Coordinated MT-(s1, s2)-Convex Functions and Their Integral Inequalities of Hermite–Hadamard Type

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                  </jats:inline-formula>-convex functions on the coordinates on the rectangle of the plane and establish some new Hermite–Hadamard-type inequalities for this kind of functions.</jats:p>


Motivations
At first, we recall several kinds of convex functions as follows.
Definition 2 (see [2,3]). Let s ∈ (0, 1] be a real number. A function f: R ⟶ R 0 is said to be s-convex in the second sense if for all x, y ∈ I and t ∈ [0, 1].
Combining the structures of Definitions 2 and 4, we introduce the notion of coordinated MT-(s 1 , s 2 )-convex functions as follows.

Remark 1.
We now discuss Examples 1 and 2 mentioned above.

Journal of Mathematics
Proof. For s 1 � 0.5 and s 2 � 0.02, for t, λ ∈ (0, 1), and for (x, y), (z, w) ∈ R 2 + , by Definition 5, we deduce Making use of the inequality λ 0.02/2 /2 is means that the function f(x, y) is MT-(0.5, 0.02)-convex on the coordinates on R 2 + . For (x, y), (z, w) ∈ R 2 + with x ≠ z, taking t � λ � 1/2 in Definition 5 leads to is means that the function f(x, y) is not MT-convex on the coordinates on R 2 + . e proof of Proposition 2 is complete.

A Lemma
In order to establish integral inequalities of the Hermite-Hadamard type for MT-(s 1 , s 2 )-convex functions on the coordinates on Δ, we need the following lemma.
Proof. Integrating by parts gives Journal of Mathematics 3

Integral Inequalities of the Hermite-Hadamard Type
In this section, we prove some new inequalities of the Hermite-Hadamard type for co-ordinated MT-(s 1 , s 2 )-convex functions.