Abstract
In this paper, based on the quadratic kernel function with three sections, which was defined by Liu in 2009, we establish some companions of perturbed Ostrowski-type inequalities for the case when , and , respectively. The special cases of these results offer better estimation than the conventional trapezoidal formula and the midpoint formula. The results we get can apply to composite quadrature rules in numerical integration and probability density functions. The effectiveness of these applications is also illustrated through several specific examples due to better error estimates. MSC:26D15, 41A55, 41A80, 65C50.
Highlights
In, Ostrowski [ ] established the following interesting integral inequality for differentiable mappings with bounded derivatives.Theorem
In [ ], Liu established some companions of an Ostrowski-type integral inequality for functions whose first derivatives are absolutely continuous and second derivatives belong to Lp ( ≤ p ≤ ∞) spaces
We will use the quadratic kernel function with three sections (see ( . ) below) which was defined by Liu in [ ]
Summary
In , Ostrowski [ ] established the following interesting integral inequality for differentiable mappings with bounded derivatives. Let f : [a, b] → R be a differentiable mapping on (a, b) whose derivative is bounded on (a, b) and denote f ∞ = supt∈(a,b) |f (t)| < ∞. The constant is sharp in the sense that it cannot be replaced by a smaller one. In [ ], Guessab and Schmeisser proved the following companion of Ostrowski’s inequality. Let f : [a, b] → R be satisfying the Lipschitz condition, i.e., |f (t) – f (s)| ≤
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