A note on Ostrowski's inequality
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This paper deals with the problem of estimating the deviation of the values of a function from its mean value. We consider the following special cases: (i) the case of random variables (attached to arbitrary probability fields); (ii) the comparison is performed additively or multiplicatively; (iii) the mean value is attached to a multiplicative averaging process. 1. Introduction The inequality of Ostrowski [7] gives us an estimate for the deviation of the values of a smooth function from its mean value. More precisely, if f : [a,b] → R is a differentiable function with bounded derivative, then 2 b x − (a + b)/2 1 f (x) − 1 f (t)dt ≤ + (b − a) f ∞ b−a 4 (b − a)2 a
(O)
for every x ∈ [a,b]. Moreover the constant 1/4 is the best possible. The proof is an application of Lagrangian’s mean value theorem: b b 1 f (x) − 1 f (t)dt = f (x) − f (t) dt b−a a b−a a b 1 f (x) − f (t)dt ≤ b−a a f ∞ b ≤ |x − t |dt b−a a (x − a)2 + (b − x)2 f = ∞ 2(b − a) 1 x − (a + b)/2 2 = + (b − a) f ∞ . 4 b−a
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:5 (2005) 459–468 DOI: 10.1155/JIA.2005.459
(1.1)
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A note on Ostrowski’s inequality
The optimality of the constant 1/4 is also immediate, checking the inequality for the family of functions fα (t) = |x − t |α · (b − a) (t ∈ [a,b], α > 1) and then passing to the limit as α → 1+. It is worth to notice that the smoothness condition can be relaxed. In fact, the Lipschitz class suffices as well, by replacing f ∞ with the Lipschitz constant of f , that is, f (x) − f (y) . f L = sup x−y x= y
(1.2)
The extension to the context of vector-valued functions, with values in a Banach space, is straightforward. Since a Lipschitz function on [a,b] is absolutely continuous, a natural direction of generalization of the Ostrowski inequality was its investigation within this larger class of functions (with refinements for f ∈ L p ([a,b]), 1 ≤ p < ∞). See Fink [2]. Also, several Ostrowski type inequalities are known within the framework of H¨older functions as well as for functions of bounded variation. The problem to estimate the deviation of a function from its mean value can be investigated from many other points of view: (i) by considering random variables (attached to arbitrary probability fields); (ii) by changing the algebraic nature of the comparison (e.g., switching to the multiplicative framework); (iii) by considering other means (e.g., the geometric mean); (iv) by estimating the deviation via other norms (the classical case refers to the sup norm, but L p -norms are better motivated in other situations). The aim of this paper is to present a number of examples giving support to this program. 2. Ostrowski type inequalities for random variables In what follows, X will denote a locally compact metric space and E a Banach space. Theorem 2.1. The following two assertions are equivalent for f : X → E a continuous mapping: (i
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