  We give an estimate of the general divided differences $[x_0,\dots,x_m;f]$,
where some of the $x_i$'s are allowed to coalesce (in which case, $f$ is
assumed to be sufficiently smooth). This estimate is then applied to
significantly strengthen Whitney and Marchaud celebrated inequalities in
relation to Hermite interpolation.
  For example, one of the numerous corollaries of this estimate is the fact
that, given a function $f\in C^{(r)}(I)$ and a set $Z=\{z_j\}_{j=0}^\mu$ such
that $z_{j+1}-z_j \geq \lambda |I|$, for all $0\le j \le \mu-1$, where
$I:=[z_0, z_\mu]$, $|I|$ is the length of $I$ and $\lambda$ is some positive
number, the Hermite polynomial ${\mathcal L}(\cdot;f;Z)$ of degree $\le
r\mu+\mu+r$ satisfying ${\mathcal L}^{(j)}(z_\nu; f;Z) = f^{(j)}(z_\nu)$, for
all $0\le \nu \le \mu$ and $0\le j\le r$, approximates $f$ so that, for all
$x\in I$, \[ \big|f(x)- {\mathcal L}(x;f;Z) \big| \le C \left( \mathop{\rm
dist}\nolimits(x, Z) \right)^{r+1} \int_{\mathop{\rm dist}\nolimits(x,
Z)}^{2|I|}\frac{\omega_{m-r}(f^{(r)},t,I)}{t^2}dt , \] where $m
:=(r+1)(\mu+1)$, $C=C(m, \lambda)$ and $\mathop{\rm dist}\nolimits(x, Z) :=
\min_{0\le j \le \mu} |x-z_j|$.
