Journal of Function Spaces
Volume 2019 (2019), Article ID 6926107, 8 pages
https://doi.org/10.1155/2019/6926107

Research Article

Conformable Integral Inequalities of the Hermite-Hadamard Type in terms of GG- and GA-Convexities

Yousaf Khurshid,¹ Muhammad Adil Khan,¹ and Yu-Ming Chu²

¹Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
²Department of Mathematics, Huzhou University, Huzhou 313000, China

Correspondence should be addressed to Yu-Ming Chu; chuyuming@zjhu.edu.cn

Received 11 October 2018; Accepted 27 December 2018; Published 10 January 2019

Academic Editor: Wilfredo Urbina

Copyright © 2019 Yousaf Khurshid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In the article, we present several conformable fractional integrals’ versions of the Hermite-Hadamard type inequalities for GG- and GA-convex functions and provide their applications in special bivariate means.

1. Introduction

Let be an interval and let be a convex function. Then the well known HH (Hermite-Hadamard) inequality [1] states that

for all . It is well known that the convexity has been playing a key role in mathematical programming, engineering, and optimization theory. Recently, many generalizations and extensions for the classical convexity can be found in the literature [2–14]. In [15, 16], Niculescu defined the GA- and GG-convex functions as follows.

Definition 1 (see [15]). A function is said to be GA-convex if the inequality

holds for all and .

Definition 2 (see [16]). A function is said to be GG-convex if the inequality

holds for all and .

Zhang, Ji, and Qi established Lemma 3 and Theorems 4–7.

Lemma 3 (see [17]). Let and be a differentiable function on . Then the identity

holds if .

Theorem 4 (see [17]). If the function satisfies the conditions of Lemma 3 and, additionally, if is GA-convex, then we have the following inequality:

where and in what follows is the logarithmic mean of and .

Theorem 5 (see [17]). If the function satisfies the conditions of Lemma 3 and, additionally, if is GA-convex, then one has

Theorem 6 (see [17]). If the function satisfies the conditions of Lemma 3, then the inequality

holds if is GA-convex.

Theorem 7 (see [17]). If with and the function satisfies the conditions of Lemma 3 and, additionally, if is -convex, then we have

where is the arithmetic mean of and .

The conformable fractional derivative of order at for a function was defined in [18] as follows:

is said to be -fractional differentiable if the conformable fractional derivative of of order exists. The fractional derivative at is defined as . Theorem 8 for the conformable fractional derivative can be found in the literature [18].

Theorem 8 (see [18]). Let and be -differentiable at . Then

(i) for all (ii) if is a constant(iii) for all (iv)(v)(vi) if is differentiable at .

In addition,

if is differentiable.

Definition 9 (see [18], conformable fractional integral). Let and . A function is -fractional integrable on if the integral

exists and is finite. All -fractional integrable functions on are indicated by

Remark 10.

where the integral is the usual Riemann improper integral and

Recently, the conformable integrals and derivatives have been the subject of intensive research; many remarkable properties and inequalities involving the conformable integrals and derivatives can be found in the literature [19–37].

Anderson [38] provided the conformable integral version of the HH inequality as follows.

Theorem 11 (see [38]). If and is an -fractional differentiable function such that is increasing, then the inequality

holds. Moreover if is decreasing on , then we have

If , then this reduces to the classical HH inequality.

In this paper, we shall establish the Hermite-Hadamard type inequalities for GA and GG-convex functions via conformable fractional integrals and give their applications in the special bivariate means.

2. Main Results

In order to establish our main results, we need a lemma which we present in this section.

Lemma 12. Let , , and be an -fractional differentiable function on . Then the identity

holds if .

Proof. Using integration by parts, we have

By the change of the variable and integration by parts, we have

Now multiplying by , we obtain the required result.

Remark 13. Let , then Lemma 12 reduces to Lemma 3.

Theorem 14. If the function satisfies the conditions of Lemma 12 and, additionally, if is GG-convex, then we have

Proof. It follows from the -convexity and Lemma 12 that

The desired result can be obtained by evaluating the above integral.

Theorem 15. If with and the function satisfies the conditions of Lemma 12 and, additionally, if is GG-convex, then one has

Proof. It follows from Lemma 12, the property of the modulus, the GG-convexity of , and the Hölder inequality that

The desired result can be obtained by evaluating the above integral.

Theorem 16. If the function satisfies the conditions of Lemma 12 and, additionally, if is GG-convex, then we have the inequality

Proof. By using Lemma 12 we clearly see that

Since , we can choose such that . Applying the Hölder integral inequality and the GG-convexity of we have

The desired result can be obtained by evaluating the above integral.

Theorem 17. If the function satisfies the conditions of Lemma 12 and, additionally, if is GG-convex, then we have

Proof. From the GG-convexity of , the power mean inequality, and the property of the modulus together with Lemma 12 we get

The desired result can be obtained by evaluating the above integral.

Theorem 18. If the function satisfies the conditions of Lemma 12 and, additionally, if is GA-convex, then

Proof. It follows from the -convexity of and the property of the modulus together with Lemma 12 that

The desired result can be obtained by evaluating the above integrals.

Remark 19. By setting in inequality (27), we regain inequality (5).

Theorem 20. If the function satisfies the conditions of Lemma 12 and, additionally, if is GA-convex, then we have the following inequality:

Proof. From the GA-convexity of , the power mean inequality, the property of the modulus, and Lemma 12 we clearly see that

The desired result can be obtained by evaluating the above integrals.

Remark 21. By setting in inequality (29), we regain inequality (6).

Theorem 22. If the function satisfies the conditions of Lemma 12 and, additionally, if is GA-convex, then we have the following inequality:

Proof. With the help of the GA-convexity of , the power mean inequality, the property of the modulus, and Lemma 12, we can write

The desired result can be obtained by evaluating the above integrals.

Remark 23. By setting in inequality (31), we regain inequality (7).

Theorem 24. If with and the function satisfies the conditions of Lemma 12 and, additionally, if is -convex, then we have the following inequality:

where represents the arithmetic mean of and .

Proof. With the help of the -convexity of , the Hölder integral inequality, the property of the modulus, and Lemma 12, we can write

The desired result can be obtained by evaluating the above integrals.

Remark 25. By setting in inequality (33), we regain inequality (8).

3. Applications to Special Means

A bivariate function is said to be a bivariate mean if for all . Recently, the bivariate mean has attracted the attention of many researchers; in particular, many remarkable inequalities for the bivariate means and their related special functions can be found in the literature [39–42].

In this section, we use the results obtained in Section 2 to present several applications to the arithmetic mean

logarithmic mean

and -th generalized logarithmic mean

Proposition 26. Let , , and . Then

Proof. Let

for . Then is a GA-convex function on for . Let . Then making use of function (39) in Theorem 18, we obtain the required result.

Proposition 27. Let , , , and . Then

Proof. Using function (39) in Theorem 20, we obtain the required result.

Proposition 28. Let , , , and . Then

Proof. Using function (39) in Theorem 22, we obtain the required result.

Proposition 29. Let , , , and with . Then

Proof. Using function (39) in Theorem 24, we obtain the required result.

4. Conclusions

In the article, we derive the conformable fractional integrals’ versions of the Hermite-Hadamard type inequalities for GG- and GA-convex functions. Our approach is based on an identity involving the conformable fractional integrals, the Hölder inequality, and the power mean inequality. The proven results generalized some previously obtained results. As applications, we provide several inequalities for some special bivariate means. The present idea may stimulate further research in the theory of inequalities for other generalized integrals, for example, as presented in [35–37].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project was funded by the Natural Science Foundation of China (Grant nos. 11701176, 11626101, and 11601485) and the Science and Technology Research Program of Zhejiang Educational Committee (Grant no. Y201635325).

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