Abstract： In Ref.[1]， the authors presented some properties of E-level sets both for the E-convex functions and E-quasiconvex functions. In this paper， we give some examples to illustrate that some results in [1] are incorrect. Afterwards， some new properties of E-convex functions and E-quasiconvex functions which are the corrections of those in [1] are given.

Key words： E-convex set， E-convex function， E-quasiconvex function， level set， counter examples.

1.Introduction

E-convex functions and E-convex programming play an important role in mathematical programming， which were presented and studied recently， see [1]-[5]. In this paper， we give some examples to illustrate that some results in are incorrect. Afterwards， some new properties of E-convex functions and E-quasiconvex functions which are the corrections of those in are given.

At first， we recall the related definitions and some results given in and which will be used in our study.

Definition 1.1 [5] A set is said to be a E-convex set， if there exists a map E： such that

（1）

Proposition 1.1 [5] If set is an E-convex set， then 。

Definition 1.2 [5] A function is said to be E-convex on a set，if there exists a map E ： such that is an E-convex set and

（2）

Definition 1.3 [1] A function is said to be E-quasiconvex on a set ，if there exists a map E：such that X is an E-convex set and

（3）

Definition 1.4 [1] If X is a nonempty subset of Rn， and E：are two functions， then， for any ，the set

（4）

is called the E -level set of f ，which is simplified R -level set.

2.Counterexample

Wang， Song and Bai give the following results of characterization of an E-convex function and an E-quasiconvex function in terms of their E-level sets.

Theorem 2.1[1] Let and E：are two functions， is an E -convex set， if is E-convex， then for any ， is an E-convex set.

Theorem 2.2[1] Let and E： are two functions， is an E-convex set， then f is E-quasiconvex iff for any ， is an E-convex set，

where

.

The following examples illustrate that both Theorem 2.1 and Theorem 2.2 are not correct.

Example 2.1

This example illustrates that Theorem 2.1 is not correct.

Let be defined by ， and be defined by

The set is E-convex， becausewe have

The function f is E-convex on X，because，we have

Namely，

Then， in view of Theorem 2.1， the set is E-convex.

This means that， for anywe have

.

Namely，， （5）

and. （6）

But， now let ， we have

，

and ，which contradicts （6）.This implies that Theorem 2.1 is incorrect.

Example 2.2

This example illustrates that the necessity of Theorem 2.2 is not correct.

Let and maps be defined as

Then is an E-convex set， and

， we have

So f is an E-quasiconvex function. But there exists an such that the E-level set is not E-convex on R.

In fact， let ， we have

.

So .However，

.

This implies that the necessity of Theorem 2.2 is incorrect.

Remark 2.1 In fact， we can use the Counterexample 2.2 to see that Theorem 2.1 is not correct. At the same time， we can also illustrate that the necessity of Theorem 2.2 is not correct by the Counterexample 2.1.

3.Properties

In this section， some new characterizations of E-convex functions and E-quasiconvex functions which are the corrections of those in are given.

If X is a nonempty subset of Rn， and E：are two functions， then， for any ，the set

（7）

is called the E*-level set of f.

Under the definition of ， we give a necessary condition for f to be an E-convex function.

Theorem 3.1 Let and E： are two functions，

is an E-convex set， is E-convex， if is convex， then for anyis a convex set.

Proof For any ， let，.Since X is an E-convex set， we have

Since is convex， we have .

Since f is E-convex， we have

.

It follows that .So is a convex set.

Corollary 3.1 Let and E： are two functions， is an E-convex set， f is E-quasiconvex， if is convex， then for anyis a convex set.

The following two theorems give a sufficient condition for f to be an E-quasiconvex function using the set .

Theorem 3.2 Let and E：are two functions， is an E-convex set， if for any is a convex set， then is E-quasiconvex.

Proof Let. Since X is an E-convex set， we have

.

Let ，we have .Since is a convex set， then .

It follows that So f is E-quasiconvex on X.

References：

[1] Wang Jian-yong， Song Ying， Bai Xiao-lu.E-Quasiconvex Function[J]. Journal of Liaocheng University（Nat.Sci）， 2003， 16（3）： 17-19.（in chinese）.

[2] Jian jin-bao. Incorrect Results for E-Convex Functions and E-Convex Programming[J].Journal of Mathematical Research Exposition， 2003， 23（3）： 461-466.

[3] Chen Xiu-su. Some Properties of Semi-E-Convex Functions[J]. J.Math.Anal.Appl.， 2002，275： 251-262.

[4] Yang，X M. On E-Convex Sets， E-Convex Functions， and E-Convex Progamming[J]. J.Optim.Theory Appl.， 2001， 109（3）： 699-704.

[5] Youness，E A.E-Convex Sets， E-Convex Functions， and E-Convex Progamming[J]. J.Optim.Theory Appl.， 1999， 102（2）： 439-450.

作者簡介：李海龍（1981-），男，天津大學(xué)仁愛學(xué)院數(shù)學(xué)教學(xué)部講師。