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

In comparing a signal with its amplitude-distorted form , where 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 . Generalizations are presented for the cases of finite domains and of signals defined in two-dimensional space.