Inequalities for convex and s-convex functions on Δ = [a, b] × [c, d]
© Ozdemir et al.; licensee Springer. 2012
Received: 1 May 2011
Accepted: 1 February 2012
Published: 1 February 2012
In this article, two new lemmas are proved and inequalities are established for co-ordinated convex functions and co-ordinated s-convex functions.
Mathematics Subject Classification (2000): 26D10; 26D15.
KeywordsHadamard-type inequality co-ordinates s-convex functions.
is well known in the literature as Hermite-Hadamard inequality. Both inequalities hold in the reversed direction if f is concave.
In , Orlicz defined s-convex function in the second sense as following:
for all x, y ∈ [0, ∞), α, β ≥ 0 with α + β = 1 and for some fixed s ∈ (0, 1]. We denote bythe class of all s-convex functions.
Obviously one can see that if we choose s = 1, the above definition reduces to ordinary concept of convexity.
In , Dragomir defined convex functions on the co-ordinates as following:
for all (x, y), (z, w) ∈ Δ and λ ∈ [0, 1].
In , Dragomir established the following inequalities of Hadamard-type for co-ordinated convex functions on a rectangle from the plane ℝ2.
The above inequalities are sharp.
In , Alomari and Darus defined co-ordinated s-convex functions and proved some inequalities based on this definition. Another definition for co-ordinated s- convex functions of second sense can be found in .
holds for all (x, y), (z, w) ∈ Δ with λ ∈ [0, 1] and for some fixed s ∈ (0, 1].
In , Sarıkaya et al. proved some Hadamard-type inequalities for co-ordinated convex functions as following:
where A, J are as in Theorem 2 and.
where A, J are as in Theorem 2.
In , Barnett and Dragomir proved an Ostrowski-type inequality for double integrals as following:
for all (x, y) ∈ [a, b] × [c, d].
In , Sarıkaya proved an Ostrowski-type inequality for double integrals and gave a corollary as following:
In , Pachpatte established a new Ostrowski type inequality similar to inequality (1.5) by using elementary analysis.
The main purpose of this article is to establish inequalities of Hadamard-type for co-ordinated convex functions by using Lemma 1 and to establish some new Hadamard-type inequalities for co-ordinated s-convex functions by using Lemma 2.
2. Inequalities for co-ordinated convex functions
To prove our main results, we need the following lemma which contains kernels similar to Barnett and Dragomir's kernels in , (see the article [17, proof of Theorem 2.1]).
for each x ∈ [a, b] and y ∈ [c, d].
Using the change of the variable and , then dividing both sides with (b - a) × (d - c), this completes the proof.
By a simple computation, we get the required result.
which is the inequality in (1.7).
where C is in the proof of Theorem 7.
This completes the proof.
which is the inequality in (1.3) with
where C is in the proof of Theorem 7.
we obtained the desired result.
3. Inequalities for co-ordinated s-convex functions
To prove our main results we need the following lemma:
for some fixed r1, r2 ∈ [0, 1].
Using the change of the variable x = tb + (1 - t) a and y = λd + (1 - λ) c for t, λ ∈ [0, 1] and multiplying the both sides by , we get the required result.
By using these in (3.2), we obtain the inequality (3.1).
- (1)If we choose r 1 = r 2 = 1 in (3.1), we have(3.3)
- (2)If we choose r 1 = r 2 = 0 in (3.1), we have
for some fixed r1, r2 ∈ [0, 1], where.
In above inequality using the s-convexity on the co-ordinates of on Δ and calculating the integrals, then we get the desired result.
- (1)Under the assumptions of Theorem 11, if we choose r 1 = r 2 = 1 in (3.4), we have(3.5)
- (2)Under the assumptions of Theorem 11, if we choose r 1 = r 2 = 0 in (3.4), we have
Remark 4. If we choose s = 1 in (3.5), we obtain the inequality in (1.3)