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 inequalityholds for all and .

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

Definition 3. (see [4, 5]). A function is said to be convex on the coordinates on ifholds 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 inequalityis 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 inequalityholds 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 haveAccordingly, 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 deduceMaking use of the inequality and letting result inThis means that the function is MT--convex on the coordinates on .
For with , taking in Definition 5 leads toThis 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 givesThe 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 , thenwhere denotes the well-known beta function which may be defined by

Proof. For all , we haveLetting , , and in (4) and using the MT--convexity of , we obtainIntegrating the above inequality over on and utilizing the change of the variable for result inBy the MT--convexity of , we obtainFrom (16) and (17), it follows thatSimilarly, we haveA combination of (18) and (19) gives the desired inequality (12).
Putting for all and using the MT--convexity of revealsTaking for and employing the MT--convexity of leads toApplying inequalities between (19) and (22) arrives atBy similar argument, we can findThe 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 , thenwhere is the Beta function.

Proof. From Lemma 1 and Hölder’s integral inequality, it follows thatBy the coordinated MT--convexity of , we haveCombining (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 , thenwhere is the Beta function.

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