Abstract
Several new inequalities for differentiable coordinated convex and concave functions in two variables which are related to the left side of Hermite Hadamard type inequality for coordinated convex functions in two variables are obtained.
Mathematics Subject Classification (2000): 26A51; 26D15
Keywords:
convex function; coordinated convex function; HermiteHadamard's inequality; Jensen's integral inequality1. Introduction
The following definition is well known in literature:
A function f: , is said to be convex on I if the inequality
holds for all x, y ∈ I and λ ∈ [0, 1].
Many important inequalities have been established for the class of convex functions, but the most famous is the HermiteHadamard's inequality (see for instance [1]). This double inequality is stated as:
where f: a convex function, a, b ∈ I with a < b. The inequalities in (1.1) are in reversed order if f is a concave function.
The inequalities (1.1) have become an important cornerstone in mathematical analysis and optimization and many uses of these inequalities have been discovered in a variety of settings. Moreover, many inequalities of special means can be obtained for a particular choice of the function f. Due to the rich geometrical significance of HermiteHadamard's inequality (1.1), there is growing literature providing its new proofs, extensions, refinements and generalizations, see for example [25] and the references therein.
Let us consider now a 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
holds for all (x, y), (z, w) ∈ Δ and λ ∈ [0, 1].
A modification for convex functions on Δ, which are also known as coordinated convex functions, was introduced by Dragomir [6,7] as follows:
A function f: Δ → ℝ is said to be convex on the coordinates on Δ if the partial mappings f_{y}: [a, b] → ℝ, f_{y}(u) = f(u, y) and f_{x}: [c, d] → ℝ, f_{x}(v) = f(x, v) are convex where defined for all x ∈ [a, b], y ∈ [c, d].
A formal definition for coordinated convex functions may be stated as follows:
Definition 1. [8]A function f: Δ → ℝ is said to be convex on the coordinates on Δ if the inequality
holds for all t, s ∈ [0, 1] and (x, u), (y, w) ∈ Δ.
Clearly, every convex mapping f: Δ → ℝ is convex on the coordinates. Furthermore, there exists coordinated convex function which is not convex, (see for example [6,7]). For recent results on coordinated convex functions we refer the interested reader to [6,813].
The following HermiteHadamrd type inequality for coordinated convex functions on the rectangle from the plane ℝ^{2 }was also proved in [6]:
Theorem 1. [6]Suppose that f: Δ → ℝ is coordinated convex on Δ. Then one has the inequalities:
The above inequalities are sharp.
In a recent article [13], Sarikaya et al. proved some new inequalities that give estimate of the difference between the middle and the rightmost terms in (1.2) for differentiable coordinated convex functions on rectangle from the plane ℝ^{2}. Motivated by notion given in [13], in the present article, we prove some new inequalities which give estimate between the middle and the leftmost terms in (1.2) for differentiable coordinated convex functions on rectangle from the plane ℝ^{2}.
2. Main results
The following lemma is necessary and plays an important role in establishing our main results:
Lemma 1. Let f: Δ ⊆ ℝ^{2 }→ ℝ be a partial differentiable mapping on Δ: = [a, b] × [c, d] with a < b, c < d. If , then the following identity holds:
where
Proof. Since
Now by integration by parts, we have
If we make use of the substitutions x = ta + (1  t)b and y = sc + (1  s)d, (t, s) ∈ [0, 1]^{2}, in (2.3), we observe that
Similarly, by integration by parts, we also have that
and
Substitution of the I_{1}, I_{2}, I_{3}, and I_{4 }in (2.2) gives the desired identity (2.1).
Theorem 2. Let f: Δ ⊆ ℝ^{2 }→ ℝ be a partial differentiable mapping on Δ:= [a, b] × [c, d] with a < b, c < d. If is convex on the coordinates on Δ, then the following inequality holds:
where
Proof. From Lemma 1, we have
Since is convex on the coordinates on Δ, we have
Substitution of (2.6) in (2.5) gives the following inequality:
Evaluating each integral in (2.7) and simplifying, we get (2.4). Hence the proof of the theorem is complete.
Theorem 3. Let f: Δ ⊆ ℝ^{2 }→ ℝ be a partial differentiable mapping on Δ: = [a, b] × [c, d] with a < b, c < d. If is convex on the coordinates on Δ and p, q > 1, , then the following inequality holds:
where A is as given in Theorem 2.
Proof. From Lemma 1, we have
Now using the wellknown Hölder inequality for double integrals, we obtain
Since is convex on the coordinates on Δ, we have
Also, we notice that
Using (2.11) and (2.12) in (2.10), we obtain
Utilizing the last inequality in (2.9) gives us (2.8). This completes the proof of the theorem.
Now we state our next result in:
Theorem 4. Let f: Δ ⊆ ℝ^{2 }→ ℝ be a partial differentiable mapping on Δ: = [a, b] × [c, d] with a < b, c < d. If is convex on the coordinates on Δ and q ≥ 1, then the following inequality holds:
where A is as given in Theorem 2.
Proof. By using Lemma 1, we have that the following inequality:
By the power mean inequality, we have
Using the fact that is convex on the coordinates on Δ, we get
and hence, we obtain
Therefore (2.15) becomes
Substitution of (2.16) in (2.14), we obtain (2.13). Hence the proof is complete.
Remark 1. Since 2^{p }> p + 1 if p > 1 and accordingly
and hence we have that the following inequality:
and as a consequence we get an improvement of the constant in Theorem 3.
Following theorem is about concave functions on the coordinates on Δ:
Theorem 5. Let f: Δ ⊆ ℝ^{2 }→ ℝ be a partial differentiable mapping on Δ: = [a, b] × [c, d] with a < b, c < d. If is concave on the coordinates on Δ and q ≥ 1, then we have the inequality:
where A is as defined in Theorem 2.
Proof. By the concavity of on the coordinates on Δ and power mean inequality, we note that the following inequality holds:
for all x, y ∈ [a, b], λ ∈ [0, 1] and for fixed v ∈ [c, d].
Hence,
for all x, y ∈ [a, b], λ ∈ [0, 1] and for fixed v ∈ [c, d].
Similarly, we can show that
for all z, w ∈ [c, d], λ ∈ [0, 1] and for fixed u ∈ [a, d], thus is concave on the coordinates on Δ.
It is clear from Lemma 1 that
Since is concave on the coordinates, we have, by Jensen's inequality for integrals, that:
In a similar way, we also have that
and
By making use of (2.19)(2.22) in (2.18), we get the desired result. This completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
MAL and SSD carried out the design of the study and performed the analysis. Both of the authors read and approved the final version of the manuscript.
Acknowledgements
This article is in final form and no version of it will be submitted for publication elsewhere.
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