Kazimierz Nikodem

A sandwich theorem and Hyers—Ulam stability of affine functions

It is shown that two real functionsf andg, defined on a real intervalI, satisfy the inequalitiesf(λx + (1 − λ)y) ≤ λg(x) + (1 − λ)g(y) andg(λx + (1 − λ)y) ≥ λf(x) + (1 − λ)f(y) for allx, y ∈ I andλ ∈ [0, 1], iff there exists an affine functionh: I → ℝ such thatf ≤ h ≤ g. As a consequence we obtain a stability result of Hyers—Ulam type for affine functions.

Around Jensen’s inequality for strongly convex functions

In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen–Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen’s operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if

Functions Generating Strongly Schur-Convex Sums

Sp\left( A \right) \subset \left( 1,\infty \right)

An Integral Jensen Inequality For Convex Multifunctions

SpA⊂1,∞, then

Continuity properties of midconvex set-valued maps

\begin{aligned} {{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right) ,\quad r\ge 2 \end{aligned}

Remarks on Bárány's theorem and affine selections

Ax,xr≤Arx,x-r2-r2A2x,x-Ax,x2,r≥2and if