Abstract
We define a new generalized class of harmonically preinvex functions named harmonically -preinvex functions, which includes harmonic -preinvex functions, harmonic -preinvex functions, harmonic -preinvex functions, and -convex functions as special cases. We also investigate the properties and characterizations of harmonically -preinvex functions. Finally, we establish some integral inequalities to show the applications of harmonically -preinvex functions.
1. Introduction
Theory of convex functions had not only stimulated new and deep results in many branches of mathematical and engineering sciences, but also provided us with a unified and general framework to study a wide class of unrelated problems. For applications, generalizations, and other aspects of convex functions, see [1–10] and the references therein.
It is well known that a function is a convex function on an interval , if and only if, the function satisfies the inequalitywhich is known as the Hermite-Hadamard inequality. For the applications in various fields of pure and applied sciences, we refer the reader to [6, 7, 11–17].
The concepts of the convex sets and convex functions have been extended and generalized in several directions by using innovative ideas and techniques. Hanson [13] introduced the invex functions in mathematical programming. Note that the invex functions are not convex functions. Ben-Israel and Mond [18] introduced the concept of invex sets and peinvex functions. They have shown that the differentiable preinvex functions are invex functions, but the inverse may not be true. It is well known that the preinvex functions may not be convex functions (see [19–21]). For example, the function is not a convex function, but it is a preinvex function with respect to , where
Pitea and Postolache [16, 22, 23] investigated the properties of quasi-invexity in theoretical mechanics and nonlinear optimization. This shows that the preinvexity and its variant generalizations play an important and significant role in the developments of various fields of pure and applied sciences. Noor [19] has shown that the following inequalityholds for preinvex function . For more integral inequalities of type (3), see [20].
In recent years, harmonic convex functions, which can be viewed as an important and significant extension of the convex functions, are being used to develop several iterative methods for solving nonlinear equations. Anderson et al. [1] and İşcan [5] have investigated various properties of harmonic convex functions. It has been shown that if is a harmonic convex function, then the following inequality holds:which is also called Hermite-Hadamard inequality for harmonic convex function.
Later on, Noor et al. [24] introduced a new unified class of convex functions, the harmonic -preinvex function, which includes two classes of convex functions (harmonic preinvex functions and harmonic -preinvex functions) as special cases.
Recently, the functions of -preinvexity and harmonic -convexity have attracted the interest of many researchers. Motivated by the idea of Noor et al. [19, 20, 24, 25], in an earlier investigation, we have introduced a more general unified class of convex functions; this class of convex functions is called harmonic -preinvex functions. One can easily show that harmonic -preinvex functions include harmonic preinvex, harmonic -convex, and preinvex functions. Following this way, the aim of this paper is to introduce a new generalized class of harmonically preinvex functions which is called harmonically -preinvex functions; also we will discuss the properties of these classes of functions. Our results include several previous results as special cases. We hope that the interested readers may discover new and innovative applications of these harmonic -preinvex functions.
2. Preliminaries
We begin with recalling some basic concepts and notions in the convex analysis. For more details, we refer the reader to [24, 26–28] and the references therein.
Definition 1. A set in is said to be a convex set, if
Definition 2. A function on the convex set is said to be a convex function, if
Definition 3. A set is said to be invex set with respect to the bifunction , if
Definition 4. The function on the invex set is said to be preinvex with respect to the bifunction , ifThe function is said to be preconcave if is preinvex.
The invex set is also called -connected set. Note that if , then invex set becomes the convex set. Clearly, every convex set is an invex set but converse is not true in general.
Definition 5. Let be an invex set in , and let be a nonnegative function. Then, a function is said to be -preinvex function with respect to the bifunction , if
Definition 6. A set is said to be a harmonically -convex set, if
Definition 7. Let be a nonnegative function. A function is said to be harmonically -convex function, if
Definition 8. A set is said to be a harmonic invex set with respect to the bifunction , if
Definition 9. Let be a nonnegative function. A function is said to be harmonically -preinvex function with respect to an arbitrary bifunction , if
Definition 10. Let . The function on the -invex set is said to be -preinvex function with respect to , where , iffor all and .
Definition 11. The function , , is said to be -convex, where , if we havefor all and . We say that is -concave if is -convex.
3. Definitions and Properties of Harmonically -Preinvex Functions
In this section, we introduce some definitions and properties related to harmonically -preinvex functions, which is a component part of our main results.
Definition 12. A function is said to be harmonically -preinvex functions with respect to the bifunction where iffor all and .
Remark 13. Notice that for , harmonic -preinvexity reduces to harmonic preinvexity; for , harmonic -preinvexity reduces to harmonic -convexity; for , harmonic -preinvexity reduces to preinvexity.
Definition 14. Two functions are said to be similarly ordered ( is -monotone) on interval , if and only if,
Proposition 15. If are two similarly ordered harmonic -preinvex functions, then the product is also a harmonic -preinvex function.
Proof. Since are two positive harmonic -preinvex functions, by using Definition 12, one haswhere, we have used the fact that two harmonic -preinvex functions are similarly ordered functions. This shows that product of two similarly ordered harmonic -preinvex functions is a harmonic -preinvex function.
Definition 16. Let be a nonnegative function. A function is said to be harmonically -preinvex functions with respect to the bifunction or belongs to the class , where , iffor all , , and .
Let be a nonnegative function. A function is said to be harmonically -preconcave functions with respect to the bifunction or belongs to the class , where , iffor all , , and .
Remark 17. Note that for and harmonically -preinvex functions reduce to the harmonically -preinvex functions; for and harmonically -preinvex functions reduce to the harmonically -preinvex functions; for and harmonically -preinvex functions reduce to the harmonically -convex functions; for and , harmonically -preinvex functions reduce to the -convex functions.
Now, we discuss some interesting results of harmonically -preinvex (preconcave) functions, which includes linearity, product, composition properties, and the order property of and .
Proposition 18. If and , then . Similarly, if and , then , .
Proof. The proof follows immediately from the definitions of the classes and .
Proposition 19. (a) Let and . If is monotone decreasing (monotone increasing) and (), then .
(b) Let and . If is monotone decreasing (monotone increasing) and (), then .
Proof. Let us recall a well-known result on the monotonicity of weighted mean (see [29]).
Letwhere , () andIn [29], it is proved that is a strictly increasing function for .
LetBy using the monotonicity of weighted mean for , we conclude that is a strictly decreasing function for . Thus, for and , we have Since is monotone decreasing and , we then obtainTherefore, we get . The result of (b) follows by the similar arguments as discussed above.
Proposition 20. Let and be nonnegative functions, and let in their domains . If , then . Similarly, if , then .
Proof. Since form the definition of the classes and the constraint conditions for , , , and , we havewhich implies that . The result in the second part of Proposition 20 can be deduced in a similar way as above.
Proposition 21. (a) Let be similarly ordered positive functions on , that is,for all . If , , and for all , where and is a fixed positive number, then the product belongs to .
(b) Let and be opposite ordered positive functions, that is,for all . If , , and for all , where and is a fixed positive number, then the product belongs to .
Proof. We only give a proof of the first part and the result of the second part of this proposition followed by a similar argument. Since and are similarly ordered, we haveLet and be positive numbers such that . We obtainThis completes the proof of Proposition 21.
Proposition 22. Let and be functions with .
(a) If the function is -convex and increasing (decreasing) and with for , then belongs to .
(b) If the function is -concave and increasing (decreasing) and with for , then belongs to .
Proof. In view of and is an increasing function, we haveSince, and is -convex, we obtainwhich leads toWe conclude that . The rest of the proof of Proposition 22 can be dealt with in the same way as described above.
4. Integral Inequalities for Harmonically -Preinvex Functions
We illustrate the applications of the definitions and properties of harmonically -preinvex functions stated in foregoing section; we will establish some integral inequalities via harmonically -preinvex functions. Our results include, as special cases, some related inequalities for harmonic -preinvex functions, harmonic -preinvex functions, and -convex functions.
Theorem 23. Let be an integrable harmonic -preinvex function, and let be a concave function on Then we have the following inequality:
Proof. Let and Using the definition of harmonic -preinvex function along with a simple computation gives where the last inequality follows from the assumption that is a concave function on This completes the proof of Theorem 23.
Choosing , in Theorem 23, we get the following.
Corollary 24. Let be an integrable harmonic -preinvex function. Then we have the inequality
Theorem 25. Let be a harmonic -preinvex function with . If , are integrable, then
Proof. Since is harmonic -preinvex function, we haveIntegrating both sides of the above inequality with respect to over , we obtain On the other hand, by settingthen the left-hand side of integral expression can be simplified toThusThe proof of Theorem 25 is complete.
Taking , in Theorem 25, we obtain the following.
Corollary 26. Let be a harmonic -preinvex function with . If is integrable, then
Theorem 27. Let , be harmonic -preinvex functions with If are integrable, then the following inequality holds:where and .
Proof. Since are harmonic -preinvex functions, it follows thatOn the other hand, by using a substitution,and then a straightforward calculation givesThe proof of Theorem 27 is completed.
Putting , into Theorem 27, we have the following.
Corollary 28. Let be harmonic -preinvex functions with If are integrable, then the following inequality holds:where and .
Theorem 29. Let be harmonic -preinvex functions with If are integrable, then the following inequality holds:where , and is nonnegative and satisfies
Proof. Note that is harmonic -preinvex function; we haveUsing a substitutionwe get Similarly, we have From the properties of expression , it is easy to find thatBy utilizing the above inequality and identities, we deduce thatwhere The proof of Theorem 29 is completed.
Putting , into Theorem 29, we have the following.
Corollary 30. Let be harmonic -preinvex functions with If are integrable, then we have the following inequality:where is nonnegative and satisfies
Competing Interests
The authors declare that they have no competing interests.
Authors’ Contributions
All authors read and approved the final manuscript.
Acknowledgments
The work of the first author has been supported by the Natural Science Foundation of Fujian Province of China under Grant no. 2016J01023.