A Generalized Version of Polynomial Convex Functions and Some Interesting Inequalities with Applications

Abstract

In this article, we introduce a general class of convex functions and proved some of its basic properties. We establish Hermite-Hadamard type inequalities as well as fractional version of Hermite-Hadamard type inequalities by using Riemann-Liouville integral operator. At the end, some application to special means of real numbers are also given. It can be observed from the remarks given in this paper that several exiting results of ligature can be obtained immediacy from our results by taking suitable involved parameters.

1. Introduction

Due to huge applications in applied sciences, notion of convexity become important for researchers, but classical convexity is not enough to solve modern problems, so there is always a need to introduce a more general notion of convexity. In mathematics, classical convexity and concavity are two important concepts. Convexity plays a fundamental role in optimization theory, mathematical economics and engineering. Convexity of a function in classical sense is defined by the following inequality: for every

If the above inequality is reversed, then the function is said to be concave.

Using various techniques, the concept of convex functions has been generalized in many directions, see [1, 2]. Inequality theory become a very dynamic and attractive field of research [3, 4]. In recent years, using various notions of convex functions a wide class of integral inequalities has been derived [5, 6]. The most important and famous inequalities are Schur-type, Hermite-Hadamard-type, and Fejér-type inequalities [7, 8].

Toplu et al. in [9] established Hermite-Hadamard inequality for -polynomial convex functions, which is given below:

Let be -polynomial convex function. If and are two real numbers such that and , then the following double inequality hold,

In this paper we introduce a new class of convex functions, namely (,)-polynomial convex functions. We drive some basic properties of (,)-polynomial convex function and establish Hermite-Hadamard type inequalities for (,)-polynomial convex function in the setting of Riemann-Liouville integral operator. Moreover some applications of established results in special means are also presented.

2. Preliminaries and Basic Results

In this section we present some preliminary material and define a new class of convex functions, which we call (,)-polynomial convex functions. Also we give some basic properties for this new class of convex functions. From now to onward, we consider .

Definition 1 (see [4]). Let . The nonnegative function is -polynomial convex function, if holds for every and .

Definition 2. Let . A nonnegative function is said to be modified -polynomial convex function, if holds for all and .
Now we, are ready to define (,)-polynomial convex functions.

Definition 3. Let . A nonnegative function is said to be (,)-polynomial convex function, if holds for all and .

Remark 4. (1)If we take in (5), then (,)-polynomial convexity reduces to modified -convexity(2)If we take and in (5), then (,)-polynomial convexity reduces to classical convexity(3)If we take in (5), then (,)-polynomial convexity reduces to modified -polynomial convexity

It is worth mentioning here that (1,)-polynomial convexity reduces to modified -convexity, (1, 1)-polynomial convexity reduces to classical convexity and (, 1)-polynomial convexity reduces to modified -polynomial convexity.

Proposition 5. If are (,)-polynomial convex functions defined on and , then is also (,)-polynomial convex function on .

Proof. Let , then from the (,)-polynomial convexity of , we have for every , Hence is a (,)-polynomial convex function on .

Proposition 6. If be a linear function and be a (,)-polynomial convex function on , then is also a(,)-polynomial convex on .

Proof. Using the linearity of and (,)-polynomial convexity of on , we have for every and , Hence is (,)-polynomial convex function on .

Proposition 7. If and are of similarly ordered (,)-polynomial convex functions on , then is also (,)-polynomial convex function on .

Proof. Using the fact that and are of similarly ordered (,)-polynomial convex functions on , we have for every and , Hence is also (,)-polynomial convex function on .

Proposition 8. Let , where be nonnegative (,)-polynomial convex functions on , then for where , the function is (,)-polynomial convex function on , where .

Proof. For all and we have Hence is (,)-polynomial convex function on .

3. Hermite-Hadamard Type Inequalities

In this section we will develop some Hermite Hadamard type integral inequalities for (,)-polynomial convex functions.

Theorem 9. Suppose that is a (,)-polynomial convex function on , then the following inequality holds

Proof. Inserting in Definition 3, we have Taking and in above inequality, we have Integrating the inequality (12) with respect to over we have Using the change of variable technique, and we have Which is left side of the inequality (10).
Now, for the right side of the inequality (10), we start with the following integral, Since, is (,)-polynomial convex function, so which is right side of the inequality (10). The proof completed.

Remark 10. (1)If and then inequality (10) reduced to the Hermite-Hadamard inequality for classical convex functions [10].(2)If , then inequality (10) reduced to the Hermite-Hadamard inequality for modified -convex functions [11].

The following result can be easily obtain by elementary analysis.

Corollary 11. Let be the sum of (,)-polynomial convex functions on . Then

Proof. If we replace by in Theorem 9, we get the required result.

In the next theorem we will derive Hermite-Hadamard type inequality for product of two (,)-polynomial convex functions.

Theorem 12. Consider , are (,)-polynomial convex on , such that and are similarly ordered functions. If then where

Proof. As , are (,)-polynomial convex on , so Since, and are similarly ordered, so Integrating (22) w. r. t “ from 0 to 1, we obtain The LHS of (19) can be obtained easily.
To prove right hand side of (19), we will use the following inequality Since, and are similarly ordered, so Integrating (25) w. r. t “’ from 0 to 1 which is the required right hand side of (19). This completes the proof.

Remark 13. Inserting in Theorem 12 we obtain [11], Theorem 4.
The following lemma, shows that (,)-polynomial convex functions have the same property of convex functions.

Lemma 14. Let be (,)-polynomial convex functions, then where .

Proof. Let be (,)-polynomial convex functions and , we have Which completes the proof.

4. Hermite-Hadamard Type Inequalities in Riemann-Liouville Integral Operator

Now, we will derive Hermite-Hadamard inequality for (,)-polynomial convex functions via Riemann-Liouville fractional integral.

Definition 15 (see [12]). Let The right-hand side and left-hand side Riemann-Liouville integral operators of order with , are defined by respectively, where is the Gamma function defined as .
It is to be noted that

Theorem 16. Let is (,)-polynomial convex and . Then the following inequalities hold

Proof. Inserting in Definition 3, we have Taking and , we have Multiplying (33) with and integrating w. r. t. , we get implies that which is (30).
Since is (,)-polynomial convex functions, so Multiplying (33) with and integrating w. r. t. , we get Which is (4.4). This completes the proof.

Remark 17. Inserting in Theorem 16 we obtain [11], Theorem 6.

5. New Inequalities for (,)-Polynomial Convex Functions

Now we will establish some new inequalities for (,)-polynomial convex functions. To proceed we begin with the following lemma which will be needed to obtain results of our desired type.

Lemma 18 (see [13], Lemma 2.1]). Let be a differentiable map on and , then we have

Theorem 19. Let be a differentiable map on and . If the function (,)-polynomial convex on , then for , we have

Proof. Using Lemma 18 and (,)-polynomial convexity of , we get

Remark 20. If we set , and in the Theorem 19 we can immediately obtain [13], Theorem 2.2.

Theorem 21. Let be a differentiable map on and with . If the is (,)-polynomial convex on , then for , we have

Proof. Unliving Lemma 18, the Hölder’s inequality and the fact that is (,)-polynomial convex, we get

Remark 22. By setting and in the Theorem 21, one obtain [13], Theorem 2.3.

Theorem 23. Let be a differentiable map on and with . If is (,)-polynomial convex on , then for , we have

Proof. Assume that . Using Lemma 18, power mean inequality and convexity of , we achieve For , the result can be proved in a similar fashion as of Theorem 19.

Next we need the following result to refine the power-mean inequality;

Lemma 24 (see [14]). Let and . If and are real functions defined on interval and if are integrable functions on , then

The refined version of integral version of power-mean inequality is as follow:

Theorem 25 (Improved power-mean integral inequality [15]). Let . If and are real functions defined on interval and if are integrable functions on , then

Theorem 26. Let be a differentiable map on and with . If is (,)-polynomial convex on , then for , we have

Proof. Utilizing convexity of , integral version of Holder-Iscan inequality and the Lemma 18, we arrive at

Remark 27. Taking and in the Theorem 26 give [14], Theorem 3.2.

Theorem 28. Let be a differentiable map on and with . If is (,)-polynomial convex on , then for , we have

Proof. Assuming , utilizing Lemma 18, integral version of improved power-mean inequality and convexity of the , we get For , we use the estimates of Theorem 19 which also follows step by step the above estimates. This completes the proof of theorem.

6. Application to the Means

Consider are two numbers. The arithmetic, geometric, logarithmic, and -logarithmic means for and are defined by,

Proposition 29. Let with , then the following inequalities hold:

Proof. Taking in Theorem 9 we obtain the required result.

Proposition 30. Let with , then the following inequalities hold:

Proof. Taking in Theorem 9 we obtain the required result.

7. Conclusion

In this paper we introduced a new more generalized class of (,)-polynomial convex functions and gave some of its basic interesting properties. We also established Hermite-Hadamard type inequalities for (,)-polynomial convex functions. Some applications of the results to the special means are also given. The remarks presented in the paper justify that our results are extension and generalization of many existing results. It will be interesting to establish Hermite-Hadamard, Fejér, and Jensen type inequalities for the different fractional integral operators.

Data Availability

All data is included within this paper.

Conflicts of Interest

The authors have no competing interests.

Authors’ Contributions

All authors contributed equally in this paper.

Acknowledgments

This work was supported by the Innovation Capability Support Program of Shaanxi (No. 2018GHJD-21), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2020JM-646), and the Science and Technology Program of Xi’an (No. 2019218414GXRC020CG021-GXYD20.3). The second author is supported by the funds of University of Okara, Okara, Pakistan.