STOCHASTIC

(1) Suppose/is twice continuously differentiable on (a,b). Show that/is /--convex iff
[>(x)T d*f(x)
r\ —— + 2 S 0 for all x e (a,b).
(m) Suppose/is twice continuously differentiable on K (assumed open). Show that/is
/•-convex (/--concave) iff the matrix
Q(x) = r V/(x)(V/(x))' + V(V/(x))
is positive (negative) semidefinite for all x e K.
(n) Show that all differentiable /--convex (/--concave) functions are pseudo-convex
(pseudo-concave), for r = ± oo.
(o) Suppose f is twice continuously differentiable and quasi-convex on K (assumed
open). It can be shown that, if there exists a real number
,*=sup- 2 ' V(V/W) *.
1=1 = 1
whenever the denominator is not zero, then / is /-"-convex. The /--concave analog is
obtained by replacing supremum by infimum. Show that /•* = 1 if f(x) = log(x). Show
that no /•* exists iff(x) = x 3 .
17. Exercise CR-14 explores the convexity properties of the addition of convex and related
functions. It is of interest to explore such results when a function is defined by an integral.
Let E