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A note on Gpreinvex functions
Journal of Inequalities and Applications volume 2013, Article number: 169 (2013)
Abstract
With the equivalent relationships between the Ggeneralized invexities and general invexities on the hand, we present two characterizations for Gpreinvexity; we also discuss the relationships between different Ggeneralized invexities such as Gpreinvexity, strict Gpreinvexity and semistrict Gpreinvexity. Note that our results are proved by applying the results from general invexities introduced in the literatures.
1 Introduction
Recently, Antczak [1, 2] introduced the concept of the Gpreinvexity, which included the preinvexity [3] and the rpreinvexity [4] as special cases. Relation of this Gpreinvexity to preinvexity and some properties of this class of functions were studied in [2]. In another recent paper, Luo and Wu [5] introduced a new class of functions, named semistrictly Gpreinvex functions. The relationships between semistrictly Gpreinvex functions and Gpreinvex functions were investigated under mild assumptions. Their results improved and extended the existing ones in the literature. Also, the properties of semistrictly Gpreinvex functions were further considered by Peng in [6].
In this note, we are interested in the relationships between three kinds of Ggeneralized invexities. For this purpose, we firstly investigate the relation between the Ggeneralized invexities and the corresponding general generalized invexities. Then we characterize these Ggeneralized invexities by applying the wellknown results from the preinvexity, the strict preinvexity and the semistrict preinvexity. Moreover, we point out that our method is different from the one used by Luo and Wu in [5]. The rest of this note is organized as follows. In Section 2, we give some definitions and some preliminaries; moreover, we establish the useful Lemma 1. Section 3 presents two characterizations for Gpreinvex functions and proves that, under certain conditions, the Gpreinvexity is equivalent with prequasiinvexity when an intermediatepoint Gpreinvexity is required. In Section 4, we obtain relationships between different Ggeneralized invexities. Section 5 gives some conclusions.
2 Definitions and preliminaries
In this section, we provide some definitions and some notations. Moreover, we establish an important lemma.
Definition 1 [3]
Let X\subset {\mathbb{R}}^{n}, \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}. The set X is said to be invex at u\in X with respect to η if for all x\in X such that
X is said to be invex set with respect to η if X is invex at each u\in X.
Definition 2 [7]
Let X be a nonempty invex subset of {\mathbb{R}}^{n} with respect to η. A function f:X\to \mathbb{R} is said to be preinvex at u\in X with respect to η if
The function f is said to be preinvex on X with respect to η if f is preinvex at each u\in X with respect to η; f is said to be strictly preinvex on X with respect to η if the inequality (1) strictly holds for all x,u\in X such that x\ne u; f is said to be semistrictly preinvex on X with respect to η if the inequality (1) strictly holds for all x,u\in X such that f(x)\ne f(u).
Let X be a nonempty invex subset of {\mathbb{R}}^{n} with respect to η. A function f:X\to \mathbb{R} is said to be Gpreinvex at u on X with respect to η if there exists a continuous function G:\mathbb{R}\to \mathbb{R} such that G:{I}_{f}(X)\to \mathbb{R} is a strictly increasing function on its domain, and
The function f is said to be Gpreinvex on X with respect to η if f is Gpreinvex at each u\in X with respect to η; f is said to be strictly Gpreinvex on X with respect to η if the inequality (2) strictly holds for all x,u\in X such that x\ne u; f is said to be semistrictly Gpreinvex on X with respect to η if the inequality (2) strictly holds for all x,u\in X such that f(x)\ne f(u).
From Definition 3, G is a strictly increasing function because {G}^{1} must exist. Hence, let G be a strictly increasing function throughout this note. Now we present a useful lemma.
Lemma 1 Let f:X\to \mathbb{R}. Suppose G:{I}_{f}(X)\to \mathbb{R} is a strictly increasing function on its domain. Then

(i)
f is Gpreinvex on X with respect to η if and only if G(f) is preinvex on X with respect to η;

(ii)
f is strictly Gpreinvex on X with respect to η if and only if G(f) is strictly preinvex on X with respect to η;

(iii)
f is semistrictly Gpreinvex on X with respect to η if and only if G(f) is semistrictly preinvex on X with respect to η.
Proof (i) By the monotonicity of G, we know that the inequality (2) is equivalent to
Therefore, by Definitions 2 and 3, f is Gpreinvex on X with respect to η if and only if G(f) is preinvex on X with respect to η.
Similar to part (i), we can prove part (ii) and (iii). □
3 Semicontinuity and Gpreinvexity
In this section, two conditions that determine the Gpreinvexity of a function via an intermediatepoint Gpreinvexity check under conditions of upper and lower semicontinuity, respectively, are presented; moreover, equivalent relationship between Gpreinvexity and prequasiinvexity is proved under the intermediatepoint Gpreinvexity assumption. Here, we need the following Condition C, which was introduced by Mohan and Neogy in [8]. The function \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n} satisfies Condition C if
hold for any x,y\in X and for any \lambda \in [0,1].
The upper and lower semicontinuity of a real function f is defined as follows.
Definition 4 [9]
Let X be a nonempty subset of {\mathbb{R}}^{n}. A function f:X\to \mathbb{R} is said to be upper semicontinuous at \overline{x}\in X if, for every \u03f5>0, there exists a \delta >0 such that for all x\in X, if \parallel x\overline{x}\parallel <\delta, then
If −f is upper semicontinuous at \overline{x}\in X, then f is said to be lower semicontinuous at \overline{x}\in X.
We also need the following Lemma 2, which is Lemma 3.2 in [9].
Lemma 2 Let X be a nonempty, open and invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Assume that f:X\to \mathbb{R} satisfies
Moreover, there exists an \alpha \in (0,1) such that for every x,y\in X the inequality
holds. Then the set A:=\{\lambda \in [0,1]f(y+\lambda \eta (x,y))\le \lambda f(x)+(1\lambda )f(y),\mathrm{\forall}x,y\in X\} is dense in [0,1].
Under semicontinuity conditions, Yang proved from Lemma 2 that judging a function to be preinvex or not can be reduced to checking intermediatepoint preinvexity for the function; see the following Lemmas 3 and 4, which are taken from Theorems 3.1 and 3.2 in [9], respectively.
Lemma 3 Let X be a nonempty open invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Assume that the function f:X\to \mathbb{R} is upper semicontinuous on X and satisfies
Then f is a preinvex function on X if and only if there exists an \alpha \in (0,1) such that
Lemma 4 Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Assume that the function f:X\to \mathbb{R} is lower semicontinuous on X and satisfies
Then f is a preinvex function on X if and only if for any x,y\in X, there exists an \alpha \in (0,1) such that
With Lemmas 24 on hand, we can prove the following Theorems 13, respectively.
Theorem 1 Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Assume that f:X\to \mathbb{R} satisfies
Suppose the function G is increasing on {I}_{f}(X). Moreover, there exists an \alpha \in (0,1) such that for every x,y\in X the inequality
holds. Then the set A:=\{\lambda \in [0,1]G(f(y+\lambda \eta (x,y)))\le \lambda G(f(x))+(1\lambda )G(f(y)),\mathrm{\forall}x,y\in X\} is dense in [0,1].
Proof
From the assumption of this theorem, we have
Hence, we can deduce the result from Lemma 2. □
Theorem 2 Let X be a nonempty open invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Assume that a function f:X\to \mathbb{R} is upper semicontinuous on X and satisfies
Moreover, the function G is both continuous and increasing on {I}_{f}(X). Then f is Gpreinvex on X if and only if there exists an \alpha \in (0,1) such that
Proof By assumption, we know that the function G(f) is upper semicontinuous on X and it satisfies
Replacing f by G(f) in Lemma 3 and combining Lemma 1(i), we obtain the desired result. □
If f is continuous on X, then the above Theorem 2 is Theorem 10 in [1]. However, our proof is simpler than the proof of Theorem 10 in [1], since we apply the result pertaining to the preinvexity as defined in Definition 2.
Theorem 3 Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Assume that the function f:X\to \mathbb{R} is lower semicontinuous on X and satisfies
Moreover, the function G is both continuous and increasing on {I}_{f}(X). Then f is Gpreinvex on X if and only if for any x,y\in X, there exists an \alpha \in (0,1) such that
Proof
By the assumption of the theorem, it is easy to check that
Moreover, G(f) is lower semicontinuous on X. Now, with Lemma 1(i) and Lemma 4, we derive the desired result. □
The above Theorems 2 and 3 illustrate that, to justify Gpreinvexity of a function, it is sufficient to check intermediatepoint Gpreinvexity for the function. Our development extends the results of general preinvexity to the Gpreinvexity. Note that Theorems 13 generalize Lemmas 24 from the preinvex case to the Gpreinvex situation, respectively.
On the relationship between the preinvexity and prequasiinvexity, where the prequasiinvexity concept is presented in Definition 5, Yang et al. obtained an interesting result (see Lemma 5).
Definition 5 Let X be a nonempty invex subset of {\mathbb{R}}^{n} with respect to η. A function f:X\to \mathbb{R} is said to be prequasiinvex on X if
Remark 1 If the function G is strictly increasing on {I}_{f}(X), then f is prequasiinvex on X if and only if {G}_{f}(f) is prequasiinvex on X.
Lemma 5 [9]
Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Then a function f:X\to \mathbb{R} is preinvex on X if and only if it is a prequasiinvex function on X and there exists an \alpha \in (0,1) such that
Next, we extend the result obtained by Yang et al. to the Gpreinvex situation in the following theorem, which reveals that, under an intermediatepoint Gpreinvexity assumption, the Gpreinvexity is equivalent with prequasiinvexity.
Theorem 4 Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n} and f:X\to \mathbb{R}, where η satisfies Condition C. Suppose that G is a strictly increasing function on {I}_{f}(X). Then f is a Gpreinvex function on X if and only if it is a prequasiinvex function on X and there exists an \alpha \in (0,1) such that
Proof By Remark 1, one obtains that f is a prequasiinvex function on X if and only if G(f) is a prequasiinvex function on X. Thus, we have the desired result from Lemma 1(i) and Lemma 5. □
4 Relationships among Ggeneralized preinvexities
In this section, we discuss the relationships between Ginvexities under Condition C. To this end, we will use the following results proved in the literatures.
Theorem 5 [[10], Theorem 2.3]
Let X be nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Suppose function f:X\to \mathbb{R} is semistrictly preinvex on X with respect to η. If there exists a \lambda \in (0,1) such that
then f is a preinvex function on X with respect to the same η.
Theorem 6 [[10], Theorem 2.5]
Let X be nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Suppose that f:X\to \mathbb{R} is a preinvex function on X with respect to η. For each pair x,y\in X, x\ne y, if there exists a \lambda \in (0,1) such that
then f is a strictly preinvex function on X with respect to the same η.
Theorem 7 [[7], Theorem 4.2]
Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Assume that f:X\to \mathbb{R} is a preinvex function on X with respect to η. If there exists a \lambda \in (0,1) such that for every x,y\in X, f(x)\ne f(y), the inequalities
hold, then f is a semistrictly preinvex function on X with respect to the same η.
Theorem 8 Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Suppose that f:X\to \mathbb{R} is a semistrictly Gpreinvex function on X with respect to η. If there exists a \lambda \in (0,1) such that
then f is a Gpreinvex function on X with respect to the same η.
Proof Since f is a semistrictly Gpreinvex function on X with respect to η. Then, by Lemma 1(iii), G(f) is a semistrictly preinvex function on X with respect to η. Replacing f by G(f) in Theorem 5, we deduce that G(f) is a preinvex function on X with respect to η. Again, from Lemma 1(i), f is a Gpreinvex function on X with respect to the same η. □
Recall that Theorem 8 was also presented in [5]. But our method of proof is different from [5]. Note that we establish the result by applying the above Theorem 8, which is an existed result for semistrictly preinvex function.
Theorem 9 Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Suppose that f:X\to \mathbb{R} is a Gpreinvex function on X with respect to η. For each pair x,y\in X, x\ne y, if there exists a \lambda \in (0,1) such that
then f is a strictly Gpreinvex function on X with respect to the same η.
Proof Note that f is Gpreinvex on X with respect to η. By Lemma 1(i), G(f) is preinvex on X with respect to η. Now, we deduce from Theorem 6 that G(f) is strictly preinvex on X with respect to η. Therefore, one obtains from Lemma 1(ii) that f is strictly Gpreinvex on X with respect to the same η. □
Lemma 6 Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Suppose function f:X\to \mathbb{R} is Gpreinvex on X with respect to η. If there exists an \alpha \in (0,1) such that for every x,y\in X, f(x)\ne f(y),
Then for every x,y\in X, f(x)\ne f(y)
Proof (i) If \alpha =\frac{1}{2}, the inequality (8) is the inequality (9).

(ii)
If \alpha <\frac{1}{2}, then \alpha <1\alpha <1. Denote by {u}_{1}=1, {u}_{2}=\alpha and \beta =\frac{12\alpha}{1\alpha}, then 1\alpha =\beta {u}_{1}+(1\beta ){u}_{2}.
From Condition C, we have
Note that the identity (3) in Condition C is used in the second, third and fourth equalities. Hence, from (8) and the Gpreinvexity of f, we obtain

(iii)
If \alpha >\frac{1}{2}, then 0<1\alpha <\alpha. Denote by {u}_{1}=\alpha, {u}_{2}=0 and \beta =\frac{1\alpha}{\alpha}, then 1\alpha =\beta {u}_{1}+(1\beta ){u}_{2}. Similar to (ii), we can prove that the inequality (9) still holds. □
Theorem 10 Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}, where η satisfies Condition C. Assume that f:X\to \mathbb{R} is a Gpreinvex function on X with respect to η. If there exists a \lambda \in (0,1) such that for every x,y\in X, f(x)\ne f(y), the inequality
holds, then f is a semistrictly Gpreinvex function on X with respect to the same η.
Proof By Lemma 1(iii), it is sufficient to prove that G(f) is semistrictly preinvex on X with respect to η. From Lemma 6 and the assumption of Theorem 10, we know that the assumption of Theorem 7 holds. Using Theorem 7 and Lemma 1(iii), we can deduce the result. □
Theorem 11 Let X be a nonempty invex set in {\mathbb{R}}^{n} with respect to \eta :X\times X\to {\mathbb{R}}^{n}, where η satisfies Condition C; suppose the function f:X\to \mathbb{R} is lower semicontinuous and satisfies
for any x,y\in X. Moreover, the function G is both continuous and increasing on {I}_{f}(X). If there exists an \alpha \in (0,1) such that for every x,y\in X, f(x)\ne f(y), the inequality
holds, then f is both Gpreinvex and semistrictly Gpreinvex on X.
Proof Firstly, we shall prove that f is a Gpreinvex function on X. Recalling Theorem 4, we need to show that there exists a \lambda \in (0,1) such that for every x,y\in X the inequality (7) holds. Assume, by a contradiction, there exist x,y\in X such that
Since G is both continuous and increasing, then G(f) is lower semicontinuous and satisfies
We need to consider the following cases.
Case (i) f(x)\ne f(y). According to (11), we must have
which contradicts to (12).
Case (ii) f(x)=f(y). Since \alpha \in (0,1), then 2\alpha {\alpha}^{2}=1{(\alpha 1)}^{2}\in (0,1). Let \lambda =2\alpha {\alpha}^{2} and \lambda =\alpha in (12), respectively. Then we have
From (14), we obtain G(f(y+\alpha \eta (x,y)))\ne G(f(x)). Therefore, according to (11), we obtain from Condition C the following inequality:
which contradicts to (13). Therefore, from Theorem 4, f is a Gpreinvex function on X with respect to η.
Further, from the above Theorem 10, f is also a semistrictly Gpreinvex function on X with respect to η. □
5 Conclusions
In this note, our purpose is to investigate the Ggeneralized invexities introduced by researchers in the past few years. To apply the existed results from the general invexities to deal with the Ggeneralized ones, we have established the useful Lemma 1, which discloses the relationships between Ggeneralized invexities and the general invexities. With this important lemma on hand, we have extended the acknowledged results pertaining to the general invexities to the corresponding Ggeneralized invexities. More exactly, some characterizations for Gpreinvex functions have been deduced; when an intermediatepoint Gpreinvexity is satisfied, two equivalent relationships between Gpreinvexity and prequasiinvexity have been established (see Theorems 2 and 3). Using the existed results (Theorems 5, 6 and 7) relating to the general invexities, we deduce the similar results for Ggeneralized invexities; see Theorems 8, 10 and 11. Note that Theorems 8, 10 and 11 are also presented in [5]. However, our method is different from the one used by Luo and Wu in [5]. Here, we prove the results by applying the wellknown results of the general invexities presented in the literatures.
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Acknowledgements
The authors are grateful to the referees for their valuable suggestions that helped to improve the paper in its present form. This research is supported by Science Foundation of Hanshan Normal University (LT200801).
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All authors carried out the proof. XL and DY conceived of the study, and participated in its design and coordination; CX polished the English of the manuscript. All authors read and approved the final manuscript.
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Liu, X., Yuan, D. & Xu, C. A note on Gpreinvex functions. J Inequal Appl 2013, 169 (2013). https://doi.org/10.1186/1029242X2013169
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DOI: https://doi.org/10.1186/1029242X2013169
Keywords
 invex set
 Ggeneralized invexity
 invexity