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Abstract
A new auxiliary result based on a three step quadratic kernel utilizing the concepts of local fractional calculus is obtained. Using this new auxiliary result we have several new Newton type inequalities whose power q of local fractional derivative in modulus is a generalized harmonic convex function.
Highlights
The following result is well known in the literature as error estimation, Simpson’s second formula.Theorem 1.1 Let φ : [c, d] → R be a four times continuously differentiable mapping on (c, d) and φ(4) ∞ = supx∈(c,d) |φ(4)| < ∞, 2c + d φ(c) + 3φ + 3φ ≤1 6480 φ(4) ∞(d – c)5. c + 2d 3 + φ(d) – φ(x) dx d–c cThis result is known as a Newton type of inequality
Some researchers focused on new Simpson type inequalities based on a two step quadratic kernel and Simpson’s second type results based on three step quadratic kernel via different classes of functions
Alomari et al [4] provided Simpson type inequalities via s-convex functions and they gave some applications to special means and numerical quadrature rules
Summary
Sarikaya et al [21] gave some new inequalities of Simpson type based on s-convexity and their applications to special means of real numbers. Gao and Shi [9] obtained new inequalities of Newton type for functions whose absolute values of second derivatives are convex. Some researchers obtained Hermite–Hadamard, Simpson and Simpson’s second type/Newton type inequalities via harmonically convex mappings (see [10, 14, 15]).
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