The paper is devoted to extend some mean value inequalities from the
function setting to the multifunction one. Let (M,d) be a metric space,
let F be a multifunctions defined on D \subset R and taking
values in the family of nonempty subsets of M, and let g: D\rightarrow
R be a strictly increasing function. The author proves the
following inequality:
\frac{\delta(F(b),F(a))}{g(b)-g(a)} \leq \sup_{s\in [a,b)\cap D}
\sup_{S\in F(s)}
\sup_{t\in (s,b)\cap D} \frac{\delta(F(t),S)}{g(t)-g(s)},
where a and b are two points of D with a<b and, if Q
and P are nonempty subsets of M, then \delta(Q,P)=\sup_{p\in P} \inf_{q\in Q}d(q,p).
An application of the previous inequality to the Dini derivatives
of a multifunction is also given.
Reviewed by L. Di Piazz