Two estimates concerning classical Diophantine approximation constants

In this paper we aim to prove two inequalities involving the classical approximation constants $w_n^{\mathrm{\prime }}(\zeta ),{\widehat{w}}_n^{\mathrm{\prime }}(\zeta )$ that stem from the simultaneous approximation problem $\mathrm{\mid }{\zeta }^{j}x-y_j\mathrm{\mid }$, $1\le j\le n$, on the one side and the constants $w_n^{\ast }(\zeta ),{\widehat{w}}_n^{\ast }(\zeta )$ connected to approximation with algebraic numbers of degree $\le n$ on the other side. We concretely prove $w_n^{\ast }(\zeta ){\widehat{w}}_n^{\mathrm{\prime }}(\zeta )\ge 1$ and ${\widehat{w}}_n^{\ast }(\zeta )w_n^{\mathrm{\prime }}(\zeta )\ge 1$. The first result is due to W. Schmidt, however our method of proving it allows to derive the other inequality as a dual result. Finally we will discuss estimates of $w_{n}^{\ast}(\zeta), \hat{w}_{n}^{\ast}(\zeta)$uniformlyin$\zeta$dependingonlyon$n$ as an application.