On the dual of Ces\`aro function space

The goal of this paper is to present an isometric representation of the dual space to Ces`aro function space $C_{p,w}$, $1<p<\mathrm{\infty }$, induced by arbitrary positive weight function $w$ on interval $(0,l)$ where $0<l⩽\mathrm{\infty }$. For this purpose given a strictly decreasing nonnegative function $\mathrm{\Psi }$ on $(0,l)$, the notion of essential $\mathrm{\Psi }$-concave majorant $\widehat{f}$ of a measurable function $f$ is introduced and investigated. As applications it is shown that every slice of the unit ball of the Ces`aro function space has diameter 2. Consequently Ces`aro function spaces do not have the Radon-Nikodym property, are not locally uniformly convex and they are not dual spaces.