Some generalizations of the Hermite–Hadamard integral inequality

In this article we give two possible generalizations of the Hermite–Hadamard integral inequality for the class of twice differentiable functions, where the convexity property of the target function is not assumed in advance. They represent a refinement of this inequality in the case of convex/concave functions with numerous applications.


Introduction
A function f : I ⊂ R → R is said to be convex on an nonempty interval I if the inequality holds for all x, y ∈ I. If inequality (1.1) reverses, then f is said to be concave on I [1]. Let f : I ⊂ R → R be a convex function on an interval I and a, b ∈ I. Then This double inequality is known in the literature as the Hermite-Hadamard (HH) integral inequality for convex functions.
It has plenty of applications in most different areas of pure and applied mathematics (see [2][3][4] and the references therein).
As an example we quote an improvement by arbitrary means given in [24]. Let f : I ⊂ R + → R be a convex function and S = S(a, b), T = T(a, b) be some means of positive numbers a, b ∈ I. Then For any means S and T, approximations (1.4) and (1.5) are better than original (1.2).
In this article we investigate the possibility of a form of the Hermite-Hadamard inequality for functions that are not necessarily convex/concave on I. This has already been attempted in [25] where the convexity/concavity of the second derivative was shown to be crucial in managing improvements of the HH inequality as a linear combination of its endpoints.
We derive here two forms of the Hermite-Hadamard inequality under the sole condition that the second derivative of the target function f exists locally on an interval I. Thus, f ∈ C (2) These numbers will play an important role in the sequel.

Results and proofs
We begin with an improved variant of the Hermite-Hadamard inequality.

Lemma 2.1 Let f : I ⊂ R → R be a convex function on an interval I and a, b ∈ I. Then
It is shown in [4] that this improvement is best possible of the form Our first main result is the following.
we find out that f is a convex function on E. Therefore, applying the form of Hermite-Hadamard inequality given by (2.1), we obtain On the other hand, taking the auxiliary function f as f (t) = Mt 2 /2g(t), we see that it is also convex on E.
There are plenty of applications of Theorem 2.2. For instance, an improvement of the assertion from Lemma 2.1 is given in the following.

Corollary 2.3 Let f ∈ C (2) (E). Then
Proof Putting r = 0, s = 1; p = q = 1/2, we get the desired result. Note that m ≥ 0 if f is a convex function on E.
Of great importance in the theory of numerical integration is the so-called Simpson's rule (cf. [26]).
There is a problem how to apply Simpson's rule if f / ∈ C (4) (E). A possible answer for twice differentiable functions is given in the following.
Another refinement of the Hermite-Hadamard inequality is given in the following.
3. In the case p ≤ 0, we have 2pt ≤ 0; φ (x), φ (y) ≥ m. Therefore, Remark 2.9 The approximations from Theorems 2.2 and 2.8 can be compared if r = p, s = q; 0 ≤ p ≤ 1/2. It is not difficult to see that they coincide for p = 0 and p = 1/2. In other cases the second approximation is better.
For example, if p = 1/3, we obtain an improvement of Corollary 2.5, i.e., another estimation of Simpson's rule for twice differentiable functions. We conjecture that the constant 1/162 is best possible.