Ostrowski’s type inequalities for complex functions defined on unit circle with applications for unitary operators in Hilbert spaces

Abstract

Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral R b a f eit du (t) of continuous complex valued integrands f : C (0; 1)! C de ned on the complex unit circle C (0; 1) and various subclasses of integrators u : [a; b] [0; 2 ] ! C of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well. 1. Introduction The problem of approximating the Riemann-Stieltjes integral R b a f (t) du (t) by the quantity f (x) [u (b) u (a)] ; which is a natural generalization of the Ostrowski problem analyzed in 1937 (see [17]), was apparently rst considered in the literature by the author in 2000 (see [9]) where he obtained the following result: [u (b) u (a)] f (x) Z b a f (t) du (t) (1.1)