tailieunhanh - A generalized trapezoid inequality for functions of bounded variation

We establish a generalization of a recent trapezoid inequality for functions of bounded variation. A number of special cases are considered. Applications are made to quadrature formulae, probability theory, special means and the estimation of the beta function. | Turk J Math 24 (2000) , 147 – 163. ¨ ITAK ˙ c TUB A Generalized Trapezoid Inequality for Functions of Bounded Variation P. Cerone, S. S. Dragomir, C. E. M. Pearce Abstract We establish a generalization of a recent trapezoid inequality for functions of bounded variation. A number of special cases are considered. Applications are made to quadrature formulæ, probability theory, special means and the estimation of the beta function. Key Words: Trapezoid inequality, bounded variation, numerical integration, beta function. 1. Introduction In [1], Dragomir proved the following trapezoid inequality for functions of bounded variation. Here and subsequently in the paper, if f is of bounded variation on [a,b], we denote its total variation on that interval by b W (f). a Theorem A. Let f : [a, b] → R be of bounded variation on [a, b]. Then b Z b _ 1 f (a) + f (b) ≤ (b − a) (f) . f (t) dt − (b − a) 2 2 a () a The constant 1/2 is best possible. We introduce the notation In : a = x0 0, that is, b a + b _ (f) |J(x)| ≤ c (b − a) + x − 2 a for all x ∈ [a, b] . For x = (a + b)/2, we get b Z b _ f (a) + f (b) ≤ c (b − a) (f) . f (t) dt − (b − a) 2 a () a Define f : [a, b] → R by 0 1 f (x) = 0 if x = a if x ∈ (a, b) if x = b. Then f is of bounded variation on [a, b] and Zb f (x) dx = b − a, b _ (f) = 2. a a For this choice of f, () provides b − a ≤ 2c (b − a) or c ≥ 1/2, concluding the proof. 150 2 CERONE, DRAGOMIR, PEARCE Remark 1 a) The choice x = b supplies the “left rectangle” inequality Zb b _ f (x) dx − f (a) (b − a) ≤ (b − a) (f) . a a b) Setting x = a yields the “right rectangle” inequality b Z b _ f (x) dx − f (b) (b − a) ≤ (b − a) (f) . a a c) For x = (a + b)/2 we obtain the known “trapezoid” inequality (). This is the best possible inequality we can derive from () in the sense that the constant 1/2 is .