Ivan Perić

Sharp integral inequalities based on general Euler two-point formulae

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.

On bounds for weighted norms for matrices and integral operators

In this note we consider inequalities of the form ∥Ax∥ω,q⩽λ∥Bx∥v,p, where A and B are matrices or integral operators, x decreasing sequence or function and ω and v are weights. Obtained results are generalizations of results of G. Bennett [Linear Algebra Appl. 82 (1986) 81] and P.E. Renaud [Bull. Aust. Math. Soc. 34 (1986) 225].

Differences of Weighted Mixed Symmetric Means and Related Results

Some improvements of classical Jensens inequality are used to define the weighted mixed symmetric means. Exponential convexity and mean value theorems are proved for the differences of these improved inequalities. Related Cauchy means are also defined, and their monotonicity is established as an application.

Some New Results Related to Favard's Inequality

Log-convexity of Favards difference is proved, and Dreschers and Lyapunovs type inequalities for this difference are deduced. The weighted case is also considered. Related Cauchy type means are defined, and some basic properties are given.

On families of quadrature formulas based on Euler identities

Abstract A family consisting of quadrature formulas which are exact for all polynomials of order ⩽5 is studied. Changing the coefficients, a second family of quadrature formulas, with the degree of exactness higher than that of the formulas from the first family, is produced. These formulas contain values of the first derivative at the end points of the interval and are sometimes called “corrected”.

Estimates for the Gauss four-point formula for functions with low degree of smoothness

The aim of this work is to derive the Gauss four-point quadrature formula using Euler-type identities. The advantage of this approach is that it enables us to obtain estimates of the error for functions with low degree of smoothness and also to produce quadrature formulae which contain values of derivatives at the end points of the interval.

DISCRETE MULTIPLE HILBERT TYPE INEQUALITY WITH NON-HOMOGENEOUS KERNEL

Multiple discrete Hilbert type inequalities are established in the case of non-homogeneous kernel function by means of Laplace integral representation of associated Dirichlet series. Using newly derived integral expressions for the Mordell-Tornheim Zeta function a set of subsequent special cases, interesting by themselves, are obtained as corollaries of the main inequality.

Note on an Iyengar type inequality

Abstract Using Hayashi’s inequality, an Iyengar type inequality for functions with bounded second derivative is obtained. This result improves a similar result from [N. Elezovic, J. Pecaric, Steffensen’s inequality and estimates of error in trapezoidal rule, Appl. Math. Lett. 11 (6) (1998) 63–69] and, for some classes of functions, the result from [X.L. Cheng, The Iyengar type inequality, Appl. Math. Lett. 14 (2001) 975–978].

Exponentially convex functions generated by Wulbert’s inequality and Stolarsky-type means

Abstract Let − ∞ a b ∞ . If f is concave on [ a , b ] and ψ ′ is convex on the interval of integration, then Wulbert proved that 1 δ + − δ − ∫ δ − δ + ψ ( u ) d u ≥ 1 b − a ∫ a b ψ ( f ( x ) ) d x , where δ − = f − 3 ( ‖ f ‖ 2 2 − ( f ) 2 ) 1 / 2 , δ + = f + 3 ( ‖ f ‖ 2 2 − ( f ) 2 ) 1 / 2 , f = 1 b − a ∫ a b f ( x ) d x and ‖ f ‖ p = ( 1 b − a ∫ a b | f ( x ) | p d x ) 1 / p . We define new Cauchy type means using a functional defined via above inequality and give some related results as applications.

Refinement of integral inequalities for monotone functions

In this paper, we give refinements of some inequalities for generalized monotone functions by using log-convexity of some functionals.