On monotonicity of first and second-order differential quotients (Q1961281)

The paper concerns some properties of those functions \(f:X \to \overline R\), where \(X\) is a topological linear space, whose first or second-order differential quotients have a monotonic behaviour. More precisely, put \[ \delta_t f(x;h):=\displaystyle{\frac{f(x+th)-f(x)}{t}} \;, \;\delta_t^2 f(x,y;h):= \displaystyle{\frac{2}{t} \left ( \frac{f(x+th)-f(x)}{t}- \langle y,h \rangle \right) }. \] Characterizations of those functions for which \(\{\delta_t f(x;\cdot)\}_{t>0}\) or \(\{\delta_t^2 f(x,y;\cdot)\}_{t>0}\) are monotone, are provided. Moreover, it is proved that the first and second-order epiderivatives of such functions, which are defined as epigraphical limits of first and second-order differential quotients, respectively, can be easily computed, contrary to what happens in more general situations. Finally, a characterization of the convex lower-semicontinuous functions such that \(\{\delta_t^2 f(x,y;\cdot)\}_{t>0}\) is monotone, is proved together with a formula involving the second-order subdifferential.