General Opial Type Inequality and New Green Functions

In this paper we provide many new results involving Opial type inequalities. We consider two functions—one is convex and the other is concave—and prove a new general inequality on a measure space (Ω, Σ, μ). We give an new result involving four new Green functions. Our results include Grüss and Ostrowski type inequalities related to the generalized Opial type inequality. The obtained inequalities are of Opial type because the integrals contain the function and its integral representation. They are not a direct generalization of the Opial inequality.


Introduction
First, we start with the Opial inequality. Opial [1] proved in 1960 the following inequality: If f ∈ C 1 [0, h] is such that f (0) = f (h) = 0 and f (x) > 0 for x ∈ (0, h), then: where h/4 is the best possible. This inequality has been generalized and extended in many different directions (for more details see e.g., [2][3][4][5][6][7][8][9]). Now we continue with the following result. In 2009, Krulić, K. et al., in [10] , observed two measure spaces (Ω 1 , Σ 1 , µ 1 ), (Ω 2 , Σ 2 , µ 2 ) and the general integral operator A k defined by: where f : Ω 2 −→ R is a measurable function, k : Ω 1 × Ω 2 → R is measurable and non-negative, and The authors proved the weighted inequality by using Jensen's inequality and Fubini's theorem. Their result is: where u : Ω 1 → R is a non-negative measurable function, x → u(x) k(x,y) K(x) is integrable on Ω 1 for each fixed y ∈ Ω 2 , v is defined on Ω 2 by v(y) = Φ is a convex function on an interval I ⊆ R, and f : Ω 2 → R is such that f (y) ∈ I for all y ∈ Ω 2 . We mention that inequality (4) unifies and generalizes many of the results of this type (including the classical ones by Hardy, Hilbert and Godunova).
In the sequel, let (Ω, Σ, µ) be a measure space and let k : Ω × Ω → R be a symmetric non-negative or nonpositive function such that K(x) is defined by: K(x) := Ω k(x, y)dµ(y), K(x) = 0, a.e. x ∈ Ω, (6) and |K(x)| < ∞. In the rest of the paper we assume that all integrals are well defined. We continue with the following result that is given in [11].
Theorem 1. Let k : Ω × Ω → R be a symmetric nonpositive or non-negative function. If f is a positive convex function, and g a positive concave function on an interval I ⊆ R, v : Ω → R is either nonpositive or non-negative, such that Im|v| ⊆ I and u is defined by: The following inequality: holds, where K is defined by (6).
In our main results, we will use the following generalized Montgomery identity: 12]). Let n ∈ N, φ : I → R be such that φ (n−1) is absolutely continuous, I ⊂ R an open interval, α, β ∈ I and α < β. Then the following identity holds: where In case n = 1 the sum ... is empty, so the identity (9) reduces to the well-known: Montgomery identity where P(x, s) is the Peano kernel, defined by: Now we recall the definition of new Green functions. For any function φ : [α, β] → R, φ ∈ C 2 ([α, β]), we can easily show by integrating by parts that the following is valid: where the function G 1 : [α, β] × [α, β] → R is Green's function of the boundary value problem and is defined by: The function G 1 is convex under u and s, it is a symmetric nonpositive function and it is continuous under s and continuous under u.
Here we give three new types of Green's functions defined on [α, β] × [α, β] as follows: All three functions are continuous, symmetric and convex with respect to both variables u and s. Lemma 1. Let G k (·, s), s ∈ [α, β], k = 2, 3, 4 be defined by (13)- (15). Then for every function φ ∈ C 2 ([α, β]), it holds that: In paper [11], you can see results involving the Green function defined by: Motivated by those results we give general Opial type inequalities. The new inequalities are not direct generalizations of the Opial inequality. They are of Opial type because the integrals contain function and its integral representation. There are many papers involving Green functions; here we mention only a few of them. In [13] you can find results involving Sherman's inequality and new Green's functions. Here we also mention new results about Hilbert-type inequalities; see [14][15][16]. This paper is organized in the following way: after the Introduction, where we recall the original Opial inequality from 1960 and also provide newer results involving two measure spaces, Section 2 follows. There we give our main results. They contain two functions-convex and concave-and four new Green's functions. In this section there are many new results, six new Theorems and many new Corollaries. In Section 3, titled Grüss and Ostrowski type inequalities related to the generalized Opial type inequality, we also provide many new results. We conclude our paper with the Discussion.

The Main Results
We give our first result, which involves two functions, one positive convex and the other a positive concave function.
holds for all nonpositive or non-negative functions φ : Proof. Function G 1 defined by (12) is a nonpositive symmetric function so we can apply and inequality (8) becomes (20) so the proof is complete.
Now we give a special case of Theorem 3 for φ(a) = φ(b) = 0.
We continue with the other three new Green's functions. We will give the result without the proof since the proof is similar to the proof of Theorem 3.

Corollary 1.
If f is a positive convex function and g is a positive concave function on an interval I ⊆ R, then the following inequalities: The results given in Theorem 3 and Corollary 1 are new. Similar results can be found in paper [11].
We continue with the following result.  (7), K(x) is defined by (6) and G 1 (., s) is defined by (12). Then the following result follows: Proof. For every function f ∈ C 2 ([a, b]), the following is valid: where G 1 is Green's function defined by (12). Now we insert (26) to (8) and we get: Now we rearrange the integrals and get (25).
We continue with analogue results with three other Green functions.

Remark 3.
If f (a) = f (b) = 0 the inequalities (29)-(31) reduce to: We continue with the following result. It holds only for G 3 and G 4 since they are non-negative functions.
Theorem 5. If f is a positive convex function f ∈ C 2 ([a, b]) such that f (a) = f (b) = 0, g a positive concave function on an interval [a, b] ⊆ R, v : Ω → R is either non-negative or nonpositive such that Im|v| ⊆ [a, b], u is defined by (7), K(x) is defined by (6) and G i (., s), i = 3, 4 are defined by (14) and (15) then the following statements are equivalent: Proof. We only give the proof for i = 3; the proof for i = 4 is similar.
Notice that in Theorem 6 we calculated, under some conditions, the difference between the right-hand and left-hand side of inequality 8.

Theorem 7.
Suppose that all assumptions of Theorem 6 hold. Let for even n the function f : I → R be n-convex and Then the following inequalities hold: Proof. (i) Since the function f is n-convex, we have f (n) ≥ 0. It is also obvious that if n is even thenT n−2 ≥ 0 because: Case I : If a ≤ t ≤ s, then s − t ≥ 0 and hence (s−t) n−2 n−2 ≥ 0. Also (s − a) ≥ 0 and (s − t) n−3 ≥ 0. So in this case from (39) we haveT n−2 ≥ 0.
Case II : If s < t ≤ b, then (s − t) n−3 and (t − b) are non positive. As n is even so we have (t − b)(s − t) n−3 ≥ 0; also (s−t) n−2 n−2 ≥ 0. So in this case from (39) we havẽ T n−2 ≥ 0. Now using (45) and the positivity ofT n−2 and f (n) in (38) we get (46); (ii) The proof is similar to the proof of part (i).
We continue with the last result in this section. Theorem 8. Let n ∈ N, n ≥ 4, f : I → R be such that f (n−1) is absolutely continuous, I ⊂ R an open interval, a, b ∈ I, a < b and f (a) = f (b) = 0. Let g be a positive concave function on an interval [a, b] ⊆ R, v : Ω → R is either non-negative or nonpositive, such that Im|v| ⊆ [a, b], u is defined by (7), K(x) is defined by (6) and G i (., s), i = 1, 2, 3, 4 are defined by (12)- (15). If n is even and f is an n-convex function, then (46) and (47) hold. Moreover, if (46) and (47) hold and the functions defined by: where ω = 1, 2, 3, 4 are convex on [a, b], then Proof. Since the functions G w (., t), w ∈ {1, 2, 3, 4}, t ∈ [a, b], are convex, so it holds that . Applying Theorem 6, we obtain (46) and (47). Since (46) holds, the right hand side of (46) can be rewritten in the form: where L 1 is defined by (49). Since L 1 is convex, therefore by Theorem 1 we have: i.e., the right hand side of (46) is non-negative, so the inequality (8) immediately follows. Similarly, we may get (8) by using the convexity of L 2 .

Grüss and Ostrowski Type Inequalities Related to the Generalized Opial Type Inequality
Cerone et al. [17] consideredČebyšev functional for Lebesgue integrable functions f , g : [α, β] → R, proving the following two results which contain the Grüss and Ostrowski type inequalities. Then The constant 1 √ 2 in (52) is the best possible.
The constant 1 2 in (53) is the best possible.
Using the previous two theorems we obtain upper bounds for the identities related to generalizations of the Opial type inequality.
We recall that the symbol L p [a, b] (1 ≤ p < ∞) denotes the space of p−power integrable functions defined on the interval [a, b], equipped with the norm |φ(t)|. Theorem 12. Let n ∈ N, n ≥ 4, f : [a, b] → R be such that f (n) is monotonic nondecreasing on [a, b] and let P 1,w , P 2,w , w = 1, 2, 3, 4, be defined as in (54) and (55) respectively. Then: (i) The remainder κ 1 ( f ; a, b) defined by (57) satisfies the estimation (ii) The remainder κ 2 ( f ; a, b) defined by (60) satisfies the estimation Proof. (i) Since Applying Theorem 10 for f → P 1,w , g → f (n) and usingČebyšev functional, we get: We calculated In the following theorem we present Ostrowski type inequality related to generalizations of Opial's inequality.
present new results involving Green functions and Hardy's inequality. Results proved in this paper are theoretical but we are open to all suggestions involving applications and further investigation.