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In this paper, new sharp weighted generalizations of Ostrowski andgeneralized trapezoid type inequalities for the Riemann--Stieltjes integralsare proved. Several related inequalities are deduced and investigated. NewSimpson's type inequalities for $\mathcal{RS}$--integral are pointed out.Finally, as application; an error estimation of a general quadrature rule for$\mathcal{RS}$--integral via Ostrowski--generalized trapezoid quadratureformula is given.

New sharp inequalities of Ostrowski and generalized trapezoid type for the Riemann-Stieltjes integrals and applications