We provide some square-free criteria for primitive polynomials over unique factorization domains, which do not make use of derivatives or discriminants. Using some ideas of Ostrowski we establish nonvanishing conditions for determinants of matrices with polynomial entries and deduce squarefree criteria for polynomials in several variables.

Here we derive multivariate weighted fractional representation formulae involving ordinary partial derivatives of …rst order. Then we present related multivariate weighted fractional Ostrowski type inequalities with respect to uniform norm. 2010 AMS Mathematics Subject Classi…cation : 26A33, 26D10, 26D15.

a f (x) dx− b− a n n ∑ i=1 f (a+ xi (b− a)) ∣∣∣∣ 5 2m+ 5 4 (b− a)m+1 (m+ 1)! (S − s) (?) holds, where S := supa5x5b f (m)(x), s := infa5x5b f (m)(x) and for suitable x1, x2, . . . , xn. It is worth noticing that n,m are arbitrary numbers. This means that the estimate in (?) is more accurate whenm is large enough. Our approach is also elementary. © 2010 Elsevier Ltd. All rights reserved.

In this paper we present a historical review of the investigation of two Ostrowski inequalities and describe several distinct streams for their generalizations. Also we point out some new methods to obtain known results and give a number of new results related to Ostrowski’s inequalities.

The classical Ostrowski inequality for functions on intervals estimates the value of the function minus its average in terms of the maximum of its first derivative. This result is extended to higher order over shells and balls of R , N ≥ 1, with respect to an extended complete Tschebyshev system and the generalized radial derivatives of Widder type. We treat radial and non-radial functions.

In this paper we present sharp estimates for the difference of general integral means with respect to even different finite measures. This is achieved by the use of the Ostrowski and Fink inequalities and the Geometric Moment Theory Method. The produced inequalities are with respect to the supnorm of a derivative of the involved function.

We generalize Ostrowski inequality for higher order derivatives, by using a generalized Euler type identity. Some of the inequalities produced are sharp, namely attained by basic functions. The rest of the estimates are tight. We give applications to trapezoidal and mid-point rules. Estimates are given with respect to L∞ norm. c © 2006 Elsevier Ltd. All rights reserved.

Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral R b a f eit du (t) of continuous complex valued integrands f : C (0; 1)! C de…ned on the complex unit circle C (0; 1) and various subclasses of integrators u : [a; b] [0; 2 ] ! C of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well. 1. Introduction The ...