Multifractals

Tau(q) is a decreasing function of q

Differentiating

piqritau(q) = 1

with respect to q gives

piqritau(q)(ln(pi) + ln(ri) dtau/dq) = 0

Solving for dtau/dq,

dtau/dq = -( piqritau(q)(ln(pi)))/( piqritau(q)(ln(ri)))

Because each piq > 0, ritau(q) > 0, ln(pi) < 0, and ln(ri) < 0, we see dtau/dq < 0.

In the special case that all the ri take on a common value r, we see

alpha = - dtau/dq = ( piq(ln(pi)))/(ln(r) piq)