
Journal of Convex Analysis 20 (2013), No. 2, 531543 Copyright Heldermann Verlag 2013 Strongly Midquasiconvex Functions Jacek Tabor Institute of Computer Science, Jagiellonian University, Lojasiewicza 6, 30348 Kraków, Poland tabor@ii.uj.edu.pl Józef Tabor Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35959 Rzeszów, Poland tabor@univ.rzeszow.pl Marek Zoldak Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35959 Rzeszów, Poland marek_z2@op.pl [Abstractpdf] \def\dim{\mathrm{dim\,\,}} \def\R{{\mathbb R}} \def\e{\varepsilon} Let $V$ be a nonempty convex subset of a normed space $X$ and let $\e>0$ and $p>0$ be given. A function $f: V \to \R$ is called {\em $(\e,p)$strongly midquasiconvex} if $$ f(\frac{x+y}{2}) \leq \max [f(x), f(y)]\e(\frac{\xy\}{2})^p \text{\ \ for\ \ } x,y \in V. $$ We call $f$ $p$strongly midquasiconvex if it is $(\e,p)$strongly midquasiconvex with a certain $\e>0$. We show that if either $p<1$ and $\dim V=1$ or $p<2$ and $\dim V>1$ then there are no $p$strongly midquasiconvex functions defined on $V$. On the other hand if $X$ is an inner product space with $\dim X \geq 2$, $p \geq 2$, then there exists an $(1,p)$strongly midquasiconvex function defined on an arbitrary ball in $X$. \medskip Consequently, the case when $p=1$ and $\dim V=1$ is of a special interest. Under this assumptions we characterize lower semicontinuous $1$strongly midquasiconvex functions. Keywords: Quasiconvexity, midquasiconvex function, strongly midquasiconvex function. MSC: 26B25, 39B62 [ Fulltextpdf (137 KB)] for subscribers only. 