## Multidirectional mean value inequalities and weak

monotonicity

### Yu. S. Ledyaev and Q. J. Zhu

**Abstract**: Multidirectional mean value inequalities provide estimates of
the difference of the extremal value of a function on a given bounded set and
its value at a given point in terms of its (sub)-gradient at some intermediate
point. We demonstrate that such multidirectional mean value inequalities and their

generalizations can be obtained by using sufficient conditions for the approximate
weak monotone

decrease of a function along approximate trajectories of differential inclusions
which allows us to remove a traditional assumption of lower boundedness on the
function. We also obtain criteria for the approximate weak monotonicity and r-growth
of lower semicontinuous functions

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