On some refinements of companions of Fejér’s inequality via superquadratic functions

In this paper some companions of Fejér’s inequality for superquadratic functions are given, we also get refinements of some known results proved in [18]. Subjclass : [2000] 26D15.


Introduction
Let ∅ 6 = I ⊆ R, a, b ∈ I with a < b, let f : I → R be a convex function and p : [a, b] → R be a non-negative integrable and symmetric about x = a+b 2 .The following two inequalities are of great significance in literature: the first known as Hermite-Hadamard inequality: with the reversed inequality for the concave function f , and the second, known as Fejér's inequality: These inequalities attracted the attention of many mathematicians over the decades and they generalize, improve and extend these inequalities in a number of ways, see [6,7,8,9,11,19].Let us now define some mappings and quote the results established by K.L. Tseng, S. R. Hwang and S.S. Dragomir in [18]: Companions of Fejér's Inequality For Superquadratic Functions 311 and S p (t) = 1 4 where f : [a, b] → R is a convex function and p : [a, b] → R is non-negative integrable and symmetric about x = a+b 2 , t ∈ [0, 1].Now we quote some results from [18]: Theorem 1. [18] Let f , p, I be defined as above.Then: 1.The following inequality holds: then for all t ∈ [0, 1] we have the inequality where then for all t ∈ [0, 1] we have the inequality (1.5) Theorem 2.
[18] Let f , p, G, I be defined as above.Then: 1.The following inequality holds for all t ∈ [0, 1] : we have the inequality where Theorem 3.
[18] Let f , p, G, I, S p be defined as above.Then we have the following results: 1. S p is convex on [0, 1].

The following equality holds:
They used the following Lemma to prove the above results: Let us now recall the definition, some of the properties and results related to superquadratic functions to be used in the sequel.
For examples of superquadratic functions see [2, p. 1049].Theorem 6. [3, Theorem 2.3] The inequality holds for all probability measure µ and all non-negative µ-integrable function g, if and only if f is superquadratic.
The following discrete version of the above theorem will be helpful in the sequel of the paper: Lemma 7. [2, Lemma A, p.1049] Suppose that f is superquadratic.Let x r ≥ 0, 1 ≤ r ≤ n, and let x = P n r=1 λ r x r where λ r ≥ 0 and The following Lemma shows that positive superquadratic functions are also convex: 3. If f ≥ 0, then f convex and f (0) = f 0 (0) = 0.
In [4] a converse of Jensen's inequality for superquadratic functions was proved: The discrete version of this theorem is: For recent results on Fejér and Hermite-Hadamard type inequalities for superquadratic functions, we refer interested readers to [4], [5] and [2].In this paper we deal with mappings G(t), I(t), S p (t) and L(t) when f is superquadratic function.In case when superquadratic function f is also non-negative and hence convex we get refinements of some parts of Theorem 1, Theorem 2 and of Theorem 3.

Main Results
In this section we prove our main results by using the same techniques as used in [17] and [2].Moreover, we assume that all the considered integrals in this section exist.
In order to prove our main results we go through some calculations.From Lemma 2 and Theorem 6 for n = 2, we get that and Therefore from (2.1) and (2.2), we have (2.3) Now for 0 ≤ t ≤ 1 2 and 0 ≤ a ≤ x ≤ a+b 2 , we obtain from (2.3) the following inequalities: 3), we observe that

3), we get that
Companions of Fejér's Inequality For Superquadratic Functions 317 holds.Now we are ready to state and prove our main results based on the calculations done above.
Theorem 1.Let f be superquadratic integrable function on [0, b] and p(x) be non-negative integrable and symmetric about x = a+b 2 , 0 ≤ a < b.Let I be defined as above, then we have the following inequalities: Proof.
Using simple techniques of integration and by the assumptions on p, we have Therefore from (2.4), we get that From (2.12), (2.13) and by the change of variable x → x+a 2 , we get (2.9).From (2.5), (2.6) and (2.13), we have 2 Companions of Fejér's Inequality For Superquadratic Functions 319 By the change of variables t → 1 − t and x → x+a 2 in (2.16), we get (2.10).
This completes the proof of the theorem as well. 2 Remark 2. If the superquadratic function f is non-negative and hence convex, then from (2.9) we get refinement of the first inequality of (1.3) in Theorem 1; from (2.10) we get refinement of the middle inequality of (1.3) in Theorem 1 and from (2.11) we get refinement of the last inequality of (1.3) in Theorem 1. where Companions of Fejér's Inequality For Superquadratic Functions 321 , and therefore the proof of the corollary follows directly from the above theorem.2 Remark 4. If the superquadratic function f is non-negative and therefore convex, then the inequalities in Corollary 1 refine the inequalities in (1.3) of Theorem B from [18, p.2 ].
To proceed to our next result, we go again through the similar calculations as given before Theorem 7.
For 0 ≤ a ≤ x ≤ a+b 2 , t ∈ [0, 1], we have that Therefore, by replacing holds.Now we are ready to state and prove our next result based on the above calculations.
Theorem 5. Let f be superquadratic integrable function on [0, b] and p(x) be non-negative integrable and symmetric about x = a+b 2 , 0 ≤ a < b.Let I and G be defined as above, then the following inequality holds for all t ∈ [0, 1] :

Proof.
Using simple techniques of integration and by the assumptions on p, we have that the following identity holds for all t ∈ [0, 1]: Arguing similarly as in obtaining (2.15), by using (2.22) and (2.24), we get that b−a , 0 ≤ a < b and G, H be defined as above.Then for all t ∈ [0, 1], we have the following inequality Proof.This is a direct consequence of the above theorem, since for p 3), we obseve that (2.28) holds.
Theorem 9. Let f be superquadratic integrable function on [0, b] and p(x) be non-negative integrable and symmetric about x = a+b 2 , 0 ≤ a < b.Let S p and G be defined as above, then the following inequality holds for all t ∈ [0, 1] : Proof.By the simple techniques of integration and by the assumptions on p, we have the following identity for all t ∈ [0, 1]: From (2.27), (2.28) and (2.30), we have that holds for all t ∈ [0, 1].From (2.24) and by the change of variable x → a+x 2 , we get from (2.31) that ) and this completes the proof of the theorem as well. 2 Remark 10.The result of the above theorem refines the first inequality of Theorem 3, when superquadratic function f is non-negative and hence convex.
Let G be defined as above, then the following inequality holds for all t ∈ [0, 1] : Therefore the proof of the crollary follows directly from the above theorem.2

Inequalities for differentiable superquadratic functions
In this section we give results when f is a differentiable superquadratic function.Those results give refinements of (1.4) and (1.5) in Theorem 1 and refine (1.7) of Theorem 2 when superquadratic function f is nonnegative and hence convex.Here we quote very important result which will be helpful in the sequel of the paper.
Theorem 1. [14, Theorem 10, p. 5] Let f be superquadratic integrable function on [0, b] and p(x) be non-negative integrable and symmetric about x = a+b 2 , 0 ≤ a < b.Let I be defined as above and let fp be integrable on [a, b], then for 0 ≤ s ≤ t ≤ 1, t > 0, we have the following inequality: Now we state and prove the first result of this section.then the following inequalities hold for all t ∈ [0, 1]: where kpk ∞ = sup Proof.By integration by parts, we have that Using the substitution rules for integration, under the assumptions on p, we have for all t ∈ [0, 1].Now by the assumptions on f , we have that By the assumptions on f and from Lemma 3, we get that Adding these inequalities we get that From (3.1), for s = 0, we have From (3.8) and (3.9), we get (3.3).This completes the proof of the theorem. 2 Now we give our last result and summarize the results related to it in the remark followed by Theorem 12.

2 Remark 6 .Corollary 7 .
dx, for all ∈ [0, 1].By the change of variable x → x+a 2 in (2.25), we get (2.23).This completes the proof of the theorem.If the superquadratic function f is non-negative and hence convex, then the inequality (2.23) represents a refinement of the inequality (1.6) in Theorem 2. Let f be superquadratic integrable function on [0, b], let p(x) = 1

Theorem 2 .
Let f be superquadratic function on [0, b] and p(x) be nonnegative integrable and symmetric about x = a+b 2 , 0 ≤ a < b.Let f be differentiable on [a, b] such that f (0) = f 0 (0) = 0 and p is bounded on [a, b],
Remark 8.If the superquadratic function f is non-negative and hence convex, then the inequality (2.26) represents refinement of the inequality (1.6) in [18, Theorem C, p. 2].