tailieunhanh - A perturbed version of the Ostrowski inequality for twice differentiable mappings
A generalisation of a perturbed version of the Ostrowski inequality for twice differentiable mappings is studied. It is shown that the error bounds are better than those obtained previously. Applications for general quadrature formulae are also given. | Turk J Math 25 (2001) , 379 – 412. ¨ ITAK ˙ c TUB A Perturbed Version of the Ostrowski Inequality for Twice Differentiable Mappings A. Sofo, S. S. Dragomir Abstract A generalisation of a perturbed version of the Ostrowski inequality for twice differentiable mappings is studied. It is shown that the error bounds are better than those obtained previously. Applications for general quadrature formulae are also given. Key Words: Ostrowski Integral Inequality, Quadrature Formulae. 1. Introduction The following theorem was proved by Ostrowski [9, p. 469] in 1938. Theorem 1 Let f : [a, b] → R be a differentiable mapping on (a, b) whose derivative is bounded on (a, b) and denote kf 0 k∞ = sup |f 0 (t)| 1, 1 p + 1 q = 1; + ν(h) max |δi | 2 i=0,.,k−1 kf 00 k1 ≤ 38 ν 2 (h) kf 00 k1 , where f 00 ∈ L1 [a, b] . 381 SOFO, DRAGOMIR Here hi := xi+1 − xi , ν (h) : = max {hi |i = 0, ., k − 1} , xi + xi+1 , δi : = αi+1 − 2 ρ (δ) : = max {δi |i = 0, ., k − 1} . Proof. Consider the kernel K : [a, b] → R given by K (t) := (t−α1)2 , 2 t ∈ [a, x1) (t−α2)2 , 2 t ∈ [x1 , x2 ) (t−αk−1 )2 , 2 t ∈ [xk−2, xk−1) (t−αk )2 , 2 t ∈ [xk−1, b]. . Successively integrating by parts, we have that Z b K (t) f 00 (t) dt a Z b f (t) dt + = a − k−1 o 1 Xn 2 2 (xi+1 − αi+1 ) f 0 (xi+1 ) − (xi − αi+1 ) f 0 (xi ) 2 i=0 k X (αi+1 − αi ) f (xi ) i=0 and Z k−1 b o 1 Xn 2 2 f (t) dt + (xi+1 − αi+1 ) f 0 (xi+1 ) − (xi − αi+1 ) f 0 (xi ) a 2 i=0 k X (αi+1 − αi ) f (xi ) − i=0 Z b K (t) f 00 (t) dt . = a 382 (5) SOFO, DRAGOMIR In the first case, consider f 00 ∈ L∞ [a, b], hence Z Z b b 00 00 K (t) f (t) dt ≤ kf k∞ |K (t)| dt. a a Z Z 2 |K (t)| dt = a x1 (t − α1 ) dt + . + 2 Z 2 (t − αk ) dt 2 a xk−1 (k−1 ) i 1 Xh 3 3 (αi+1 − xi ) + (xi+1 − αi+1 ) . = 6 b b i=0 n n n Using the inequality (A − B) + (C − A) ≤ (C − B) , Z b |K (t)| dt .
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