Convex Functions

By D. Varberg. 1973

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Multivariate Archimedean copulas, $d$-monotone functions and $\ell_1$-norm symmetric distributions (2009, background): “te that d-monotonicity as deﬁned here is diﬀerent from the notion of k-monotonicity as used, for example, in approximation. More speciﬁcally, a function f:(a,b)→R is called k-monotone (or k-convex in [33] and elsewhere) if its divided diﬀerence of order kis nonnegative. Without going into more detai”
Integral Probability Metrics and Their Generating Classes of Functions (1997, methodology): “Q-+-OO b-+oo For the case where [a, b] is a compact interval, a thorough treatment of the functional K~ can be found in Section 14 of Roberts and Varberg (1973).”
Yield criteria for quasibrittle and frictional materials (2004, methodology): “7 As prescri conseq implies f ðp; qÞ 1⁄4 p 4 a4 p 2 a2 þ q 2 b2 ; 06 p a 6 1; ð25Þ (where a and b are material parameters having the dimension of stress) which corresponds to a convex yield surface f ðp; qÞ 1⁄4 0, Fig. 13. After the pioneering work of De Finetti (1949), it became clear that convexit”