  For $\zeta$ a transcendental real number, we consider the classical
Diophantine exponents $w_{n}(\zeta)$ and $\widehat{w}_{n}(\zeta)$. They measure
how small $| P(\zeta)|$ can be for an integer polynomial $P$ of degree at most
$n$ and naive height bounded by $X$, for arbitrarily large and all large $X$,
respectively. The discrepancy between the exponents $w_{n}(\zeta)$ and
$\widehat{w}_{n}(\zeta)$ has attracted interest recently. Studying parametric
geometry of numbers, W. Schmidt and L. Summerer were the first to refine the
trivial inequality $w_{n}(\zeta)\geq \widehat{w}_{n}(\zeta)$. Y. Bugeaud and
the author found another estimation provided that the condition
$w_{n}(\zeta)>w_{n-1}(\zeta)$ holds. In this paper we establish an
unconditioned version of the latter result, which can be regarded as a proper
extension. Unfortunately, the new contribution involves an additional exponent
and is of interest only in certain cases.
