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We consider families of general two-point quadrature formulae, using the extension of Montgomery’s identity via Taylor’s formula. The formulae obtained are used to present a number of inequalities for functions whose derivatives are from Lp spaces and Bullen-type inequalities.
The results obtained by A. J. Roberts and N. Ujević in a recent paper are generalised. A number of inequalities for functions whose derivatives are either functions of bounded variation or Lipschitzian functions or R-integrable functions are derived. Also, some error estimates for the derived formulae are obtained.
We consider a family of two-point quadrature formulae, using some Euler-type identities. A number of inequalities, for functions whose derivatives are either functions of bounded variation, Lipschitzian functions or R-integrable functions, are proved.
Modified versions of the Euler midpoint formula are given for functions whose derivatives are either functions of bounded variation, Lipschitzian functions or functions in Lp-spaces. The results are applied to quadrature formulae.
We explore the role of weighted functional Stolarsky means in providing bounds in generalised Hadamard-type inequalities for g-convex functions. Refinements are given for Levinson's inequality and the generalised Hadamard inequality. Applications are made to multivariate weighted functional Stolarsky means.
We consider some general switching inequalities of Brenner and Alzer. It is shown that Brenner's Theorem B below does not hold in general without further conditions. A simple proof is given of Alzer's Corollary D.
An analysis is made of quadrature viatwo-point formulae when the integrand is Lipschitz or of bounded variation. The error estimates are shown to be as good as those found in recent studies using Simpson (three-point) formulae.
A new version of Jensen's inequality is established for probability distributions on the non-negative real numbers which are characterized by moments higher than the first. We deduce some new sharp bounds for Laplace-Stieltjes transforms of such distribution functions.