New ？eby？ev Type Inequalities and Applications for Functions of Self-Adjoint Operators on Complex Hilbert Spaces

Several new error bounds for the ？eby？ev functional under various assumptions are proved. Applications for functions of self-adjoint operators on complex Hilbert spaces are provided as well. 1. Introduction In recent years the approximation problem of the Riemann-Stieltjes integral via the famous ？eby？ev functional increasingly became essential. In 1882, ？eby？ev [1] derived an interesting result involving two absolutely continuous functions whose first derivatives are continuous and bounded and is given by and the constant is the best possible. In 1935, Grüss [2] proved another result for two integrable mappings , such that and ; the inequality holds, and the constant is the best possible. In [3, p 302] Beesack et al. have proved the following ？eby？ev inequality for absolutely continuous functions whose first derivatives belong to spaces: where , , and . For the constant we have for all , . Furthermore, we have the following particular cases in (4).(1)If , we have (2)If , we have In 1970, Ostrowski [4] has proved the following combination of the ？eby？ev and Grüss results: where is absolutely continuous with and is Lebesgue integrable on and satisfying , for all . The constant is the best possible. In 1973, Lupa？ [5] has improved Beesack et al. inequality (7), as follows: provided that , are two absolutely continuous functions on with , where . The constant is the best possible. More recently, and using the identity ([3], page 246), Dragomir [6] has proved the following inequality. Theorem 1. Let be of bounded variation on and a Lebesgue integrable function on ; then where denotes the total variation of on the interval . The constant is best possible in (12). Another result when both functions are of bounded variation was considered in the same paper [6], as follows. Theorem 2. If are of bounded variation on , then The constant is best possible in (13). Many authors have studied the functional (1) and, therefore, several bounds under various assumptions have been obtained; for more new results and generalizations the reader may refer to [6–21]. On other hand and in order to study the difference between two Riemann integral means, Barnett et al. [22] have proved the following estimates. Theorem 3. Let be an absolutely continuous function with the property that ; that is, Then for , we have the inequality The constant in the first inequality and in the second inequality are the best possible. After that, Cerone and Dragomir [23] have obtained the following three results as well. Theorem 4. Let be an absolutely continuous mapping. Then for , we have the