Straightforward Proofs of Ostrowski Inequality and Some Related Results

It is known as the Hadamard inequality. History of this inequality begins with the papers of Hermite [1] and Hadamard [2] in the years 1883–1893 (see [3]). A rich literature of mathematical inequalities is due to convex functions equally determined by the Hadamard inequality, inspired from this inequality many closely related results have been established which have their applications in approximately all fields of mathematical analysis up to some extent (see, [3–8]). In 1935 Grüss proved an inequality well known as the Grüss inequality stated in the following theorem [9]. Theorem 2. Let f, g : [a, b] → R be integrable functions such that φ ≤ f(x) ≤ Φ and γ ≤ g(x) ≤ Γ for all x ∈ [a, b], where φ, Φ, γ, Γ are constants. Then we have 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 1 b − a ∫ b a f (x) g (x) dx − 1 b − a ∫ b a f (x) dx ⋅ 1 b − a ∫ b a g (x) dx󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 1 4 (Φ − φ) (Γ − γ) , (3)


Introduction
A function is convex on an interval [ , ] if for all 1 , 2 ∈ [ , ] where 0 ≤ ≤ 1, The most classical result for convex functions can be seen in the following theorem. Theorem 1. Let : → R be a convex function defined on the interval of real numbers and , ∈ with < . Then following double inequality holds.
It is known as the Hadamard inequality. History of this inequality begins with the papers of Hermite [1] and Hadamard [2] in the years 1883-1893 (see [3]). A rich literature of mathematical inequalities is due to convex functions equally determined by the Hadamard inequality, inspired from this inequality many closely related results have been established which have their applications in approximately all fields of mathematical analysis up to some extent (see, [3][4][5][6][7][8]).
In 1935 Grüss proved an inequality well known as the Grüss inequality stated in the following theorem [9].
where the constant 1/4 is sharp.
In 1938, Ostrowski established the following inequality known as the Ostrowski inequality stated. . Several quadrature rules of numerical integration have been estimated using Ostrowski and Ostrowski-Grüss type inequalities (see [4,5,8,10,12]). In [4] Cerone and Dragomir have estimated differences of the Hadamard inequality as follows.
Theorem 4. Suppose that : [ , ] → R be a twice differentiable function on ( , ) and suppose that ≤ ( ) ≤ Γ for all ∈ ( , ). Then we have the double inequality: Theorem 5. Under the assumptions of Theorem 4 we have Ujević in [8] also estimated differences of the Hadamard inequality.

Theorem 7. Under the assumptions of Theorem 6 we have
The aim of this paper is in fact to establish proof of well known Ostrowski inequality in a very straightforward way, and to establish bounds of a difference of the Hadamard inequality given in [4,8] in very simple way, here there is no need to define a two variable kernel. In the last by involving a parameter a similar but general result have been found and some particular bounds of a difference of the Hadamard inequality are calculated, also an Ostrowski-Grüss type inequality is obtained by elementary calculation.

Some Alternative Proofs
First we give a proof of well-known Ostrowski inequality, and then proofs of Theorems 4 and 7 are given.

Proof of Theorem 3
Proof. It is clear that Integrating by parts we have Also Integrating by parts we have By adding (10) and (12) one has On the other hand using positivity of ( − )( ( ) + ) and From which one can have From inequalities in (13) and (15) we have Using the following identity one can get inequality in (4) International Journal of Analysis 3

Some Related Results
In this section we give some more results in a very simple way. First by involving a parameter, we prove a general result that provides bounds of a nonnegative difference of the Hadamard inequality and gives particular bounds, and then an Ostrowski-Grüss type inequality is proved. Proof. It is clear that From which one has Further we can say for some > 0 Adding From (32) and (34) we have the required inequality.
Corollary 9. If one selects, for example, = 24 and = 48 in Theorem 8, then In the following, adopting the pattern of proofs we give the following Ostrowski-Grüss type inequality. It is remarkable to mention here that in [11] Cheng has proved an improved result adopting a comparatively different method.