  Let $n \ge 2$ be an integer and $\xi$ a transcendental real number. We
establish several new relations between the values at $\xi$ of the exponents of
Diophantine approximation $w_n, w_{n}^{\ast}, \hat{w}_{n}$, and
$\hat{w}_{n}^{\ast}$. Combining our results with recent estimates by Schmidt
and Summerer allows us to refine the inequality $\hat{w}_{n}(\xi) \le 2n-1$
proved by Davenport and Schmidt in 1969.
