ON HERMITE-HADAMARD-TYPE INEQUALITIES FOR DIFFERENTIABLE QUASI-CONVEX FUNCTIONS ON THE CO-ORDINATES†
ON HERMITE-HADAMARD-TYPE INEQUALITIES FOR DIFFERENTIABLE QUASI-CONVEX FUNCTIONS ON THE CO-ORDINATES†
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(3_4): 303-314
• Received : June 17, 2013
• Accepted : September 16, 2013
• Published : September 28, 2014
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Feixiang Chen

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. AMS Mathematics Subject Classification : 26A51, 26D15.
Keywords
1. Introduction
Let f : I ⊆ ℝ → ℝ be a convex function and a, b I with a < b , we have the following double inequality
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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 ℝ 2 with a < b and c < d , a mapping f : Δ → ℝ is said to be convex on Δ if the inequality
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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 ℝ 2 .
Theorem 1.3. Suppose that f : Δ = [ a, b ] × [ c, d ] → ℝ is convex on the coordinates on Δ. Then one has the inequalities:
PPT Slide
Lager Image
The concept of quasi-convex function on the co-ordinates was introduced by Ö zdemir et al. in ( [9] , 2012).
Let us consider the bidimensional interval Δ := [ a, b ]×[ c, d ] in ℝ 2 with a < b and c < d , a mapping f : Δ → ℝ is said to be a quasi-convex function on Δ if the inequality
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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 ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ := [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
then the following equality holds:
PPT Slide
Lager Image
In ( [7] , 2012), M. E. Ö zdemir 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 ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ = [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
is a quasi-convex function on the co-ordinates on Δ, then the following inequality holds:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Theorem 1.7. Let f : Δ ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ = [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
is a quasi-convex function on the co-ordinates on Δ and q > 1, then:
PPT Slide
Lager Image
where A is defined in Theorem 1.6 and
PPT Slide
Lager Image
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 ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ := [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
then the following equality holds:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Theorem 1.9 ( [6] ). Let f : Δ ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ := [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
is convex on the co-ordinates on Δ, then the following inequality holds:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Theorem 1.10 ( [6] ). Let f : Δ ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ := [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
is convex on the co-ordinates on Δ and
PPT Slide
Lager Image
then the following inequality holds:
PPT Slide
Lager Image
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 ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ := [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
then the following equality holds:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Proof . Since
PPT Slide
Lager Image
Thus, by integration by parts, it follows that
PPT Slide
Lager Image
Similarly, we can get
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Now
PPT Slide
Lager Image
Multiplying the both sides by
PPT Slide
Lager Image
and using Lemma 1.8, which completes the proof.
Theorem 2.2. Let f : Δ ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ := [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
is a quasi-convex function on the co-ordinates on Δ, then the following inequality holds :
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Proof . From Lemma 2.1, we obtain
PPT Slide
Lager Image
Because
PPT Slide
Lager Image
is quasi-convex on the co-ordinates on Δ, then one has
PPT Slide
Lager Image
On the other hand, we have
PPT Slide
Lager Image
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 ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ := [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
is a quasi-convex function on the co-ordinates on Δ and q > 1, then:
PPT Slide
Lager Image
where A is defined in Theorem 3.1 and
PPT Slide
Lager Image
Proof . From Lemma 2.1, we obtain
PPT Slide
Lager Image
By using the well known H ö lder’s inequality for double integrals, then one has
PPT Slide
Lager Image
Because
PPT Slide
Lager Image
is quasi-convex on the co-ordinates on Δ, then one has
PPT Slide
Lager Image
We note that
PPT Slide
Lager Image
Hence, it follows that
PPT Slide
Lager Image
So, the proof is completed.
Theorem 2.4. Let f : Δ ⊆ ℝ 2 → ℝ be a partial differentiable mapping on Δ := [ a, b ] × [ c, d ] in 2 with a < b and c < d .
PPT Slide
Lager Image
is a quasi-convex function on the co-ordinates on Δ and q > 1, then:
PPT Slide
Lager Image
where A is defined in Theorem 3.1.
Proof . From Lemma 2.1, we obtain
PPT Slide
Lager Image
By using the well known power mean inequality for double integrals, then one has
PPT Slide
Lager Image
Because
PPT Slide
Lager Image
is quasi-convex on the co-ordinates on Δ, then one has
PPT Slide
Lager Image
Thus, it follows that
PPT Slide
Lager Image
Thus, we get the following inequality
PPT Slide
Lager Image
which complete the proof.
BIO
Feixiang Chen received his MS degree from Xi’an Jiaotong University. Since 2009 he has been teaching at Chongqing Three Gorges University. His research interests include integral inequalities on several kinds of convex functions and applications.
School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, 404000, P.R. China.
e-mail: cfx2002@126.com
References
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Alomari M. , Darus M. 2009 On the Hadamard’s inequality for log-convex functions on the coordinates. J. Inequal. Appl 2009 283147 -    DOI : 10.1155/2009/283147
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