Abstract

We give proofs of some known results in very simple and antique way. Also we find some general bounds of a nonnegative difference of the Hadamard inequality and an Ostrowski-Grüss type inequality is proved.

1. Introduction

A function is convex on an interval if for all , where ,The most classical result for convex functions can be seen in the following theorem.

Theorem 1. Let 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, [3–8]).

In 1935 Grüss proved an inequality well known as the Grüss inequality stated in the following theorem [9].

Theorem 2. Let be integrable functions such that and for all , where , , , are constants. Then we havewhere the constant is sharp.

In 1938, Ostrowski established the following inequality known as the Ostrowski inequality stated.

Theorem 3. Let , where is an interval in , be a mapping differentiable in the interior of and , . If , for all , then we havefor all .

By using Grüss inequality some results have been established which are well known as Ostrowski-Grüss or Ostrowski-Grüss type inequalities (see [10–12] and references therein). 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 be a twice differentiable function on and suppose that 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 6. Suppose that be a twice differentiable function on and suppose that for all . Then we have where .

Theorem 7. Under the assumptions of Theorem 6 we havewhere .

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.

2. Some Alternative Proofs

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

2.1. Proof of Theorem 3

Proof. It is clear that Integrating by parts we haveAlso Integrating by parts we haveBy adding (10) and (12) one hasOn the other hand using positivity of and we have From which one can haveFrom inequalities in (13) and (15) we have Using the following identity one can get inequality in (4)

2.2. Proof of Theorem 4

Proof. It is clear that Integrating by parts we haveNow using and integrating on we haveAdding (19) and (20) one can haveSimilarly using and integrating on we haveNow using , integrating on we haveAdding (22) and (23) we haveFrom (21) and (24) we have (5).

2.3. Proof of Theorem 7

Proof. It is clear that Integrating by parts we haveSimilarly using and integrating on we haveFrom (26) and (27) we have (8).

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.

Theorem 8. Suppose that be a twice differentiable function on and suppose that for all . Then we have where .

Proof. It is clear that From which one has Further we can say for some Adding (21) and (31) one hasOn the other hand , which gives for Adding (24) and (33) one hasFrom (32) and (34) we have the required inequality.

Corollary 9. If one selects, for example, and 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.

Theorem 10. Let , where is an interval in be a mapping differentiable in , the interior of and , If for all , then we havefor all

Proof. It is clear that and for all Therefore After some computation we haveAlso it is easy to see and for all Therefore After some computation it can be seenBy adding (38) and (40) one hasOn the other hand we have This givesAlso we have which givesAdding (43) and (45), then combining with (41) one can get (36).

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.