Iva Franjić

On the refinements of the Jensen-Steffensen inequality

In this paper, we extend some old and give some new refinements of the Jensen-Steffensen inequality. Further, we investigate the log-convexity and the exponential convexity of functionals defined via these inequalities and prove monotonicity property of the generalized Cauchy means obtained via these functionals. Finally, we give several examples of the families of functions for which the results can be applied.2010 Mathematics Subject Classification. 26D15.

Schur-Convexity of Averages of Convex Functions

The object is to give an overview of the study of Schur-convexity of various means and functions and to contribute to the subject with some new results. First, Schur-convexity of the generalized integral and weighted integral quasiarithmetic mean is studied. Relation to some already published results is established, and some applications of the extended result are given. Furthermore, Schur-convexity of functions connected to the Hermite-Hadamard inequality is investigated. Finally, some results on convexity and Schur-convexity involving divided difference are considered.

GENERALISATION OF A CORRECTED SIMPSON'S FORMULA

The results obtained by A. J. Roberts and N. Ujevic in a recent paper are generalised. A number of inequalities for functions whose derivatives are either functions of bounded variation or Lipschitzian functions or R-integrable functions are derived. Also, some error estimates for the derived formulae are obtained.

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”.

Schur-convexity and the Simpson formula

The main objective of this work is to give a necessary and sufficient condition for the function defined as the difference of the Simpson quadrature rule and the arithmetic integral mean to be Schur-convex.

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.

CORRECTED EULER-MACLAURIN'S FORMULAE

The aim of this paper is to derive corrected Euler-Maclaurin’s formulae, i.e. open type quadrature formulae where the integral is approximated not only with the values of the function in points (5a+b)/6, (a+b)/2 and (a+5b)/6, but also with values of the first derivative in end points of the interval. These formulae will have a higher degree of exactness than the ones obtained in [2]. Using the derived formulae, a number of inequalities for various classes of functions are obtained.

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].

Refinements of the Lower Bounds of the Jensen Functional

The lower bounds of the functional defined as the difference of the right-hand and the left-hand side of the Jensen inequality are studied. Refinements of some previously known results are given by applying results from the theory of majorization. Furthermore, some interesting special cases are considered.

Ostrowski type inequalities for functions whose higher order derivatives have a single point of non-differentiability

Ostrowski type inequalities for the class of functions whose ( n - 1 ) th order derivatives are continuous, of bounded variation and have a single point of non-differentiability are derived. Special attention is given to functions whose first derivative has a single point of non-differentiability. Improvements of some previously obtained results are provided.