1. Introduction
Convex functions play an important role in economics, management science, engineering, finance, and optimization theory. Many interesting generalizations and extensions of classical convexity have been used in optimization and mathematical inequalities. Generalized convex functions called b-vex functions were introduced by Bector and Singh [
1], and some of their basic properties have been discussed. Choa et al. [
2] investigated a new class of functions called sub-b-convex functions and proved the sufficient conditions of optimality for both unconstrained and inequality-constrained sub-b-convex programming. Hudzik and Maligranda [
3] studied certain classes of functions introduced by Orlicz [
4], namely, the classes of s-convex functions. Meftah [
5] introduced a new class of non-negative functions called s-preinvex functions in the second sense with respect to
, for some
. Jiagen and Tingsong Du [
6] presented a class of generalized convex function has some similar properties of sub-b-convex function and s-convex functions.
Ben-Isreal and Mond [
7] defined preinvex functions, and, in [
8], Weir and Mond studied how and where preinvex functions could replace convex functions. Mohan and Neogy [
9] presented certain properties of preinvex functions. Suneja et al. [
10] considered a class of function called b-preinvex functions that are generalizations of preinvex and b-vex functions. A generalization of the b-vex function, called semi-b-preinvex, was given by Long et al. [
11]. Refinements of the mathematical inequalities on convex and generalized convex functions have been investigated [
12,
13,
14,
15,
16,
17,
18,
19,
20].
Motivated by earlier research works [
6,
12,
21,
22,
23], the purpose of this article is to present a new class of functions, called sub-b-s-preinvex functions, that can be reduced to sub-b-preinvex when s = 1. Some of their properties are studied. Furthermore, a new class of sets, called sub-b-s-preinvex sets, is defined. A new sub-b-s-preinvex programming is introduced, and the sufficient conditions of optimality under this type of function is established. Moreover, some examples of applications are given.
2. Preliminaries
Throughout the paper, the convention bellow will be followed:
Let denote the n-dimensional Euclidean space, and let K be a non-empty convex subset in . In addition, let and be two fixed mappings.
The following definitions about b-vex, sub-b-convex, s-convex, sub-b-s-convex, and preinvex functions that will be used throughout the paper are given:
Definition 1 ([1]) The function is called
- 1.
a b-vex function on K with respect to (w.r.t. in short) b if - 2.
and a b-linear function on K w.r.t. b if
Definition 2 ([2]) The function is called a sub-b-convex function on K w.r.t. b if Definition 3 ([3]) The function is called an s-convex function in the second sense if Definition 4 ([6]) A function is called a sub-b-s-convex function on a non-empty convex set w.r.t. b: ifand Recall [
9] that, by definition, a set
is called an invex set w.r.t
if
,
and
.
Ben-Israel and Mond [
7] defined a class of functions called preinvex in the non-empty invex set
, as follows:
Definition 5. A function is preinvex on K w.r.t.η if there exists an n-dimensional vector function η such that 3. Sub-b-s-Preinvex Function and Their Properties
In this section, the concepts of sub-b-s-preinvex function and sub-b-s-preinvex set are given. Furthermore, some of their properties are studied.
Definition 6. A function is called a sub-b-s-preinvex function on a non-empty invex set ifwhere Remark 7. - 1.
If in Equation (1), then sub-b-s-preinvex w.r.t.η, b becomes a sub-b-s-convex function. Moreover, if s = 1, then Equation (1) becomes a sub-b-convex function. - 2.
When and in Equation (1), then the sub-b-s-preinvex function becomes a convex function.
Theorem 8. If are sub-b-s-preinvex functions w.r.t.η,b, then and () are also sub-b-s-preinvex functions w.r.t.η,b.
Corollary 9. If , where are sub-b-s-preinvex functions w.r.t.η,, then the function which is , is also sub-b-s-preinvex function w.r.t.η, b where .
Proposition 10. If , where are sub-b-s-preinvex functions w.r.t.η,, then the function which is , is also a sub-b-s-preinvex function w.r.t.η,b, where .
Theorem 11. Let be a sub-b-s-preinvex function w.r.t.η, and be an increasing function. Then is a sub-b-s-preinvex function where if satisfies the following conditions:
- 1.
;
- 2.
.
Proof.
which means that
is a sub-b-s-preinvex function
. ☐
We introduce a definition of a sub-b-s-preinvex set as follows.
Definition 12. A set is called a sub-b-s-preinvex set if, , and . The epigraph of the sub-b-s-preinvex function
can be given as
Now, we are going to investigate characterizations of the sub-b-s-preinvex function in terms of their epigraph G(h), and we start with sufficient and necessary conditions for h to be a sub-b-s-preinvex function , b.
Theorem 13. is a sub-b-s-preinvex function iff its epigraph is also a sub-b-s-preinvex set .
Proof. Let h be a sub-b-s-preinvex and let . Then, by using the hypothesis, we have and .
Therefore, is sub-b-s-preinvex set .
Now, assume that
is a sub-b-s-preinvex set
. Then
where
.
, which means that
Then h is sub-b-s-preinvex function . ☐
Proposition 14. Assume that is a family of is sub-b-s-preinvex sets . Then is also a sub-b-s-preinvex set .
Proof. Consider
. Then we have
⇒
Hence, is a sub-b-s-preinvex set. ☐
According to Theorem 13 and Proposition 14, the following proposition holds:
Proposition 15. Let be a sub-b-s-preinvex function . Then a function is also a sub-b-s-preinvex function .
Theorem 16. Let be a non-negative differentiable sub-b-s-preinvex function . Then
- 1.
;
- 2.
.
Theorem 17. Let be a negative differentiable sub-b-s-preinvex function . Then Proof. We obtain the result by using the hypotheses, since
Then, by taking
, which is the maximum of
, we obtain the result. ☐
Corollary 18. Assume that is a differentiable sub-b-s-preinvex function , and
- 1.
h is a non-negative function, then - 2.
h is a negative function, then
4. Hermite–Hadamard-Type Integral Inequalities for Differentiable Sub-B-S-Preinvex Functions
There are a great deal of inequalities related to the class of convex functions. For example, Hermite–Hadamard’s inequality is one of the well-known results in the literature, which can be stated as follows.
Theorem 19. (Hermite–Hadamard’s inequality) Let h be a convex function on with . If h is an integral on , then For more properties about the above inequality, we refer the interested readers to [
24,
25]. Dragomir and Fitzpatrick [
26] demonstrated a variation of Hadamard’s inequality, which holds for
s-convex functions in the second sense.
Theorem 20. Theorem Let be an s-convex function in the second sense and , . If , then Now, we will present new inequalities of Hermite–Hadamard for functions whose derivatives in absolute value are sub-b-s-preinvex functions. Our results generalize those results presented in [
27] concerning Hermite–Hadamard type inequalities for preinvex functions.
Lemma 21 ([27]) Assume that is an open invex subset w.r.tη and with . Let be a differentiable mapping on K such that . Then the following equality holds: Theorem 22. Assume that is an open invex subset w.r.tη and with . Let be a differentiable mapping on K such that . If is a sub-b-s-preinvex function on K, then we have the following inequality: Proof. Since
is a sub-b-s-preinvex on
K, for every
,
and
, we obtain
Therefore, the proof of Theorem 22 is complete. ☐
Corollary 23. If in Theorem 22, then Inequality 4 reduces to the following inequality: Theorem 24. Assume that is an open invex subset w.r.tη and with . Let be a differentiable mapping on K such that . If is a sub-b-s-preinvex function on K for , then we have the following inequality:where . Proof. From Lemma 21 and using the Hölder’s integral inequality, we have
Since
is a sub-b-s-preinvex on
K, for every
,
and
, we obtain
Moreover, via basic calculus, we obtain . Thus, the proof of Theorem 24 is complete. ☐
Corollary 25. If in Theorem 24, then Inequality 8 reduces to the following inequality:where . 5. Application
In this section, we apply our results to the non-linear programming problem and to special means.
Let us consider the unconstraint problem (
P)
Theorem 26. Consider that is a non-negative differentiable sub-b-s-preinvex function . If andthen is the optimal solution to with respect to h on K. Proof. By using the hypothesis and the second pair of Theorem 16, we obtain
and since
That is , which means that is the optimal solution. ☐
Example 27. Let us take the following function such that , where . Additionally, let and Since , it is easy to say that h is a sub-b-s-preinvex function. Additionally, is a non-negative differentiable, and exists for every and . Thus, the following unconstraint sub-b-s-preinvex programming can be given asand Thus, we see that andholds . Hence, according to Theorem 26, the minimum value of at zero. Corollary 28. Assume that is a strictly non-negative differentiable sub-b-s-preinvex function . If and satisfies the condition of Equation (12), then is the unique optimal solution of h on K. Proof. Since
h is a strictly non-negative differentiable sub-b-s-preinvex function
and by using Theorem 16, we obtain
Let
where
be optimal solutions of (
P). Then
, and Equation (
13) yields that
By using Equation (
12), we have
, but
. Thus,
. It follows that
is the unique optimal optimal solution of
h on
K. ☐
Now, grant nonlinear programming as follows:
is the feasible set of
, which is given as
In addition, for , we define .
Theorem 29 (Karush-Kuhn-Tucker Sufficient Conditions) Assume that is a non-negative differentiable sub-b-s-preinvex function and are differentiable sub-b-s-preinvex functions . Additionally, let Ifthen is an optimal solution of (). Proof. For any
, then we obtain
. Therefore, from the sub-b-s-preinvexity of
and Theorem 17, we get
From Equation (
14), we obtain
Equations (
15) and (
17) yields that
Here, we use Equations (
16) and (
18) to obtain
and according to Theorem 26, one has
Hence, is an optimal solution of (). ☐
Now, some applications to special means are given. The following result is established in [
28].
Assume that is a non-negative convex function on . Then is s-convex on , where . For arbitrary positive real numbers , the following special means are given:
It is well known that
is monotonic non-decreasing over
with
and
. In particular, we have the following inequality
Now, some new inequalities are derived for the above means
Let
,
,
and
, Then
If
, then
where
and
6. Conclusions
In this paper, we introduce a new class of functions and sets called sub-b-s-preinvex functions and sub-b-s-preinvex sets and discuss some of their properties. In addition, the optimality conditions for a non-linear programming problem are also established. Hermite–Hadamard-type integral inequalities for differentiable sub-b-s-preinvex functions have been studied. Relationships between these inequalities and the classical inequalities have been established. The ideas and techniques of this paper may motivate further research, for example, in manifolds.