Let [$]Z[$] be a standard normal variate. I would like to see a proof that
(*) [$]E[\log (a Z^2 +b)] < 0[$] for all [$]\{(a,b): a > 0, b>0, a +b < 1\}[$].
In other words,
[$] \int_{-\infty}^{\infty} \log (a z^2 + b) \, e^{-z^2/2} \frac{dz}{\sqrt{2 \pi}} < 0[$] under the same conditions.
The connection with GARCH is that this condition is sufficient (according to others, and presumably also with [$]\omega > 0[$]) to establish that the GARCH(1,1) process
[$] \sigma^2_t = \omega + \sigma_{t-1}^2 (a Z^2_t + b)[$]
has a stationary distribution for [$]\sigma^2_t[$].
Numerically, I'm already convinced -- both in the validity of (*) and the existence of the stationary distribution under the given conditions. But, I would like to see a proof of the inequality (*).
Thanks!