Abstract

In this paper, we further study the quasi-p-convex function. The concepts of strictly quasi-p-convex function and quasi-p-convex cone are given and some new fundamental characterizations and operational properties of quasi-p-convex function are obtained.

1. Introduction

Convex function is an important kind of function in mathematics, which is widely used in mathematical programming, approximation theory, control theory and other fields. But it is found that the mathematical models of many optimization problems in practical application are non-convex, which prompts us to consider the generalized convex function with weak convexity. Generalized convex function can not only retain some good characteristics of convex function but also relax the requirement for convexity appropriately. So its application scope is wider than convex function. In 1949, De Finetti ¹ proposed the first generalized convex function, and in 1953, Fenchel ² named it quasi-convex function. Yang ^{3, 4, 5} further studied the properties of quasi-convex functions. In 2003, Wang et al. ⁶ proposed the concept of -quasi-convex function and studied its related properties. In 2011, Lin et al. ⁷ established several new Hadamard type inequalities for quasi-convex function. In 2013, Zhang ⁸ defined harmonic quasi-convex function. In 2021, Bai ⁹ established Simpson type fractional integral inequality for quasi-convex function. In the same year, Sevda et al. ¹⁰ defined p-convex functions by using the concept of epigraph on the basis of p-convex sets. In 2022, Gültekin et al. ¹¹ defined quasi-p-convex functions.

In this paper, we further study the quasi-p-convex function. The concepts of strictly quasi-p-convex function and quasi-p-convex cone are given and some new fundamental characterizations and operational properties of quasi-p-convex function are obtained.

2. Preliminaries

Definition 2.1 ¹³ Let and . If for each , ,

where , , then is called a p-convex set in .

The definition of p-convexity of can also be given as

for all and .

Definition 2.2 ¹⁴ Given , the set

is called epigraph of , it is denoted by .

Definition 2.3 ¹⁰ Let and be a function. If the set

is p-convex set, then is called a p-convex function.

Theorem 2.1 ¹⁰ Let , be a function and , then is a p-convex function if and only if is a p-convex set and for any , holds, where , .

Definition 2.4 ¹¹ Let .

(i) A function is called quasi-p-convex function if

for each ; such that or, equivalently

for every and for every .

(ii) A function is called quasi-p-concave function if is quasi-p-convex, i.e., for each ; such that , or, equivalently

for every and for every .

Example 2.1 Let be a p-convex set, . Define the function ,

where , , then is a quasi-p-convex function.

Proof. Suppose that such that , , then for each ,,

that is, is a quasi-p-convex function.

Theorem 2.2 ¹¹ Let be a p-convex set, . Ifis a p-convex function, thenis a quasi-p-convex function.

Definition 2.5 ¹⁴ Suppose that is a subset of , if for each , such that , then is called a cone.

Definition 2.6 ¹⁴ A cone is said to be convex cone if it is also a convex set.

Definition 2.7 ¹² Let be a convex cone and be a function. If for each , , such that , then the function is called a positive homogeneous function with respect to degree .

Remark 2.1 ¹² (1) Let , be a positive homogeneous function with respect to degree . If exists, then .

If , is called a linear positive homogeneous function.

Theorem 2.3 ¹⁴ The function is a linear positive homogeneous function if and only if its epigraph is a cone in .

Definition 2.8 ¹² Let , then is called a strict local minimum (maximum) of , if it exists such that

.

3. Main Results

Definition 3.1 Let , be a p-convex set and be a function. For each , if the inequality

holds for all such that , then is said to be a strictly quasi-p-convex function.

Definition 3.1 can also be expressed as follows:

Let , be a p-convex set and be a function. If for any ,

where , then is said to be a strictly quasi-p-convex function.

Definition 3.2 Let , be a p-concave set and be a function. For each , if the inequality

(3.3)

holds for all such that , then is said to be a strictly quasi-p-concave function.

Definition 3.2 can also be expressed as follows:

Let , be a p-convex set and be a function. If for any ,

(3.4)

where , then is said to be a strictly quasi-p-concave function.

Theorem 3.1 Let be a p-convex set, .

(1) If is a strictly p-convex function, theis a strictly quasi-p-convex function;

(2) If is a strictly quasi-p-convex function, thenis a quasi-p-convex function.

Proof. (1) From the strictly p-convexity of , for each , such that such that ,

,

that is, is a strictly quasi-p-convex function.

(2) It comes straight from the definition.

Definition 3.3 A cone is said to be p-convex cone if it is also a p-convex set.

Remark 3.1 If , p-convex cone is a convex cone.

Lemma 3.1 The set is a p-convex cone if and only if is closed for operations of addition and positive multiplication.

Proof. Suppose thatis a p-convex cone, , then for each, such that ,

Take , then

Also is a cone, so is obviously closed for operation of positive multiplication. Therefor

That is, is closed for operation of addition.

, if is closed for operations of positive multiplication, then is obviously a cone. For each such that , then . Also is closed for operations of addition, so

,

that is, is a p-convex set. Thus is a p-convex cone.

Lemma 3.2 For , let be a p-convex cone, be a positive homogeneous function of degree p. Thenis a p-convex function if and only if for any ,

Proof. Supposeis a p-convex function, then is a p-convex set. Also is a positive homogeneous function of degree p, for each , , then

.

Specially, take , then , namely , so the is a cone.

From lemma 3.1, for each ,

,

namely

.

, for each , then

,

namely

,

sois closed for operation of addition. Also becauseis a positive homogeneous function of degree p, taking, by Theorem 2.3, we known that is closed for operation of positive multiplication. From Lemma 3.1, is a p-convex set, so is a p-convex function.

Theorem 3.2 For , let be a p-convex cone andbe a positive homogeneous function of degree p. If for all , , then is a quasi-p-convex function if and only ifis a p-convex function.

Proof. For all , by Theorem 2.2, a p-convex function is obviously is a quasi-p-convex function.

, if is a quasi-p-convex function, by the positive homogeneity of degree p of and Lemma 2, it just need to prove thatsatisfies the inequality in Lemma 2.

Let any , . Since is a positive homogeneous function of degree p, namely

,

it follows that

Given the quasi-p-convexity of , we obtain

.

Let , then

.

If eitheroris zero, for example , remark 1 show that , then

.

This completes the proof.

Theorem 3.3 Let be a p-convex set, and be a quasi-p-convex function. If is a strict local minimum of , it is also a strict global minimum of , and the set of all minimal points of is p-convex set.

Proof. Let be a strict local minimum of , if is not a strictly global minimum of , then it exists such that .

By using the quasi-p-convexity of , for , we have

(3.5)

It exists small enough such that

,

namely , it contradicts that is a strict local minimum of . So is the strict global minimum of .

Now assume that , let be the minimum value of on , it is noticed that

By the quasi-p-convexity of and Theorem 2.4, the lower level set is a p-convex set. Thus is also a p-convex set.

Theorem 3.4 Let be a p-convex set, , and be a quasi-p-convex function. Then for each , then a quasi-p-convex function on .

Proof. For all such that , it can be given by conditions

.

Thus is a quasi-p-convex function.

Corollary 3.1 If the function is a strictly quasi-p-convex function, then a strictly quasi-p-convex function on .

Proof. It is clear from Theorem 3.4.

Theorem 3.5 Let be a p-convex set, . If is quasi-p-convex functions for , then is a quasi-p-convex function where .

Proof. Let , so . For such that , we have

.

This is, is a quasi-p-convex functions.

Theorem 3.6 If function is a quasi-p-concave function. For each , then a quasi-p-concave function on .

Proof. For all such that , it can be given by conditions

.

Thus is a quasi-p-concave function.

Corollary 3.2 If the function is a strictly quasi-p-concave function, then a strictly quasi-p-concave function on .

ACKNOWLEDGEMENTS

This work was supported by PhD Research Foundation of Inner Mongolia Minzu University (No. BS402) and Operating expenses for basic scientific research at Universities directly affiliated to Inner Mongolia Autonomous Region (No. GXKY22159).

References