In this paper, a generalized perturbed midpoint rule is estab-lished. Various error bounds for this generalization are also obtained. AMS Mathematics Subject Classification : 26D15, 26D30. In recent years a number of authors have considered error inequalities for some known and some new quadrature rules. Sometimes they have considered generalizations of these rules. For example, the well-known trapezoid and midpoint quadrature rules are considered in ( [1 , 2 , 9 , 10] ). In [2] , we can find where denotes the integer part of and For n = 1, we get the midpoint rule In [11] , a generalized trapezoid rule is derived by Ujevi ć as follows: where In [4] and [8] , the following unified treatment for generalizations of the midpoint, trapezoid, averaged midpoint-trapezoid and Simpson type inequalities is obtained by Liu and Liu, respectively, where θ ∈ [0, 1] and In [5] , Liu established the following generalized perturbed trapezoid rule. where K_n ( x ) is the kernel given by In [7] and [12] , the following generalization of the perturbed midpoint-trapezoid rule is established by Liu and Ujevi ć et al., respectively. Theorem 1.1. Let f : [ a, b ] → ℝ be a function such that f ⁽ⁿ⁻¹⁾ is absolutely continuous on [ a, b ]. Then we have where denotes the integer part of and R ( f ) = (−1) ⁿ Some sharp perturbed midpoint inequalities are proved by Liu in [6] based on the following identity: where and Theorem 1.2 ( [6] ). Let f : [ a, b ] → ℝ be a twice differentiable mapping such that f′′ is integrable with Γ ₂ = sup _x∈(a,b) f′′ ( x ) and γ ₂ = inf _x∈(a,b) f′′ ( x ). Then we have Theorem 1.3 ( [6] ). Let f : [ a, b ] → ℝ be a third-order differentiable mapping such that f′′′ is integrable with Γ ₃ = sup _x∈(a,b) f′′′ ( x ) and γ ₃ = inf _x∈(a,b) f′′′ ( x ). Then we have The purpose of this paper is to extend (2) to a more general version, that is, a generalized perturbed midpoint rule is established. Various error bounds for the generalizations are also given. Theorem 2.1. Let f : [ a, b ] → ℝ be a mapping such that the derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous on [ a, b ] and M_n = sup _x∈(a,b) | f ⁽ⁿ⁾ ( x )| < ∞. Then we have where denotes the integer part of . Proof . It is not difficult to find the identity where S_n ( x ) is the kernel given by Using the above identity, we get Now, we put It is clear that P_n ( x ) and Q_n ( x ) are symmetric with respect to the line for n even and symmetric with respect to the point for n odd. Therefore, By substitution we find that is always negative on [0, 1] for n ≥ 3. Thus for n ≥ 3, and Hence, Consequently, inequalities (9) follow from (11) and (12). Remark . Applying (10) for n = 2, 3 respectively, we get the identity (2). For convenience in further discussions, we collect some technical results which are not difficult to obtain by elementary calculus as: Before we end this section, we introduce the notations Recall that a function f : [ a, b ] → R is said to be L -Lipschitzian on [ a, b ] if for all x, y ∈ [ a, b ],where L > 0 is given, and, it is said to be ( l,L ) -Lipschitzian on [ a, b ] if for all a ≤ x ≤ y ≤ b where l,L ∈ R with l < L . From [3] , we get that if h, g : [ a, b ] → ℝ are such that h is Riemann-integral on [ a, b ] and g is L -Lipschitzian on [ a, b ], then exists and Theorem 3.1. Let f : [ a, b ] → ℝ be a mapping such that derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is ( l,L )- Lipschitzian on [ a, b ]. Then we have Proof . By (10) and (13) we get Then notice that Lipschitzian on [ a, b ] and by using (16), we have Hence, the inequality (17) follows from (16) and (12). Corollary 3.2. Let f : [ a, b ] → ℝ be a mapping such that derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is L - Lipschitzian on [ a, b ]. Then we have Theorem 4.1. Let f : [ a, b ] → ℝ be a mapping such that the ( n -1) th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous on [ a, b ]. If f ⁽ⁿ⁾ ∈ L _∞ [ a, b ], then we have where ∥ f ⁽ⁿ⁾ ∥ _∞ := ess sup _x∈[a,b] |f ⁽ⁿ⁾ ( x )| is the usual Lebesgue norm on L _∞ [ a, b ]. Proof . We can obtain the result by taking L = ∥ f ⁽ⁿ⁾ ∥ _∞ in Corollary 3.2. Theorem 4.2. Let f : [ a, b ] → ℝ be a mapping such that the ( n -1) th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous on [ a, b ]. If f ⁽ⁿ⁾ ∈ L ₁ [ a, b ], then we have where is the usual Lebesgue norm on L ₁ [ a, b ]. Proof . By using the identity (10) we get Then the conclusion follows from (15). Theorem 4.3. Let f : [ a, b ] → ℝ be a mapping such that the ( n -1) th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous on [ a, b ]. If f ⁽ⁿ⁾ ∈ L ₂ [ a, b ], then we have where is the usual Lebesgue norm on L ₂ [ a, b ]. Proof . By using the identity (10) we get Then the conclusion follows from (14). Theorem 5.1. Let f : [ a, b ] → ℝ be a mapping such that the ( n -1) th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous with γ_n ≤ f ⁽ⁿ⁾ ( x ) ≤ Γ _n a.e. on [ a, b ], where γ_n , Γ _n ∈ ℝ are constants, then we have Proof . By (10) and (13) we get then notice that a.e. on [ a, b ], we have We complete the proof from (12). Remark . Applying Theorem 5.1 for n = 2, 3, we get (3), (6), respectively. Theorem 5.2. Let f : [ a, b ] → ℝ be a mapping such that the ( n -1) th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous with γ_n ≤ f ⁽ⁿ⁾ ( x ) ≤ Γ _n a.e. on [ a, b ], where γ_n ∈ ℝ is a constant, then we have where Proof . By (10) and (13) we get then notice that f ⁽ⁿ⁾ ( x ) − γ_n ≥ 0 a.e. on [ a, b ], we have From (15), we get the desired result. Remark. Applying Theorem 5.2 for n = 2, 3, we get (4), (7), respectively. Theorem 5.3. Let f : [ a, b ] → ℝ be a mapping such that the ( n −1) th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous with f ⁽ⁿ⁾ ( x ) ≤ Γ _n a.e. on [ a, b ], where Γ _n ∈ ℝ is a constant, then we have where D_n is defined in Theorem 5.2. Proof . The proof of inequalities (18) is similar to the proof of Theorem 5.2 and so is omitted. Remark . Applying Theorem 5.3 for n = 2, 3, we get (5), (8), respectively. In this section, we derive two sharp error inequalities when n is an odd and an even integer, respectively. Theorem 6.1. Let f : [ a, b ] → ℝ be a mapping such that the ( n −1) th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous on [ a, b ]. If f ⁽ⁿ⁾ ∈ L ₂ [ a, b ] and n is an odd integer. Then we have where σ(·) is defined by Inequality (19) is the best possible in the sense that the constant can not be replaced by a smaller one. Proof . By using the identity (10) and (13) we get To prove the sharpness of (19), we suppose that (19) holds with a constant C > 0 as We may find a function f : [ a, b ] → ℝ such that the ( n − 1)th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous on [ a, b ] as It follows that Then we can find that the left-hand side of inequality (20) becomes and the right-hand side of inequality (20) becomes From (20), (21) and (22), we get which proving that the constant is the best possible in (19). Theorem 6.2. Let f : [ a, b ] → ℝ be a mapping such that the ( n −1) th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous on [ a, b ]. If f ⁽ⁿ⁾ ∈ L ₂ [ a, b ] and n is an even integer. Then we have where σ (·) is defined in Theorem 6.1. Inequality (23) is the best possible in the sense that the constant can not be replaced by a smaller one. Proof . By using the identity (10) and (13) we get We now suppose that (23) holds with a constant C > 0 as We may find a function f : [ a, b ] → ℝ such that the ( n − 1)th derivative f ⁽ⁿ⁻¹⁾ ( n ≥ 2) is absolutely continuous on [ a, b ] as It follows that Then we can find that the left-hand side of inequality (24) becomes and the right-hand side of inequality (24) becomes It follows from (24), (25) and (26) that proving that the constant is the best possible in (23). Feixiang Chen received his MS degree from Xi'an Jiaotong University. Since 2009 he has been teaching at Chongqing Three Gorges University. His research interests include integral inequalities on several kinds of convex functions and applications. School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, 404000, P.R. China. e-mail:cfx2002@126.com