Multifractals

β(q) is a decreasing function of q

Differentiating

piqriβ(q) = 1

with respect to q gives

piqriβ(q)(ln(pi) + ln(ri) dβ/dq) = 0

Solving for dβ/dq,

dβ/dq = -( piqriβ(q)(ln(pi)))/( piqriβ(q)(ln(ri)))

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

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

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