The first goal of this paper is to establish properties $(\Omega­$) and $(\underline{DN})$ for spaces of entire functions of bounded type on $(DF)$-spaces. Next we show the connection between the property $(\underline{DN})$ fulfilled on $H(E_b^\prime)$ and the existence of a non-pluripolar compact set in $E_b^\prime$ under the assumption that $E$ is a Frechet-Montel space.