1">1</xref>]).
Let G be a continuous real-valued strictly monotonic function defined on D⊂R. A real-valued function f defined on a convex set X⊂Rn is said to be G-convex if for any x,y∈X, λ∈[0,1],
(2.1)f(y+λ(x-y))≤G-1(λG(f(x))+(1-λ)G(f(y))),
where G-1 is the inverse of G, f(X)⊂D.

Remark 2.2.
Every convex functions is G-convex, but the converse is not necessarily true.

Example 2.3.
Let X=[-1,1], f:X→R, If(X) be the range of real-valued function f, and let G:If(X)→R be defined by
(2.2)f(x)=arctan(|x|+1), G(t)=tan(t).
Then, we can verify that f is a G-convex function. But f is not a convex function because the following inequality
(2.3)f(y+λ(x-y))>λf(x)+(1-λ)f(y)
holds for x=1/4, y=3/4, and λ=1/2.

Weir et al. [4, 5] presented the concepts of invex sets and preinvex functions as follows.

Definition 2.4 (see [<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr" rid="B5">5</xref>]).
A set X⊆Rn is said to be invex if there exists a vector-valued function η:X×X→Rn such that for any x,y∈X, λ∈[0,1],
(2.4)y+λη(x,y)∈X.

Definition 2.5 (see [<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr" rid="B5">5</xref>]).
Let X⊆Rn be invex with respect to vector-valued function η:X×X→Rn. Function f(x) is said to be preinvex with respect to η if for any x,y∈X, λ∈[0,1],
(2.5)f(y+λη(x,y))≤λf(x)+(1-λ)f(y).

Remark 2.6.
Every convex function is a preinvex function with respect to η=x-y, but the converse is not necessarily true.

Example 2.7.
Let X=[-1,1]. f:X→R be defined by
(2.6)f(x)=arctan(|x|+1).
Then, we can verify that f is a preinvex function with respect to η, where
(2.7)η(x,y)={-y-x2+2x,0≤x≤1, 0≤y≤1,-y-x,-1≤x<0, 0≤y≤1,-y-x,0≤x≤1, -1≤y<0,-y+x,-1≤x<0, -1≤y<0.
But f is not convex a function in Example 2.3.

Antczak [17] introduced the concept of G-preinvex functions as follows.

Definition 2.8 (see [<xref ref-type="bibr" rid="B17">17</xref>]).
Let X be a nonempty invex (with respect to η) subset of Rn. A function f:X→R is said to be (strictly) G-preinvex at y with respect to η if there exists a continuous real-valued increasing function G:If(X)→R such that for all x∈X (x≠y), λ∈[0,1],
(2.8)f(y+λη(x,y))≤G-1(λ(G(f(x)))+(1-λ)G(f(y))),(f(y+λη(x,y))<G-1(λ(G(f(x)))+(1-λ)G(f(y)))).
If (2.8) is satisfied for any y∈X, then f is said to be (strictly) a G-preinvex function on X with respect to η.

Remark 2.9.
Every preinvex function with respect to η is G-preinvex function with respect to the same η, where G(x)=x. Every G-convex function is G-preinvex function with respect to η(x,y,λ)=x-y. However, the converse is not necessarily true.

Example 2.10.
Let X=[-1,1]. f:X→R, G:If(X)→R be defined by
(2.9)f(x)=arctan(2-|x|), G(t)=tant.
Then, we can verify that f is a G-preinvex function with respect to η, where
(2.10)η(x,y)={-y-x2+2x,0≤x≤1, 0≤y≤1,-y-x2-2x,-1≤x<0, 0≤y≤1,-y-x,0≤x≤1, -1≤y<0,-y+x,-1≤x<0, -1≤y<0.
But f is not a preinvex function because the following inequality
(2.11)f(y+λη(x,y))>λf(x)+(1-λ)f(y)
holds for x=0, y=1, and λ=1/2.

And f(x) is not a G-convex function because the following inequality
(2.12)f(y+λ(x-y))>G-1(λG(f(x))+(1-λ)G(f(y)))
holds for x=1, y=-1, and λ=1/2.

Definition 2.11 (see [<xref ref-type="bibr" rid="B15">15</xref>]).
A set X⊆Rn is said to be a semiconnected set if there exists a vector-valued function η:X×X×[0,1]→Rn such that for any x,y∈X, λ∈[0,1],
(2.13)y+λη(x,y,λ)∈X.

Definition 2.12 (see [<xref ref-type="bibr" rid="B15">15</xref>]).
Let X⊆Rn be a semiconnected set with respect to a vector-valued function η:X×X×[0,1]→Rn. Function f is said to be semipreinvex with respect to η if for any x,y∈X, λ∈[0,1], limλ→0λη(x,y,λ)=0,
(2.14)f(y+λη(x,y,λ))≤λf(x)+(1-λ)f(y).
Next we present the definition of G-semipreinvex functions as follows.

Definition 2.13.
Let X⊆Rn be semiconnected set with respect to vector-valued function η:X×X×[0,1]→Rn. A function f:X→R is said to be (strictly) G-semipreinvex at y with respect to η if there exists a continuous real-valued strictly increasing function G:If(X)→R such that for all x∈X (x≠y), λ∈[0,1], limλ→0λη(x,y,λ)=0,(2.15)f(y+λη(x,y,λ))≤G-1(λG(f(x))+(1-λ)G(f(y))),(f(y+λη(x,y,λ))<G-1(λG(f(x))+(1-λ)G(f(y)))).
If (2.15) is satisfied for any y∈X, then f is said to be (strictly) G-semipreinvex on X with respect to η.

Remark 2.14.
Every semipreinvex function with respect to η is a G-semipreinvex function with respect to the same η, where G(x)=x. However, the converse is not true.

Example 2.15.
Let X=[-6,6]. Then X is a semiconnected set with respect to η(x,y,λ) and limλ→0λη(x,y,λ)=0, where
(2.16)η(x,y,λ)={x-yλ3,-6≤x<0, -6≤y<0, x>y, 0<λ≤1,λ2(x-y),0≤x≤6, 0≤y≤6, x≥y,λ2(x-y),-6≤x<0, -6≤y<0, x≤y,x-y,0≤x≤6, 0≤y≤6, x<y,x-y,0≤x≤6, -6≤y<0, x<-y,x-y,-6≤x<0, 0≤y≤6, x>-y,0,0≤x≤6, -6≤y<0, x≥-y,0,-6≤x<0, 0≤y≤6, x≤-y.
Let f:X→R, G:If(X)→R be defined by
(2.17)f(x)=arctan(x2+2), G(t)=tant.
Then, we can verify that f is a G-semipreinvex function with respect to η. But f is not a semipreinvex function with respect to η because the following inequality
(2.18)f(y+λη(x,y,λ))>λf(x)+(1-λ)f(y)
holds for x=2, and y=4, λ=1/2.

Example 2.16.
Let X=[-6,6]. Then X is a semiconnected set with respect to η(x,y,λ) and limλ→0λη(x,y,λ)=0, where
(2.19)η(x,y,λ)={x-yφ(λ),-6≤x<0, -6≤y<0, x>y, 0<λ≤1,φ(λ)(x-y),0≤x≤6, 0≤y≤6, x≥y,φ(λ)(x-y),-6≤x<0, -6≤y<0, x≤y,x-y,0≤x≤6, 0≤y≤6, x<y,x-y,0≤x≤6, -6≤y<0, x<-y, λ<φ(λ)<1x-y,-6≤x<0, 0≤y≤6, x>-y,0,0≤x≤6, -6≤y<0, x≥-y,0,-6≤x<0, 0≤y≤6, x≤-y.
Let f:X→R, G:If(X)→R be defined by
(2.20)f(x)=arctan(x2+k), G(t)=tant, ∀k∈R.
Then, we can verify that f(x) is a G-semipreinvex function with respect to classes of functions η. But f(x) is not semipreinvex function with respect to η because the following inequality
(2.21)f(y+λη(x,y,λ))>λf(x)+(1-λ)f(y)
holds for x=2, y=4, and λ=1/2.

Remark 2.17.
Every a G-convex function is G-semipreinvex function with respect to η(x,y,λ)=x-y. But the converse is not true.

Example 2.18.
Let X=(-6,6), it is easy to check that X is a semiconnected set with respect to η(x,y,λ) and limλ→0λη(x,y,λ)=0, where
(2.22)η(x,y,λ)={λ(x-y),0≤x<6, 0≤y<6, x<y,λ(x-y),-6<x<0, -6<y<0, x>y,x-yλ,0≤x<6, 0≤y<6, x≥y, 0<λ≤1,x-yλ,-6<x<0, -6<y<0, x≤y, 0<λ≤1,-x-y,0≤x<6, -6<y<0, x≥-y,-x-y,-6<x<0, 0≤y<6, x≤-y,0, 0≤x<6, -6<y<0, x<-y,0,-6<x<0, 0≤y<6, x>-y.
Let f:X→R, G:If(X)→R be defined by
(2.23)f(x)=arctan(6-|x|), G(t)=tant.
Then, we can verify that f is a G-semipreinvex function with respect to η. But f is not a G-convex function, because the following inequality
(2.24)f(y+λ(x-y))>G-1(λG(f(x))+(1-λ)G(f(y)))
holds for x=1, y=-1, and λ=1/2.

3. Some Properties of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M202"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math></inline-formula>-Semipreinvex Functions
In this section, we give some basic characterizations of G-semipreinvex functions.

Theorem 3.1.
Let f be a G1-semipreinvex function with respect to η on a nonempty semiconnected set X⊂Rn with respect to η, and let G2 be a continuous strictly increasing function on If(X). If the function g(t)=G2G1-1(t) is convex on the image under G1 of the range of f, then f is also G2-semipreinvex function on X with respect to the same function η.

Proof.
Let X be a nonempty semiconnected subset of Rn with respect to η, and we assume that f is G1-semipreinvex with respect to η. Then, for any x,y∈X, λ∈[0,1],
(3.1)f(y+λη(x,y,λ))≤G1-1(λG1(f(x))+(1-λ)G1(f(y))).
Let G2 be a continuous strictly increasing function on If(X). Then,
(3.2)G2(f(y+λη(x,y,λ)))≤G2G-1(λG1(f(x))+(1-λ)G1(f(y))).
By the convexity of g(t)=G2G1-1, it follows the following inequality
(3.3)G2G-1(λG1(f(x))+(1-λ)G1(f(y)))≤λG2G1-1(G1(f(x))+(1-λ)G2G1-1(G1f(y)))=λG2(f(x))+(1-λ)G2(f(y))
for all x,y∈X, λ∈[0,1]. Therefore,
(3.4)G1-1[λ(G1(f(x)))+(1-λ)G1(f(y))] ≤G2-1[λ(G2(f(x)))+(1-λ)G2(f(y))].
Thus, we have
(3.5)f(y+λη(x,y,λ))≤G2-1(λG2(f(x))+(1-λ)G2(f(y))).

Theorem 3.2.
Let f be a G-semipreinvex function with respect to η on a nonempty semiconnected set X⊂Rn with respect to η. If the function G is concave on If(X), then f is semipreinvex function with respect to the same function η.

Proof.
Let y,z∈If(X), from the assumption G is concave on If(X), we have
(3.6)G(z+λ(y-z))≥λG(y)+(1-λ)G(z), λ∈[0,1].
Let
(3.7)G(y)=x, G(z)=u, y=G-1(x), z=G-1(u),
then
(3.8)G(G-1(u)+λ(G-1(x)-G-1(u)))≥λG(G-1(x))+(1-λ)G(G-1(u))=λx+(1-λ)u.
It follows that
(3.9)G-1G(λG-1(x)+(1-λ)G-1(u))≥G-1(λx+(1-λ)u).
Then,
(3.10)λG-1(x)+(1-λ)G-1(u)≥G-1(λx+(1-λ)u).
This means that G-1 is convex. Let G1=G, G2=t, then g(t)=G2G1-1(t) is convex. Hence by Theorem 3.1, f is G2-semipreinvex with respect to η. But G2 is the identity function; hence, f is a semipreinvex function with respect to the same function η.

Theorem 3.3.
Let X be a nonempty semiconnected set with respect to η subset of Rn and let fi:X→R, i∈I, be finite collection of G-semipreinvex function with respect to the same η and G on X. Define f(x)=sup(fi(x):i∈I), for every x∈X. Further, assume that for every x∈X, there exists i*=i(x)∈I, such that f(x)=fi*(x). Then f is G-semipreinvex function with respect to the same function η.

Proof.
Suppose that the result is not true, that is, f is not G-semipreinvex function with respect to η on X. Then, there exists x,y∈X, λ∈[0,1] such that
(3.11)f(y+λη(x,y,λ))>G-1(λG(f(x))+(1-λ)G(f(y))).
We denote z=y+λη(x,y,λ) there exist i(z):=iz∈I, i(x):=ix∈I, and i(y):=iy∈I, satisfying
(3.12)f(z)=fiz(z), f(x)=fix(x), f(y)=fiy(y).
Therefore, by (3.11),
(3.13)fiz(z)>G-1(λG(fix(x))+(1-λ)G(fiy(y))).
By the condition, we obtain
(3.14)fiz(z)≤G-1(λG(fiz(x))+(1-λ)G(fiz(y))).
From the definition of G-semipreinvexity, G is an increasing function on its domain. Then, G-1 is increasing. Since fiz(x)≤fix(x), fiz(y)≤fiy(y), then (3.14) gives
(3.15)fiz(z)≤G-1(λG(fix(x))+(1-λ)G(fiy(y))).
The inequality (3.15) above contradicts (3.13).

Theorem 3.4.
Let f be a G-semipreinvex function with respect to η on a nonempty semiconnected set X⊂Rn with respect to η. Then, the level set Sα={x∈X:f(x)≤α} is a semiconnected set with respect to η, for each α∈R.

Proof.
Let x,y∈Sα, for any arbitrary real number α. Then, f(x)≤α, f(y)≤α. Hence, it follows that
(3.16)f(y+λη(x,y,λ))≤G-1(λG(f(x))+(1-λ)G(f(y)))≤G-1(G(α))=α.
Then, by the definition of level set we conclude that y+λη(x,y,λ)∈Sα, for any λ∈[0,1], we conclude that Sα is a semiconnected set with respect to η.

Let f is a G-semipreinvex function with respect to η, its epigraph Ef={(x,α):x∈X,α∈R,f(x)≤α} is said to be G-semiconnected set with respect to η if for any (x,α)∈Ef, (y,β)∈Ef, λ∈[0,1],
(3.17)(y+λη(x,y,λ),G-1(λG(α)+(1-λ)G(β)))∈Ef.

Theorem 3.5.
Let X⊂Rn with respect to η be a nonempty semiconnected set, and let f be a real-valued function defined on X. Then, f is a G-semipreinvex function with respect to η if and only if its epigraph Ef={(x,α):x∈X,α∈R,f(x)≤α} is a G-semiconnected set with respect to η.

Proof.
Let (x,α)∈Ef, (y,β)∈Ef, then f(x)≤α,f(y)≤β. Thus, for any λ∈[0,1],
(3.18)f(y+λη(x,y,λ))≤G-1(λG(f(x))+(1-λ)G(f(y)))≤G-1(λG(α)+(1-λ)G(β)).
By the definition of an epigraph of f, this means that
(3.19)(y+λη(x,y,λ),G-1(λG(α)+(1-λ)G(β)))∈Ef.
Thus, we conclude that Ef is a G semiconnected set with respect to η.

Conversely, let Ef be a G semiconnected set. Then, for any x,y∈X, we have (x,f(x))∈Ef, (y,f(y))∈Ef. By the definition of an epigraph of f, the following inequality
(3.20)f(y+λη(x,y,λ))≤G-1(λG(f(x))+(1-λ)G(f(y)))
holds for any λ∈[0,1]. This implies that f is a G-semipreinvex function on X with respect to η.

The following results characterize the class of G-semipreinvex functions.

Theorem 3.6.
Let X⊆Rn be a semiconnected set with respect to η:X×X×[0,1]→Rn; f:X→R is a G-semipreinvex function with respect to the same η if and only if for all x,y∈X, λ∈[0,1], and u,v∈R,
(3.21)f(x)≤u, f(y)≤v⇒f(y+λη(x,y,λ))≤G-1(λG(u)+(1-λ)G(v)).

Proof.
Let f be G-semipreinvex functions with respect to η, and let f(x)≤u, f(y)≤v, 0<λ<1. From the definition of G-semipreinvexity, we have
(3.22)f(y+λη(x,y,λ))≤G-1(λG(f(x))+(1-λ)G(f(y)))≤G-1(λG(u)+(1-λ)G(v)).
Conversely, let x,y∈X, λ∈[0,1]. For any δ>0,
(3.23)f(x)<f(x)+δ,f(y)<f(y)+δ.
By the assumption of theorem, we can get that for 0<λ<1,
(3.24)f(y+λη(x,y,λ))≤G-1(λG(f(x))+(1-λ)G(f(y)))≤G-1(λG(f(x)+δ)+(1-λ)G(f(y)+δ)).
Since G is a continuous real-valued increasing function, and δ>0 can be arbitrarily small, let δ→0, it follows that
(3.25)f(y+λη(x,y,λ))≤G-1(λG(u)+(1-λ)G(v)).