ON HERMITE-HADAMARD-TYPE INEQUALITIES FOR DIFFERENTIABLE QUASI-CONVEX FUNCTIONS ON THE CO-ORDINATES

Abstract

In this paper, a new lemma is established and several new inequalities for differentiable co-ordinated quasi-convex functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for co-ordinated quasi-convex functions in two variables are obtained.

1. Introduction

Let f : I ⊆ ℝ → ℝ be a convex function and a, b ∈ I with a < b, we have the following double inequality

This remarkable result is well known in the literature as the Hermite-Hadamard inequality for convex mapping.

Definition 1.1. A function f : [a, b] → R is said to be quasi-convex on [a, b], if

holds for all x, y ∈ [a, b] and λ ∈ [0, 1].

Clearly, any convex function is a quasi-convex function, but the converse is not generally true.

In [4], S. S. Dragomir defined convex functions on the co-ordinates as following:

Let us consider the bidimensional interval Δ := [a, b] × [c, d] in ℝ2 with a < b and c < d, a mapping f : Δ → ℝ is said to be convex on Δ if the inequality

holds for all (x, y), (z, w) ∈ Δ and λ ∈ [0, 1].

A function f : Δ → ℝ is said to be co-ordinated convex on Δ if the partial mappings fy : [a, b] → ℝ, fy(u) = f(u, y) and fx : [c, d] → ℝ, fx(v) = f(x, v) are convex for all y ∈ [c, d] and x ∈ [a, b].

A formal definition for co-ordinated convex functions may be stated as follows:

Definition 1.2. A function f : Δ → ℝ is said to be convex on co-ordinates on Δ if the inequality

holds for all (x, y), (z, y), (x, w), (z, w) ∈ Δ and t, λ ∈ [0, 1].

S. S. Dragomir in [4] established the following Hadamard-type inequalities for co-ordinated convex functions in a rectangle from the plane ℝ2.

Theorem 1.3. Suppose that f : Δ = [a, b] × [c, d] → ℝis convex on the coordinates on Δ. Then one has the inequalities:

The concept of quasi-convex function on the co-ordinates was introduced by Özdemir et al. in ([9], 2012).

Let us consider the bidimensional interval Δ := [a, b]×[c, d] in ℝ2 with a < b and c < d, a mapping f : Δ → ℝ is said to be a quasi-convex function on Δ if the inequality

holds for all (x, y), (z, w) ∈ Δ and λ [0, 1].

A function f : Δ → ℝ is said to be quasi-convex functions on the co-ordinates if the partial mappings fy : [a, b] → ℝ, fy(u) = f(u, y) and fx : [c, d] → ℝ, fx(v) = f(x, v) are convex for all y ∈ [c, d] and x ∈ [a, b].

A formal definition of quasi-convex functions on the co-ordinates as follows:

Definition 1.4. A function f : Δ → ℝ is said to be a quasi-convex function on the co-ordinates on Δ if the inequality

holds for all (x, y), (z, y), (x, w), (z, w) ∈ Δ with t,λ ∈ [0, 1].

In ([10], 2012), M. Z. Sarıkaya et al. established some inequalities for coordinated convex functions based on the following lemma.

Lemma 1.5. Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ := [a, b] × [c, d] in ℝ2 with a < b and c < d. then the following equality holds:

In ([7], 2012), M. E. Özdemir et al. established the following inequalities for quasi-convex functions on the co-ordinates based on Lemma 1.5.

Theorem 1.6. Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ = [a, b] × [c, d] in ℝ2 with a < b and c < d. is a quasi-convex function on the co-ordinates on Δ, then the following inequality holds:

where

Theorem 1.7. Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ = [a, b] × [c, d] in ℝ2 with a < b and c < d. is a quasi-convex function on the co-ordinates on Δ and q > 1, then:

where A is defined in Theorem 1.6 and

Some new integral inequalities that are related to the Hermite-Hadamard type for co-ordinated convex functions are also established by many authors.

In ([1], [2], 2008), M. Alomari and M. Darus defined co-ordinated s-convex functions and proved some inequalities based on this definition. In ([5], 2009), M. A. Latif and M. Alomari defined co-ordinated h-convex functions and proved some inequalities based on this definition. In ([3], 2009), Alomari et al. established some Hadamard-type inequalities for coordinated log-convex functions.

In ([6], 2012), M. A. Latif and S. S. Dragomir obtained some new Hadamard type inequalities for differentiable co-ordinated convex and concave functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for co-ordinated convex functions in two variables based on the following lemma:

Theorem 1.8. Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ := [a, b] × [c, d] in ℝ2 with a < b and c < d. then the following equality holds:

where

Theorem 1.9([6]). Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ := [a, b] × [c, d] in ℝ2 with a < b and c < d. is convex on the co-ordinates on Δ, then the following inequality holds:

where

Theorem 1.10([6]). Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ := [a, b] × [c, d] in ℝ2 with a < b and c < d. is convex on the co-ordinates on Δ and then the following inequality holds:

where A is as given in Theorem 1.9.

For recent results and generalizations concerning Hermite-Hadamard type inequality for differentiable co-ordinated convex functions see ([8], 2012) and the references given therein.

In this paper, we establish several new inequalities for differentiable co-ordinated quasi-convex functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for co-ordinated quasi-convex functions in two variables.

 

2. Main results

To establishing our results, we need the following lemma.

Lemma 2.1. Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ := [a, b] × [c, d] in ℝ2 with a < b and c < d. then the following equality holds:

where

Proof. Since

Thus, by integration by parts, it follows that

Similarly, we can get

and

Now

Multiplying the both sides by and using Lemma 1.8, which completes the proof.

Theorem 2.2. Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ := [a, b] × [c, d] in ℝ2 with a < b and c < d. is a quasi-convex function on the co-ordinates on Δ, then the following inequality holds:

where

Proof. From Lemma 2.1, we obtain

Because is quasi-convex on the co-ordinates on Δ, then one has

On the other hand, we have

The proof is completed.

The corresponding version for powers of the absolute value of the fourth partial derivative is incorporated in the following theorems.

Theorem 2.3. Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ := [a, b] × [c, d] in ℝ2 with a < b and c < d. is a quasi-convex function on the co-ordinates on Δ and q > 1, then:

where A is defined in Theorem 3.1 and

Proof. From Lemma 2.1, we obtain

By using the well known Hölder’s inequality for double integrals, then one has

Because is quasi-convex on the co-ordinates on Δ, then one has

We note that

Hence, it follows that

So, the proof is completed.

Theorem 2.4. Let f : Δ ⊆ ℝ2 → ℝ be a partial differentiable mapping on Δ := [a, b] × [c, d] in ℝ2 with a < b and c < d. is a quasi-convex function on the co-ordinates on Δ and q > 1, then:

where A is defined in Theorem 3.1.

Proof. From Lemma 2.1, we obtain

By using the well known power mean inequality for double integrals, then one has

Because is quasi-convex on the co-ordinates on Δ, then one has

Thus, it follows that

Thus, we get the following inequality

which complete the proof.