1995/12/01 by George A. Anastassiou · 5 citations
Mathematics · #Mathematical Inequalities and Applications #Mathematical Approximation and Integration
paper · doi:10.1090/s0002-9939-1995-1283537-3
Optimal upper bounds are given to the deviation of a function / 6 CN([a, b]), N 6 N, from its averages.These bounds are of the form A ' H/^'lIco i where A is the smallest universal constant, i.e., the produced inequalities are sharp and sometimes are attained.This work has been greatly motivated by the works of Ostrowski (1938) and Fink (1992)./ f(y)dy-f(x Ja ■ 1 , (*-*?) 1 4+ (b-a)2 where / e C ([a, b]), x £ [a, b].Inequality (1) is sharp since the function in ( ) cannot be replaced by a smaller one.One can easily notice that (2) fv*-t)2V(»-«)=(*-a>:+(*-*)2 (b-a)2 / v "' 2-(b-a)