The aim of this paper is to establish the property $LB^{\infty}$ on $[H(K_{\varepsilon})]^\prime$ under the assumption that $E$ is a Frechet space with an absolute basis and $K_{\epsilon}$ is a balanced convex compact subset of $E$. At the same time, the properties $(\tilde{\Omega},\overline{\Omega})$ for the Hartogs domains associated to an open polydisc in a nuclear Frechet space with a basis will be also proved.