On one estimate of divided differences and its applications

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 $\mathop{\rm dist}\nolimits(x, Z) := \min_{0\le j \le \mu} |x-z_j|$.