Ostrowski Type Inequalities for Higher-Order Derivatives
© M.Wang and X. Zhao. 2009
Received: 12 February 2009
Accepted: 14 July 2009
Published: 4 August 2009
This paper has shown some new Ostrowski type inequalities involving higher-order derivatives. The results generalized the Ostrowski type inequalities. Applications of the inequalities are also given.
1. Main Result and Introduction
The following inequality is well known in literature as Ostrowski's integral inequality.
Moreover the constant 1/4 is the best possible. Because Ostrowski's integral inequality is useful in some fields, many generalizations, extensions, and variants of this inequality have appeared in the literature; see [1–9] and the references given therein. The main aim of this paper is to establish some new Ostrowski type inequalities involving higher-order derivatives. The analysis used in the proof is elementary. The main result of this paper is the following inequality.
As applications of the inequality (1.2), we give more Ostrowski type inequalities.
2. The Proof of Theorem 1.1
In this section, we use the Taylor expansion to prove Theorem 1.1. Before the proof, we need the following lemmas.
From (2.2) and (2.3), we know that (2.1) holds.
Therefore we get the inequality (2.4).
Now, we give the proof of Theorem 1.1.
Using Lemmas 2.1 and 2.2 gives (1.2). Thus, we complete the proof.
3. Some Applications
In this section, we show some applications of the inequality (1.2). In fact, we can use (1.2) to derive some new Ostrowski type inequalities.
Substituting (3.6) into (3.1) gives (3.5).
Substituting (3.13) and (3.14) into (3.12) gives (3.7).
The author would like to express deep appreciation to the referees for the helpful suggestions. Mingjin Wang was supported by STF of Jiangsu Polytechnic University.
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