Several mean value theorems for higher order divided differences and approximate Peano derivatives are proved.

We analyse mean values of functions with values in the boundary of a convex two-dimensional set. As an application, reverse integral inequalities imply exactly the same inequalities for the monotone rearrangement. Sharp versions of the classical Gehring lemma, the Gurov-Resetnyak theorem and the Muckenhoupt theorem are obtained.

For a differentiable function $𝐟:I\to {ℝ}^k,$ where $I$ is a real interval and $k\in \mathbb{N}$, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean $M:I^2\to I$ such that$$𝐟\left(x\right)-𝐟\left(y\right)=(x-y){𝐟}^{\text{'}}\left(M(x,y)\right),x,y\in I,$$ are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.

The author gives a new simple proof of monotonicity of the generalized extended mean values $M(r,s)=\le {(\left(\int f^{s}d\mu \right)/\left(\int f^{r}d\mu \right))}^{1/(s-r)}$ introduced by F. Qi.