STOCHASTIC

(c) (Carathtodory's theorem) Let D r(s 2
is quasi-convex but not /--convex for any r.
(e) Suppose g(x) = exp(z/(x)). Show that / is /--convex (r-concave) with r # 0 iff g is
convex (concave) whenever /• > 0 and concave (convex) whenever /• < 0. [Hint:
Exponentiate r times the /--convex definition.]
(f) Show that/is /--convex iff —/is —/--concave.
(g) Suppose / is /--convex (/--concave). Show that /+ a is /--convex (/--concave) tor all
IE£. Show that kf is (r/A:)-convex [(/-/A:)-concave] if k > 0.
(h) Suppose /and g are /--convex (/--concave) and ai,a2 > 0. Show that
(log[a1exp[r/(x)] + a2exp[/9'(x)]} 1/ ' if r # 0,
h(x) = I [aif(x) + a2g(x) if r = 0,
is /--convex (/--concave).
(i) Suppose/is /--convex, r£0(r|O), and g is s-convex and nondecreasing on E. Show
that h(x) = g[f(x)] is s-convex (.s-concave).
(j) Show that / is r-convex (r-concave) iff for all x 1 ,x 2 eK the function /(A) =
f[Xx l + (1 — X)x 2 ] is r-convex (r-concave) on [0,1].
(k) Show that / is r-convex and differentiable iff for all x 1 , x 2 e K,
(1/r) exp [r/(x 2 )] £ (1/r) exp [r/Oc 1 )] {1 + r(x 2 - x 1 )' V/^ 1 )} if r ^ 0,
Z(* 2 ) S /(*') + (.x 2 - x 1 )' V/Ot l ) if r = 0,
where V denotes the gradient operator, i.e., {djdxi,..., 8/dx„).
MIND-EXPANDING EXERCISES 73