A simple method is presented by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials into Bernstein polynomials.

0. Introduction; notation This paper lists the essential facts about the representation of polynomials in m variables as Bernstein polynomials. An expanded version may appear elsewhere. While univariate Bernstein polynomials are well studied see, e.g., Lorentz’ classical book Lorentz (1953), the multivariate version has only attracted attention sporadically. Lorentz’ book devotes just one page ...

Matrix methods for the computation of bounds range a complex polynomial and its modulus over rectangular region in plane are presented. The approach relies on expansion given into Bernstein polynomials. results extended to multivariate polynomials rational functions.

In this survey paper we give a short account on characterizations for very classical orthogonal polynomials via extremal problems and the corresponding inequalities. Beside the basic properties of the classical orthogonal polynomials we consider polynomial inequalities of Landau and Kolmogoroff type, some weighted polynomial inequalities in L2-norm of Markov-Bernstein type, as well as the corre...

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava–Pintér addition theorem is obtained. We give new identities involving q-Bernstein polynomials.