On some extremalities in the approximate integration

Some extremalities for quadrature operators are proved for convex functions of higher order. Such results are known in the numerical analysis, however they are often proved under suitable differentiability assumptions. In our considerations we do not use any other assumptions apart from higher order convexity itself. The obtained inequalities refine the inequalities of Hadamard type. They are applied to give error bounds of quadrature operators under the assumptions weaker from the commonly used

On some inequality of Hermite-Hadamard type

It is well-known that the left term of the classical Hermite-Hadamard inequality is closer to the integral mean value than the right one. We show that in the multivariate case it is not true. Moreover, we introduce some related inequality comparing the methods of the approximate integration, which is optimal. We also present its counterpart of Fejer type.Comment: Submitted to Opuscula Mat

Support-type properties of convex functions of higher order and Hadamard-type inequalities

It is well-known that every convex function admits an affine support at every interior point of a domain. Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree no greater than the order of convexity. In this paper the attaching method is developed. It is applied to obtain the general result Theorem 2, from which the mentioned above support theorem and some related properties of convex functions of higher (both odd and even) order are derived. They are applied to obtain some known and new Hadamard-type inequalities between the quadrature operators and the integral approximated by them. It is also shown that the error bounds of quadrature rules follow by inequalities of this kind.Comment: In the journal version of the paper an example given in Remark 4 was not correct. Here we give a proper on

Hermite-Hadamard-type inequalities in the approximate integration

We give a slight extension of the Hermite-Hadamard inequality on simplices and we use it to establish error bounds of the operators connected with the approximate integration

On the classes of higher-order Jensen-convex functions and Wright-convex functions

The classes of n-Wright-convex functions and n-Jensen-convex functions are compared with each other. It is shown that for any odd natural number $n$ the first one is the proper subclass of the second one. To reach this aim new tools connected with measure theory are developed