Some new inequalities of the Hermite–Hadamard type for extended ((s1, m1)-(s2, m2))-convex functions on co-ordinates

In the paper, the authors introduce a new concept “extended ((s 1 ,m 1 )(s 2 ,m 2 ))-convex function on co-ordinates” and establish some new inequalities of the Hermite–Hadamard type for extended ((s 1 ,m 1 )-(s 2 ,m 2 ))-convex functions of 2 variables on co-ordinates. Subjects: Science; Mathematics & Statistics; Advanced Mathematics; Analysis Mathematics; Real Functions; Special Functions


Introduction
In Toader (1985) the concept of m-convex functions was introduced as follows.

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In the paper, the authors introduce a new concept of a convex function on co-ordinates and establish some new inequalities of the Hermite-Hadamard type for such a kind of convex functions of 2 variables on co-ordinates.

Let us now consider a bi-dimensional interval
In Dragomir (2002), Dragomir and Pearce (2000) considered the convexity on the co-ordinates.
Definition 1.7 (Dragomir, 2002;Dragomir & Pearce, 2000) A function f :Δ → ℝ is said to be convex on the co-ordinates on Δ = [a, b] × [c, d] ⊆ ℝ 2 with a < b and c < d if the partial functions are convex for all x ∈ (a, b) and y ∈ (c, d).
A formal definition for co-ordinated convex functions may be stated as follows. A formal definition for co-ordinated s-convex mappings may be stated as follows. holds for all t, ∈ (0, 1) and (x, y), (z, w) ∈ Δ.
The main aim of this paper is to introduce the concept "co-ordinated extended ((s 1 , m 1 )-(s 2 , m 2 ))convex function" and to establish the Hermite-Hadamard inequalities for extended ((s 1 , m 1 )-(s 2 , m 2 ))convex functions on the co-ordinates on a rectangle Δ from the plane ℝ 2 0 .

Some integral inequalities of the Hermite-Hadamard type
Now we first state a new notion "co-ordinated extended ((s 1 , m 1 )-(s 2 , m 2 ))-convex function" as follows.
Now we start off to establish some integral inequalities of the Hermite-Hadamard type for the above-introduced co-ordinated extended ((s 1 , m 1 )-(s 2 , m 2 ))-convex functions. where and for x, y, z ≥ 0.
Proof Letting y = c + (1 − )d, 0 ≤ ≤ 1 and employing the co-ordinated extended ((s 1 , m 1 )-(s 2 , m 2 ))-convexity of f (with t = = 1 2 in (2.1)), we have Since f is co-ordinated extended ((s 1 , m 1 )-(s 2 , m 2 ))-convex, using the change of variable x = (1 − t)a + tb and with t = = 1 2 in (2.1), we obtain x m 1 ;y, y dx; Substituting the above inequalities into the inequality (2.5) gives By same argument as in the proof of inequality (2.6), we obtain By adding inequalities (2.6) and (2.7) and dividing both sides by 2, we obtain the inequality (2.2). The proof of Theorem 2.1 is complete. ✷ If putting m 1 = m 2 = m in Theorem 2.1, we can obtain the following corollary. (2.10) Applying the above inequalities to (2.10), we obtain By similar argument as in the proof of the inequality (2.11), we obtain By (2.11) and (2.12), we obtain the inequality (2.8). This completes the proof. If letting s 1 = s 2 = s or m 1 = m 2 = 1 in Theorem 2.2, we can obtain the following corollaries. (2.11)