Correlation of a signal with an amplitude-distorted form of itself

In comparing a signal
f(t)
with its amplitude-distorted form
g(f(t))
, where
g(\cdot)
is a monotonically increasing function of its argument, one is led to consider the correlation function \begin{equation} R(s) \triangleq \int_{-\infty}^{\infty} dtg (f(t))f(t-s). \end{equation} A rigorous proof is given of the inequality
R(s) \leq R(O)
. Generalizations are presented for the cases of finite domains and of signals defined in two-dimensional space.