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.

For q = 1, and hence β(1) = 0, we have each riβ(q) = riβ(1) = ri0 = 1, and piq = pi1 = pi. Consequently,

alpha = -dβ/dq = (p1ln(p1) + ... + pNln(pN))/(p1ln(r1) + ... + pNln(rN))