The Unified Treatment of Some Inequalities of Ostrowski and Simpson Types
by
Vera Čuljak
Faculty of Civil Engineering, University of Zagreb
Coauthors: C.E.M. Pearce (Applied Mathematics Department,University of Adelaide), J. Pečarić (Faculty of Textile Technology, University of Zagreb)

Recently, a variety of inequalities of Ostrowski and Simpson types have been obtained which make assumptions only on f and its first derivative. Here we shall give an unified treatment of these, deriving new generalizations and subsuming a variety of known results as particular cases. We consider inequalities for lipschitzian functions, functions of bounded variation and functions with L_p derivative.

We denote

I*: a <= t1 < t2 < t3 <= b,

R(f, I*) = ó õ b a  f(x)dx-[(t1-a)f(a)+(b-t3)f(b)+(t3-t1)f(t2)].

Our basic result for functions with first derivative belonging to L_p[a, b] is as follows.

Let f:[a, b] subset or equal R --> R be differentiable. If \nu > 1, \frac1\nu+\frac1\mu=1 and f^' in L_\nu[a, b], then we have the inequality

|R(f, I*)| <= [ \frac(t1-a)\mu+1+(t2-t1)\mu+1+(t3-t2)\mu+1+(b-t3)\mu+1\mu+1] \frac1\mu||f'||\nu. \notag

Date received: March 13, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadz-63.