  We establish new inequalities involving classical exponents of Diophantine
approximation. This allows for improving on the work of Davenport, Schmidt and
Laurent concerning the maximum value of the exponent $\hat{\lambda}_{n}(\zeta)$
among all real transcendental $\zeta$. In particular we refine the estimation
$\hat{\lambda}_{n}(\zeta)\leq \lceil n/2\rceil^{-1}$ due to M. Laurent by
$\hat{\lambda}_{n}(\zeta)\leq \hat{w}_{\lceil n/2\rceil}(\zeta)^{-1}$ for all
$n\geq 1$, and for even $n$ we replace the bound $2/n$ for
$\hat{\lambda}_{n}(\zeta)$ first found by Davenport and Schmidt by roughly
$\frac{2}{n}-\frac{1}{2n^{3}}$, which provides the currently best known bounds
when $n\geq 6$.
