- Open Access
New Ostrowski-Grüss type inequalities with the derivatives bounded by functions
© Feng and Meng; licensee Springer. 2013
- Received: 2 March 2013
- Accepted: 3 September 2013
- Published: 7 November 2013
In this paper, we establish some new Ostrowski-Grüss type inequalities involving multiple interior points with the first-order derivative bounded by functions instead of constants, some of which provide sharp bounds. Then we establish a new 2D Ostrowski-Grüss type inequality involving multiple interior points with the second mixed partial derivative bounded by functions. For illustrating the applications of the Ostrowski-Grüss type inequalities established, we apply them to derive error bounds for some numerical integration formulae.
- Ostrowski-Grüss type inequality
- numerical integration
- error bound
As is known, the Ostrowski-type inequality  can be used to estimate the absolute deviation of a function from its integral mean, while the Grüss inequality  can be used to estimate the absolute deviation of the integral of the product of two functions from the product of their respective integral. Recently, various generalizations of the Ostrowski inequality and the Grüss inequality have been established (for example, see [3–21] and the references therein). These inequalities can be used to provide explicit error bounds for numerical quadrature formulae such as the Simpson quadrature formula, trapezoid quadrature formula and so on. Among the generalizations, many Ostrowski-Grüss type inequalities have been established [13–21]. Now we list some important results in the literature.
for all , where γ, Γ are two constants such that , .
This inequality is sharp in the sense that the constant cannot be replaced by a smaller one.
where , , are interior points, , , , , , , and γ, Γ are defined as before.
We notice that little attention is paid to the Ostrowski-Grüss type inequalities involving multiple interior points with bounded by functions instead of constants so far in the literature. So, in this paper, motivated by the above works, we extend the Ostrowski-Grüss type inequalities to the case involving multiple interior points with the bounds of shown as , . Some bounds will be derived based on the inequalities, and some of the bounds are sharp. A new 2D Ostrowski-Grüss type inequality will also be derived with the bounds of shown as , , . We also present some applications for the Ostrowski-Grüss type inequalities established, in which new error bounds for some numerical integration formulae are derived.
Lemma 2.1 [, Lemma 2.1]
where . Then .
Combining (8)-(10) and Lemma 2.1, we obtain the desired result. □
Then, combining (7) and (12), we get the desired inequality (11). □
Next we present a sharp Ostrowski-Grüss type inequality containing multiple interior points with bounded by functions as follows.
where is defined as in Lemma 2.2.
The inequality (13) is sharp in the sense that the constant on the right-hand side cannot be replaced by a smaller one.
Combining (14) and (15), we obtain the desired inequality (13).
So, (15) holds in the equality form, which confirms the proof. □
where , .
Remark 2.2 Under the conditions of Corollary 2.2, furthermore, assume that the conditions of [, Th. 2.4] hold. Then, proceeding in the same manner as the proof in [, Eqs. (13)-(14)], we obtain . So, Corollary 2.2 provides better bound than the inequality in (2) [, Th. 2.4].
In the following, we extend the result in Theorem 2.1 to 2D case, in which is bounded by two functions.
Combining (22)-(24), we get the desired inequality (20). □
Combining (20) and (26), we obtain the desired result. □
Remark 2.3 In Corollary 2.3, if we take , , , , , , , , , , , , , , , then Corollary 2.3 becomes the 2D Simpson-type inequality.
In this section, we present some applications of the results established above, and derive error bounds for some numerical quadrature formulae.
Combining (30), (32) and (33) and a summation with respect to i from 0 to yield (29). □
Proof Considering and , from (29) we can obtain (33). □
The desired result can be obtained by a suitable application of Theorem 2.3 (with replaced by , replaced by , , , , , , , , , , , , , , , ), and a summation with respect to i, j from 0 to .
In Theorem 3.2, after a simple computation, one can see . So, we obtain the following corollary.
We have established some new Ostrowski-Grüss type inequalities involving multiple interior points with the derivatives bounded by functions in both 1D and 2D cases and derived some sharp bounds related to them. As one can see, the inequalities are of new forms and provide better bounds than some existing results in the literature. As for applications, new error bounds for some numerical quadrature formulae are derived based on the inequalities established.
The authors would like to thank the referees very much for their valuable suggestions on improving this paper.
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