Some Improvements of the Hermite–Hadamard Integral Inequality

: We propose several improvements of the Hermite–Hadamard inequality in the form of linear combination of its end-points and establish best possible constants. Improvements of a second order for the class Φ ( I ) with applications in Analysis and Theory of Means are also given.


Introduction
A function h : I ⊂ R → R is said to be convex on a non-empty interval I if the inequality holds for all x, y ∈ I. If the inequality (1) reverses, then h is said to be concave on I [1].
Let h : I ⊂ R → R be a convex function on an interval I and a, b ∈ I with a < b. Then This double inequality is well known in the literature as the Hermite-Hadamard (HH) integral inequality for convex functions. It has a plenty of applications in different parts of Mathematics; see [2,3] and references therein.
If h is a concave function on I then both inequalities in (2) hold in the reversed direction.
Our task in this paper is to improve the inequality (2) in a simple manner, i.e., to find some constants p, q; p + q = 1 such that the relations hold for any convex h.
It can be easily seen that the condition is necessary for (3) to hold for an arbitrary convex function.
Take, for example, f (t) = Ct, C ∈ R. Since and, analogously, it follows that the inequality of the form (3) represents a refinement of Hermite-Hadamard inequality (2) for each p, q > 0, p + q = 1.
Note also that the linear form p h(a)+h(b) 2 holds for some p = p 0 , then it also holds for each p ∈ [p 0 , 1].
In the sequel we shall prove that the value p 0 = 1/2 is best possible for above inequality to hold for an arbitrary convex function on I.
Also, it will be shown that convexity/concavity of the second derivative is a proper condition for inequalities of the form (3) to hold (see Proposition 5 below).
This condition enables us to give refinements of second order and to increase interval of validity to p 0 = 1/3 as the best possible constant. In this case, coefficients p 0 = 1/3, q 0 = 2/3 are involved in the well-known form of Simpson's rule, which is of great importance in Numerical Analysis. Our results sharply improve Simpson's rule for this class of functions (Proposition 4).
Finally, we give some applications in Analysis and Numerical Analysis. Also, new and precise inequalities between generalized arithmetic means and power-difference means will be proved.

Results and Proofs
We shall begin with the basic contribution to the problem defined above. Theorem 1. Let h : I ⊂ R → R be a convex function on an interval I and a, b ∈ I. Then The constants p 0 = q 0 = 1/2 are best possible.
If h is a concave function on I then the inequality is reversed.
Proof. We shall derive the proof by Hermite-Hadamard inequality itself. Indeed, applying twice the right part of this inequality, we get Summing up those inequalities the result appears. Therefore, HH inequality has this self-improving property.
That the constants p 0 = q 0 = 1/2 are best possible becomes evident by the example For the second part, note that concavity of f implies convexity of − f on I. Hence, applying (5) we get the result.
For the sake of further refinements, we shall consider in the sequel functions from the class C (m) (I), m ∈ N i.e., functions which are continuously differentiable up to m-th order on an interval I ⊂ R.
Of utmost importance here is the class Φ(I) of functions which second derivative is convex on I. For this class we have the following Theorem 2. Let φ ∈ Φ(I) and the inequality holds for a, b ∈ I. Then p Since this inequality should be valid for each a, b ∈ I, a < b, let b → a. We obtain that Indeed, applying L'Hospital's rule 3 and 4 times to the above quotient, we get . Therefore, the result follows.
In the sequel we shall give sharp two-sided bounds of second order for inequalities of the type (3) involving functions from the class Φ with p ≥ 1/3.
Main tool in all proofs will be the following relation.

Lemma 1.
For an integrable function φ : I → R and arbitrary real numbers p, q; p + q = 1, we have the identity Proof. It is not difficult to prove this identity by double partial integration of its right-hand side.
Note that, if φ is concave on I, then the function ξ(a, b; t) is monotone increasing and the inequality (7) is reversed.
Main results of this paper are given in the next two assertions.
Proof. If p ≥ 1/2 we have that 2p − t ≥ 0. Therefore, applying Lemma 1 and the second part of Lemma 2, we obtain In the case 1/3 ≤ p < 1/2, write and apply Lemma 2 to each integral separately.
It follows that which is equivalent to the stated assertion.
For p ≤ 0 we have that 2p − t ≤ 0 and the proof develops in the same manner.
Above theorems are the source of a plenty of important inequalities which sharply refine Hermite-Hadamard inequality for this class of functions.
Some of them are listed in the sequel.
Proof. Put p = 0 in the above theorems.
The next assertion represents a refinement of Theorem 1 in the case of convex functions.

Proposition 3.
Let φ ∈ Φ(I). Then for each a, b ∈ I, a < b, If φ is concave on I, then Proof. Put p = 1/2 in Theorems 3 and 4. The second part follows from a variant of Lemma 2 for concave functions.
Note that the coefficients p = 1/3 and q = 2/3 are involved in well-known Simpson's rule which is of importance in numerical integration [5].
The next assertion sharply refines Simpson's rule for this class of functions.
If φ is concave on I, then Proof. Applying Theorems 3 and 4 with both parts of Lemma 2 for p = 1/3, the proof follows.
The next assertion gives a proper answer to the problem posed in Introduction.

Proposition 5.
If φ is a convex and φ is a concave function on I, then Analogously, let φ be concave and φ a convex function on I, then Proof. Combining Proposition 4 with the results of Theorem 1, we obtain the proof.

Applications in Analysis
Theorems proved above are the source of interesting inequalities from Classical Analysis. As an illustration we shall give here a couple of Cusa-type inequalities.
Another application can be obtained by integrating both sides of (8) on the range x ∈ [0, a], 0 < a < π/2.
By the power series expansion, we know that Hence, This estimation is effective for small values of a.

Applications in Theory of Means
A mean M(a, b) is a map M : R + × R + → R + , with the property min{a, b} ≤ M(a, b) ≤ max{a, b}, for each a, b ∈ R + .
Some refinements of HH inequality by arbitrary means is given in [6].
An ordered set of elementary means is the following family, are the harmonic, geometric, logarithmic, identric, arithmetic and Gini mean, respectively.
Generalized arithmetic mean A α is defined by Power-difference mean K α is defined by It is well known that both means are monotone increasing with α and, evidently, As an illustration of our results, we shall give firstly some sharp bounds of power-difference means in terms of the generalized arithmetic mean. Theorem 6. For a, b ∈ R + and α ≥ 1, we have For α < 1 the inequality (9) is reversed.
Proof. Let g α (t) = t 1/α , α = 0. Since g α is concave for α ≥ 1, Theorem 1 combined with the HH inequality gives 1 2 x + y 2 1/α Now, simple change of variables x = a α , y = b α yields the result. For the second part, note that g α is convex for α < 1 and repeat the procedure.
Remark 2. Note that the above inequalities are so precise that in critical points for α = 1/3, 1/2, 1 we have equality sign.
An inequality for the reciprocals follows.