Report of Meeting: The Fifteenth Katowice–Debrecen Winter Seminar Będlewo (Poland), January 28–31, 2015

Sprawozdanie z konferencji: The Fifteenth Katowice–Debrecen Winter Seminar Będlewo (Poland), January 28–31, 2015

Stability of functional equations connected with quadrature rules

We study the stability properties of the equation F(y)−F(x)=(y−x)∑i=1naif(αix+βiy) which is motivated by numerical integration. In Szostok and Wa̧sowicz (Appl Math Lett 24(4):541–544, 2011) the stability of the simplest equation of the type (0.1) was investigated thus the inequality |F(y)−F(x)−(y−x)f(x+y)|≤ε was studied. In the current paper we present a somewhat different approach to the problem of stability of (0.1). Namely, we deal with the inequality ∣∣∣F(y)−F(x)y−x−∑i=1naif(αix+βiy)∣∣∣≤ε

Ohlin’s lemma and some inequalities of the Hermite–Hadamard type

Using the Ohlin lemma on convex stochastic ordering we prove inequalities of the Hermite–Hadamard type. Namely, we determine all numbers a,α,β∈[0,1] such that for all convex functions f the inequality af(αx+(1−α)y)+(1−a)f(βx+(1−β)y)≤1y−x∫xyf(t)dt is satisfied and all a,b,c,α∈(0,1) with a + b + c = 1 for which we have af(x)+bf(αx+(1−α)y)+cf(y)≥1y−x∫xyf(t)d

Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

We write expressions connected with numerical differentiation formulas of order 2 in the form of Stieltjes integral, then we use Ohlin lemma and Levin–Stechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequality f(x+y2)≤1(y-x)2∫xy∫xyf(s+t2)dsdt≤1y-x∫xyf(t)dtsatisfied by every convex function f:R→R and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form

The Ninth Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities, Będlewo (Poland), February 4-7, 2009. Report of Meeting

The Ninth Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities was held in the Mathematical Research and Conference Center Bedlewo, Poland, from February 4 to 7, 2009. It was organized by the Institute of Mathematics of the Silesian University from Katowice

Alienation of two general linear functional equations

We study the alienation problem for two general linear equations i.e. we compare the solutions of the system of equations n i=1 αif(pix + qiy) = 0 m j=1 βjg(sjx + tjy) = 0 with the solutions of the single equation n i=1 αif(pix + qiy) = m j=1 βjg(sjx + tjy). To this end we introduce the notion of l-alienation—alienation in the class of monomial functions of order l. We use our results among others to study the alienation properties of two monomial functional equations. Mathematics Subject Classification. 39B52, 39B72

Inequalities of the Hermite-Hadamard type involving numerical differentiation formulas

We observe that the Hermite-Hadamard inequality written in the form $$f\left(\frac{x+y}{2}\right)\leq\frac{F(y)-F(x)}{y-x}\leq\frac{f(x)+f(y)}{2}$$ may be viewed as an inequality between two quadrature operators $f\left(\frac{x+y}{2}\right)$ $\frac{f(x)+f(y)}{2}$ and a differentiation formula $\frac{F(y)-F(x)}{y-x}.$ We extend this inequality, replacing the middle term by more complicated ones. As it turns out in some cases it suffices to use Ohlin lemma as it was done in a recent paper \cite{Rajba} however to get more interesting result some more general tool must be used. To this end we use Levin-Ste\v{c}kin theorem which provides necessary and sufficient conditions under which inequalities of the type we consider are satisfied

On a functional equation connected to Gauss quadrature rule

In this paper, the authors characterize the solutions of the functional equation F(y)−F(x)=(y−x)[f(αx+βy)+f(βx+αy)] for real functions F,f and real coefficients α,β . Some of the results obtained by the authors are also valid on integral domains