Abstract
In this paper, we study a generalized version of strongly reciprocally convex functions of higher order. Firstly, we prove some basic properties for addition, scalar multiplication, and composition of functions. Secondly, we establish Hermite-Hadamard and Fejér type inequalities for the generalized version of strongly reciprocally convex functions of higher order. We also include some fractional integral inequalities concerning with this class of functions. Our results have applications in optimization theory and can be considered extension/generalization of many existing results.
1. Introduction
Convexity is a very simple and ordinary concept. Due to its massive applications in industry and business, convexity has a great influence on our daily life. In the solution of many real-world problems, the concept of convexity is very decisive. Problems faced in constrained control and estimation are convex. Geometrically, a real-valued function is said to be convex if the line segment joining any two of its points lies on or above the graph of the function in Euclidean space.
Convexity of a function in classical sense is defined as a function is convex if we have
If the above inequality is reversed, then the function is said to be concave.
Using different techniques, the notion of convexity is being extended day by day [1–3]. Many extensions and generalizations are made speedily due to its applications in modern engineering, optimization, economics, and nonlinear programming [4–7]. For recent generalizations, one can see [8, 9] and the references therein.
Using the definition of convex functions, several important inequalities can be proved, and the Hermite-Hadamard inequality is one of them. The Hermite-Hadamard inequality is for any convex function with and , the Hermite-Hadamard double inequality is
In [9], using the weight function , Fejér gave a generalization of the Hermite-Hadamard inequality as follows:
Let be a convex function and is nonnegative, integrable, and symmetric about , then we have
In [10], the notations of -convex set and -convex functions are introduced. The strongly convex functions of modulus are introduced in [11]. In [12, 13], the strongly -convex and harmonic convex functions were introduced, respectively. The -harmonic convex set and -harmonic convex functions were studied in [14], and in [15], the strongly reciprocally convex of modulus are introduced. The strongly reciprocally -convex and -convex functions were introduced in [16, 17], respectively. The (,)-convex functions are introduced in [18], and the higher-order strongly convex with modulus are introduced in [19]. Now, we present the notation of strongly reciprocally (,)-convex functions of higher order (SRHO).
Definition 1. Let and is any interval. Then the function is SRHO of modulus on the interval , if we have for all , , and , where .
Remark 2. Inserting in Def. 1 with same as defined above, we obtain strongly reciprocally (,)-convex functions. Similarly, inserting and in Def. 1, we obtain strongly reciprocally -convex functions, and for , , and , Def. 1 reduces to the strongly reciprocally convex function of modulus .
As we know that is a Norm space under the usual modulus norm, thus, for any ,
Using (5), the inequality 1 can be written as and with , where is same as in Definition 1.
The aim of this paper is to study a generalized version of strongly reciprocally convex functions of higher order and establish the Hermite-Hadamard and Fejér type inequalities for this new class of convex functions. We also presented fractional versions of the above mentioned inequalities for the strongly reciprocally (,)-convex of higher order. It is worthy to mention here that the results presented in this paper are more generalized and can be considered extensions of many existing results.
2. Basic Results
Now, we present some basic properties for strongly reciprocally (,)-convex of higher order.
Proposition 3. For any two SRHO with modulus on the interval the is also SRHO with modulus on the interval , where .
Proof. By definition, we have
which in turns implies that
where and .
This completes the proof.
Proposition 4. For any SRHO with modulus and any , is also SRHO with modulus on the interval , where .
Proof. Let , then by definition of we obtain where and . This completes the proof.
Proposition 5. Consider a sequence of SRHO, are defined on an interval , provide , then for positive constants , the function is SRHO with nonnegative modulus .
Proof. For a -harmonic convex set , we have and where . Hence, the result is proved.
Proposition 6. Consider a sequence of SRHO, are defined on an interval , provide , then for positive constants , the function is SRHO of modulus .
Proof. Let be a -harmonic convex set. Then, and we have This completes the proof.
3. Hermite-Hadamard Type Inequality
In this section, we establish Hermite-Hadamard’s type inequality for the function belonging to .
Theorem 7. Consider an interval not containing zero and SRHO of nonnegative modulus and then for , we have
Proof. Substituting in Definition 1, gives
Considering and and integrating (13)
which is left side of the inequality (12).
Finally, for the right side of the inequality (12), setting and in Definition 1 gives
Integrating (15)
that is right hand side of (12) and proof is completed.
4. Fejér Type Inequality
Now, we are going to develop the Fejér type inequality for the function belonging to .
Theorem 8. Consider an interval not containing zero and real-valued SRHO defined on of nonnegative modulus then for , we have holds for with and , where the nonnegative real-valued function defined on satisfies
Proof. Substituting in Definition 1, yields
Considering and and integrating (19),
By the properties of ,
Integrating inequality (21),
which is left side of the inequality (17).
Finally, for the right side of the inequality (17), setting in Definition 1 gives
By the properties of ,
Integrating inequality (25),
that is right hand side of (17) and the proof is completed.
5. Fractional Integral Inequalities
Lemma 9 ([20], Lemma 2.1). Let be a differentiable function on the interior of . If and , then
Theorem 10. Let be a -harmonic convex set and let be a differentiable function on the interior of . If and are strongly reciprocally (,)-convex function of higher order on , , and then where
Proof. Using Lemma 9, we have Using power mean inequality, Since is in , so Hence, the desired result is obtained.
For , Theorem 10 reduces to the following result.
Corollary 11. Let be a -harmonic convex and let be a differentiable function on the interior of . If and are in on and then where , , , , , and are given by (31) to (36).
Theorem 12. Let be a -harmonic convex and let be a differentiable function on the interior of . If and are strongly reciprocally (,)-convex function of higher order on , , and then where
Proof. Using Lemma 9, we have Applying Hölder’s integral inequality, Since is in , so Hence, proved.
For Theorem 12 reduces to the following result.
Corollary 13. Let be a -harmonic convex set and let be a differentiable function on the interior of . If and are in on , , , and then where , , , , ,and are given by (42)–(47). For Theorem 12 reduces to the following result.
Corollary 14. Let be a-harmonic convex set and let be a differentiable function on the interior of . If and are strongly reciprocally (,)-convex function of higher order on , , and then, where , , , , ,and are given by (42)–(47). For Theorem 12 reduces to the following result.
Corollary 15. Let be a -harmonic convex set and let be a differentiable function on the interior of . If and is strongly reciprocally (,)-convex function of higher order on , , and then, where , , , , , and are given by (42)–(47). For Theorem 12 reduces to the following result.
Corollary 16. Let be a -harmonic convex set and let be a differentiable function on the interior of . If and is strongly reciprocally (,)-convex function of higher order on , , and then, where , , , , , and are given by (42)–(47).
Remark 17. Inserting and with in Corollary 16, we obtained ([20], Corollary 3.8).
6. Conclusion
In this paper, a new definition of convex functions “namely strongly reciprocally (,)-convex functions of higher order” is introduced. This new definition extends almost all the existing versions of convex functions. For the strongly reciprocally (,)-convex functions of higher order, we established several interesting inequalities which have applications in optimization theory, probability theory, as well as pure and applied mathematics. We also established several fractional versions of the Hermite-Hadamard type inequalities. The remarks presented in the paper justify the validity of our results and prove that our results are more general than almost every existing result.
Data Availability
All data required for this research is included within this paper.
Conflicts of Interest
The authors declare that they do not have any competing interests.
Authors’ Contributions
Limei Liu proposed and proved the main results, verified and analyzed the results, and arranged the funding. Muhammad Shoaib Saleem also verified and analyzed the results, arranged the funding for this paper, and prepared the final version of the manuscript. Faisal Yasin wrote the introduction and related the results of this paper to the existing literature. In particular, he showed that several results in the literature can be obtained directly from the results of this paper by suitable substitution of the involved parameters. Kiran Naseem Aslam supervised this work and proved some of its main results. Pengfei Wang proposed the problem and supervised this work.
Acknowledgments
This research is supported by Department of Mathematics, University of Okara, Okara Pakistan.