We consider the functions hi(q) = piqritau(q)
for i = 1, ..., N. These are natural functions because the equation defining tau(q) |
p1qr1tau(q) + ...
+ pNqrNtau(q) = 1 |
can be written |
h1(q) + ... + hN(q) = 1. |
Differentiating, |
dhi/dq = piqritau(q)
(ln(pi) + ln(ri)(dtau/dq)) |
Suppose ln(pm)/ln(rm) = alphamin and
ln(pM)/ln(rM) = alphamax |
Recalling alpha = -dtau/dq, the bounds
alphamin <= alpha <= alphamax become |
-ln(pm)/ln(rm) >= dtau/dq >= -ln(pM)/ln(rM) |
Because ln(ri) < 0 for each i, we have |
-ln(ri)(ln(pm)/ln(rm)) <= ln(ri)(dtau/dq)
<= -ln(ri)(ln(pM)/ln(rM)) |
Adding ln(pi) to each side |
ln(pi) - ln(ri)(ln(pm)/ln(rm)) <=
ln(pi) + ln(ri)(dtau/dq)
<= ln(pi) - ln(ri)(ln(pM)/ln(rM)) |
Taking i = m
gives dhm/dq >= 0; taking
i = M gives
dhM/dq <= 0. |
For all i, 0 <= hi(q) <= 1
for all q. |
Because hm(q) is a nondecreasing function bounded between 0 and 1, both limits |
limq -> infinityhm(q) and limq -> -infinityhm(q) |
exist. These limits exist also for hM(q). |
Moreover, limq -> -infinityhM(q) = cM and
limq -> infinityhm(q) = cm, with both cM and cm
positive. |
In fact, each dhi/dq has
at most one zero,
and so the limits limq -> infinityhi(q) and limq -> -infinityhi(q)
exist |
If limq -> infinityhi(q) = ci > 0, then
ln(pi)/ln(ri)
=
ln(pm)/ln(rm). |
Here is a
picture
of some typical hi. |
From this we
deduce
limq -> infinitydtau/dq = -alphamin and similarly
limq -> -infinitydtau/dq = -alphamax |
Because dtau/dq appraoches -alphamax as q -> -infinity and
-alphaminas q -> infinity, we deduce the graph of tau(q) has oblique
asymptotes of these slopes. |