Journal of Mathematics
Volume 2021 (2021), Article ID 5586377, 10 pages
https://doi.org/10.1155/2021/5586377

Research Article

Coordinated MT--Convex Functions and Their Integral Inequalities of Hermite–Hadamard Type

Hua Mei,¹ Aying Wan,² and Bai-Ni Guo³

¹College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China
²College of Mathematics and Statistics, Hulunbuir University, Hailaer, Hulunbuir 021008, Inner Mongolia, China
³School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, Henan, China

Correspondence should be addressed to Bai-Ni Guo; bai.ni.guo@hotmail.com

Received 26 January 2021; Revised 11 April 2021; Accepted 15 April 2021; Published 29 April 2021

Academic Editor: Basil Papadopoulos

Copyright © 2021 Hua Mei 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 paper, the authors introduce a new concept of MT--convex functions on the coordinates on the rectangle of the plane and establish some new Hermite–Hadamard-type inequalities for this kind of functions.

1. Motivations

At first, we recall several kinds of convex functions as follows.

Definition 1. (see [1]). Let be an interval. A nonnegative function is said to be MT-convex if the inequality

holds for all and .

Definition 2. (see [2, 3]). Let be a real number. A function is said to be -convex in the second sense if

for all and .

Definition 3. (see [4, 5]). A function is said to be convex on the coordinates on if

holds for all and . If the inequality (3) is reversed, then is said to be concave on the coordinates on .

Definition 4. (see [6]). We say that a function is MT-convex on the coordinates on with and , if the inequality

is valid for all and .
In the papers [7–12] and closely related references therein, the HT-convexity, GT-convexity, and the --convexity on the coordinates were introduced and investigated.
Combining the structures of Definitions 2 and 4, we introduce the notion of coordinated MT--convex functions as follows.

Definition 5. For , a function is said to be MT--convex on the coordinates on with and , if the inequality

holds for all and . If inequality (5) is reversed, then is said to be a MT--concave function on the coordinates on .

2. Simple Properties of MT--Convex Functions

After introduced Definition 5, now we are in a position to investigate in this section simple properties of MT--convex functions on the coordinates on .

Proposition 1. Let and . If is nonnegative and convex on the coordinates on , then is MT-convex on the coordinates on , while is also MT--convex on the coordinates on .

Proof. This follows from for and .

Example 1. (see [13], p. 104). When , the functions and for are MT-convex, but they are not convex on .
For , the function for is MT-convex, but it is not convex on .

Example 2. (see [6], p. 259). When , the function for is MT-convex on the coordinates on , but it is not convex on the coordinates on .

Remark 1. We now discuss Examples 1 and 2 mentioned above.
For , , and being a nonempty interval, the function is concave on . Therefore, for and all with , we have

Accordingly, the function is not MT-convex on . From this, we conclude as follows:

(1)For , the functions and with respect to are not MT-convex on (2)For , the function with respect to is not MT-convex, but it is convex on and is concave on (3)For , the function with respect to is not MT-convex on the coordinates on

Proposition 2. Let for . Then, the function is MT--convex, but not MT-convex, on the coordinates on .

Proof. For and , for , and for , by Definition 5, we deduce

Making use of the inequality and letting result in

This means that the function is MT--convex on the coordinates on .
For with , taking in Definition 5 leads to

This means that the function is not MT-convex on the coordinates on . The proof of Proposition 2 is complete.

3. A Lemma

In order to establish integral inequalities of the Hermite–Hadamard type for MT--convex functions on the coordinates on , we need the following lemma.

Lemma 1. Let have partial derivatives of the second order and let and . If , then

Proof. Integrating by parts gives

The proof of Lemma 1 is complete.

4. Integral Inequalities of the Hermite–Hadamard Type

In this section, we prove some new inequalities of the Hermite–Hadamard type for co-ordinated MT--convex functions.

Theorem 1. Let for and . If is coordinated MT--convex on for and , then

where denotes the well-known beta function which may be defined by

Proof. For all , we have

Letting , , and in (4) and using the MT--convexity of , we obtain

Integrating the above inequality over on and utilizing the change of the variable for result in

By the MT--convexity of , we obtain

From (16) and (17), it follows that

Similarly, we have

A combination of (18) and (19) gives the desired inequality (12).
Putting for all and using the MT--convexity of reveals

Taking for and employing the MT--convexity of leads to

Applying inequalities between (19) and (22) arrives at

By similar argument, we can find

The proof of Theorem 1 is complete.

Corollary 1. Under the conditions of Theorem 1, if , then

Theorem 2. Let for and have the second partial derivatives and and let . If for is coordinated MT--convex functions on , then

where is the Beta function.

Proof. From Lemma 1 and Hölder’s integral inequality, it follows that

By the coordinated MT--convexity of , we have

Combining (27) and (28) results in (26). Theorem 2 is thus proved.

Corollary 2. Under the assumptions of Theorem 2, if , then

Theorem 3. Let for and have the second partial derivatives and and let . If for is co-ordinated MT--convex functions on , then

where is the Beta function.

Proof. From Lemma 1, Hölder’s integral inequality, and the coordinated MT--convexity of , it follows that

Theorem 3 is thus proved.

Corollary 3. Under the conditions of Theorem 3, if , then

5. Conclusion

In this paper, we conclude the following:

(1)From Definition 5, we introduced a new concept of MT--convex functions on the coordinates on the rectangle of the plane (2)From Propositions 1 and 2, we investigated simple properties of MT--convex functions on the coordinates on (3)With the help of the integral identity in Lemma 1, via Theorems 1, 2, and 3, and via Corollaries 1, 2, and 3, we established some new Hermite–Hadamard type inequalities for MT--convex functions on the coordinates on

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

The authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant No. 12061033), by the Natural Science Foundation of Inner Mongolia (Grants Nos. 2018MS01023 and 2019MS01007), and by the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (Grant No. NJZY20119).

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