Shahar Hod: bound on relaxation times

Shahar Hod has an interesting paper

• gr-qc/0611004

that argues that the relaxation time "TAU" of a physical system is never smaller than a bound:

• TAU >= BETA / PI

where BETA is the inverse temperature. Black holes may saturate it. In terms of quasinormal frequencies, there must exist a quasinormal frequency whose imaginary part satisfies

• PI . TEMPERATURE >= Im (OMEGA)

which is obtained by inverting the previous inequality. I feel that there should exist a proof. The temperature kind of causes the quasinormal modes to be periodic in "Im (OMEGA)" with the period "2.PI.TEMPERATURE", and there should exist a mode in the lower half of the horizontal periodic strip. Can you complete the proof?

Update: I just learned that the book

• Quantum Phase Transitions, Cambridge University Press (1999)

by Prof Subir Sachdev of Harvard University has conjectured such an equality in eqn (3.14) (pi), using very different considerations.