STOCHASTIC

for the case (3); and
for the case (4).
«(*)•=
Pix)
o{x)
(B) Reciprocal functions [6, 2, 10]. Let T be a convex set in R", let a
be a numerical function defined on J\ and let 0(x) = \la(x). Then
(1) a concave > 0 on T => 0 convex > 0 on T,
(2) a convex < 0 on r => 0 concave < 0 on J\
The above follows from (Iiv) and (1'iv') respectively if we define
q>(y,z) = l.ly, (y,z)eRxR and note that V, 0, (3 concave > 0 => 0 convex >0
(2) a concave < 0, fl concave > 0 => 8 convex > 0
(3) a linear, ft concave > 0 =» 6 convex > 0
(1') a concave < 0, 0 convex < 0 => 0 concave 0 concave < 0.
If we observe that the function