  In this paper we aim to prove two inequalities involving the classical
approximation constants $w_{n}^{\prime}(\zeta),\hat{w}_{n}^{\prime}(\zeta)$
that stem from the simultaneous approximation problem $|\zeta^{j}x-y_{j}|$,
$1\leq j\leq n$, on the one side and the constants
$w_{n}^{\ast}(\zeta),\hat{w}_{n}^{\ast}(\zeta)$ connected to approximation with
algebraic numbers of degree $\leq n$ on the other side. We concretely prove
$w_{n}^{\ast}(\zeta)\hat{w}_{n}^{\prime}(\zeta)\geq 1$ and
$\hat{w}_{n}^{\ast}(\zeta)w_{n}^{\prime}(\zeta)\geq 1$. 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 $w_{n}^{\ast}(\zeta),
\hat{w}_{n}^{\ast}(\zeta)$ uniformly in $\zeta$ depending only on $n$ as an
application.
